CoCalc Public Fileswww / talks / harvard-talk-2004-12-08 / e2heights / tmpOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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Total time: 213.469 seconds, Total memory usage: 5.24MB
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[email protected]:~/papers/padic_cyclotomic_height/e2heights$ me
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Magma V2.11-1 Wed Sep 29 2004 23:46:36 on form [Seed = 135448808]
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Type ? for help. Type <Ctrl>-D to quit.
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Loading startup file "/home/was/magma/local/emacs.m"
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Loading "/home/was/magma/local/init.m"
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> E := EC("37A");
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> prec := 30; sigma30 := sigma_alg3(E, p, prec : e2prec := prec);
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>> prec := 30; sigma30 := sigma_alg3(E, p, prec : e2prec := p
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^
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User error: Identifier 'p' has not been declared or assigned
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> p := 5;
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> prec := 30;
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> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2;
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>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2;
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^
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User error: Identifier 'E2' has not been declared or assigned
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>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2;
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^
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User error: Identifier 'E2' has not been declared or assigned
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>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2;
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^
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User error: Identifier 'f1' has not been declared or assigned
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> Attach("kedlaya.m");
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> Attach("formal.m");
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> >> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2;
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.041
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Reducing differentials modulo cohomology relations.
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.039
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Reducing differentials modulo cohomology relations.
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>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2
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^
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Runtime error in '-': Arguments are not compatible
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Argument types given: AlgMatElt, AlgMatElt
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> e1;
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-2767125139749567853
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> e2;
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6650601347913228947772
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> f1;
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[2657944961933597168*5 + O(5^28) 437853581402776789*5 + O(5^28)]
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[-1600106773161651063 + O(5^27) 1611436384179670408 + O(5^27)]
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> f2;
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[5747055585190205081543*5 + O(5^33) -8396366479151751520086*5 +
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O(5^33)]
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[10831544081154084442687 + O(5^32) -5452213560564062517092 +
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O(5^32)]
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> f1[1,1] - f2[1,1];
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>> f1[1,1] - f2[1,1];
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^
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Runtime error in '-': Arguments are not compatible
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Argument types given: FldPadElt, FldPadElt
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> Parent(f1);
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Full Matrix Algebra of degree 2 over pAdicField(5, 27)
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> Parent(f2);
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Full Matrix Algebra of degree 2 over pAdicField(5, 32)
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> Trace(f1);
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-2 + O(5^27)
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> Trace(f2);
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-2 + O(5^32)
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> prec := 10; s := sigma(E, p, prec : e2prec := prec);
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e2prec = 10
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.001
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Reducing differentials modulo cohomology relations.
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e2 = -101130353
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K = 5-adic field mod 5^12
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r2 = 0
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d1 = 0
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0
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-75421447 + O(5^12)
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> prec := 20; sigma20 := sigma_alg3(E, p, prec : e2prec := prec);
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e2prec = 20
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.041
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Reducing differentials modulo cohomology relations.
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e2 = 914563619572772
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K = 5-adic field mod 5^22
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r2 = 0
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d1 = 0
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0
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435471115984803 + O(5^22)
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> sigma - sigma20;
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O(5^12)*t + O(5^12)*t^2 + O(5^12)*t^3 + O(5^12)*t^4 + O(5^11)*t^5
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+ O(5^9)*t^6 + O(5^8)*t^7 + O(5^7)*t^8 + O(5^7)*t^9 + O(t^10)
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> h := height_function(E,5,30);
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e2prec = 30
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.041
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Reducing differentials modulo cohomology relations.
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e2 = 6650601347913228947772
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K = 5-adic field mod 5^32
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r2 = 0
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d1 = 0
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0
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-11364423793197096905822 + O(5^32)
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> G, f := MordellWeilGroup(E);
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> P := f(G.1);
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> h(P);
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-1305176752965909410953*5 + O(5^32)
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> h40 := height_function(E,5,40);
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e2prec = 40
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.079
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Reducing differentials modulo cohomology relations.
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e2 = -38243125061477700243509333478
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K = 5-adic field mod 5^42
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r2 = 0
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d1 = 0
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0
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55249955316579777359543719178 + O(5^42)
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> h40(P);
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28398557172280389803479672*5 + O(5^38)
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> a40 := h40(P);
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> a30 := h(P);
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> a30;
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-1305176752965909410953*5 + O(5^32)
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> a40;
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28398557172280389803479672*5 + O(5^38)
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> Parent(a30)!a40 - a30;
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-22*5^29 + O(5^32)
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> E := EC("1058C");
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> Order(P);
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0
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> E := EC("1058C");
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> regulator(E,5, 20);
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>> regulator(E,5);
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^
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Runtime error in 'regulator': Bad argument types
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Argument types given: CrvEll[FldRat], RngIntElt
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> regulator(E,5, 20);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.02
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Reducing differentials modulo cohomology relations.
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regulator(
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E: E,
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p: 5,
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prec: 20
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)
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In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\
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al.m", line 584, column 17:
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>> B := [f(P) : P in G | Order(P) eq 0];
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^
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Runtime error in for: Iteration is not possible over this object
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> regulator(E,5, 20);
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In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\
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al.m", line 584, column 22:
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>> B := [f(P) : P in Gens(G) | Order(P) eq 0];
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^
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Runtime error: Undefined reference 'Gens' in package
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"/home/was/papers/padic_cyclotomic_height/e2heights/formal.m"
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> regulator(E,5, 20);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.02
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Reducing differentials modulo cohomology relations.
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-736185286609*5^2 + O(5^20)
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[158952545746*5 + O(5^19) 54853902771*5^2 + O(5^19)]
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[54853902771*5^2 + O(5^19) 1264328610646*5 + O(5^19)]
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> regulator(E,5, 20);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.031
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Reducing differentials modulo cohomology relations.
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-736185286609 + O(5^18)
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[158952545746 + O(5^18) 54853902771*5 + O(5^18)]
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[54853902771*5 + O(5^18) 1264328610646 + O(5^18)]
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> E := EC("1058C");
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> E;
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Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
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Field
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> regulator(E,5, 20);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.029
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Reducing differentials modulo cohomology relations.
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-736185286609 + O(5^18)
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[158952545746 + O(5^18) 54853902771*5 + O(5^18)]
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[54853902771*5 + O(5^18) 1264328610646 + O(5^18)]
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> P := f(G.1); Q := f(G.2); R := P+Q;
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>> P := f(G.1); Q := f(G.2); R := P+Q;
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^
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Runtime error in '.': Argument 2 (2) should be in the range [-1
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.. 1]
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>> P := f(G.1); Q := f(G.2); R := P+Q;
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^
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Runtime error in '+': Bad argument types
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Argument types given: PtEll[FldRat], FldRat
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> G, f := MordellWeilGroup(E);
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> >> P := f(G.1); Q := f(G.2); R := P+Q;
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> h(P);
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-321842139675355243103*5 + O(5^32)
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> P;l
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(-1 : 1 : 1)
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> ;
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>> P;l
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^
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User error: Identifier 'l' has not been declared or assigned
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> height_of_point(P,5,30);
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Time: 0.760
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s1 = -47683715820312 + O(5^20)
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Time: 0.010
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s2 = 3 + O(5^2)
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.079
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Reducing differentials modulo cohomology relations.
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Time: 0.180
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s2b = -28994174169036 + O(5^20)
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s2 - s2b = -11 + O(5^2)
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5 + O(5^2)
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> height_of_point(P,5,30)/5;
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Time: 0.750
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s1 = -47683715820312 + O(5^20)
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Time: 0.010
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s2 = 3 + O(5^2)
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.07
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Reducing differentials modulo cohomology relations.
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Time: 0.190
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s2b = -28994174169036 + O(5^20)
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s2 - s2b = -11 + O(5^2)
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1 + O(5)
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> prec := 10; s := sigma(E, p, prec : e2prec := prec);
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>> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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^
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Runtime error: Attempting to call something that is not callable
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> Dettach("formal.m");
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>> Dettach("formal.m");
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^
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User error: Identifier 'Dettach' has not been declared or
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assigned
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0
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Reducing differentials modulo cohomology relations.
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> s;
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t - (122070312 + O(5^12))*t^2 - (47836256 + O(5^12))*t^3 +
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(22270217*5 + O(5^12))*t^4 + (13762429 + O(5^11))*t^5 -
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(153097*5 + O(5^9))*t^6 + (915486*5^-1 + O(5^8))*t^7 +
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(24036*5^-1 + O(5^7))*t^8 + (3956*5 + O(5^7))*t^9 + O(t^10)
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> s1 := sigma_alg1(E,5,50);
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Time: 7.250
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Time: 0.080
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> s - s1;
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O(5^3)*t + O(5^3)*t^2 - (1 + O(5^3))*t^3 + (61 + O(5^3))*t^4 -
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(48 + O(5^3))*t^5 + (6 + O(5^3))*t^6 + (26*5^-1 + O(5^2))*t^7
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+ (5^-1 + O(5^2))*t^8 + (2 + O(5^2))*t^9 + O(t^10)
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.01
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Reducing differentials modulo cohomology relations.
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s1 = 25/2
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s2 = 33543952 + O(5^12)
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> E := EC("37A");
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0
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Reducing differentials modulo cohomology relations.
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s1 = 0
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s2 = -75421447 + O(5^12)
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> s1 := sigma_alg1(E,5,50);
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Time: 5.210
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Time: 0.070
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> s1 - s;
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O(5^3)*t + O(5^3)*t^2 + O(5^3)*t^3 + O(5^3)*t^4 + O(5^3)*t^5 +
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O(5^3)*t^6 + O(5^2)*t^7 + O(5^3)*t^8 + O(5^2)*t^9 + O(t^10)
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> E;
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Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
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> E := EC("1058C");
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Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
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Field
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> E := EC("1058C");
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> E;
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Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
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Field
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.01
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Reducing differentials modulo cohomology relations.
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s1 = 25/2
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s2 = 33543952 + O(5^12)
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> s2;
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>> s2;
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^
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User error: Identifier 's2' has not been declared or assigned
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> s1;
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(1 + O(5^3))*t + O(5^3)*t^2 + (53 + O(5^3))*t^3 - (62 +
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O(5^3))*t^4 - (23 + O(5^3))*t^5 + (17 + O(5^3))*t^6 - (6 +
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O(5^2))*t^7 - (58 + O(5^3))*t^8 - (5 + O(5^2))*t^9 + (11 +
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O(5^2))*t^10 - (2 + O(5))*t^11 + (11 + O(5^2))*t^12 - (1 +
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O(5))*t^13 + O(5)*t^14 - (1 + O(5))*t^15 + (1 + O(5))*t^16 +
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O(1)*t^17 + (2 + O(5))*t^18 + O(1)*t^19 + O(1)*t^20 +
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O(5^-1)*t^21 + O(1)*t^22 + O(5^-1)*t^23 + O(5^-1)*t^24 +
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O(5^-1)*t^25 + O(5^-1)*t^26 + O(5^-2)*t^27 + O(5^-1)*t^28 +
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O(5^-2)*t^29 + O(5^-2)*t^30 + O(5^-3)*t^31 + O(5^-2)*t^32 +
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O(5^-3)*t^33 + O(5^-3)*t^34 + O(5^-3)*t^35 + O(5^-3)*t^36 +
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O(5^-4)*t^37 + O(5^-3)*t^38 + O(5^-4)*t^39 + O(5^-4)*t^40 +
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O(5^-5)*t^41 + O(5^-4)*t^42 + O(5^-5)*t^43 + O(5^-5)*t^44 +
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O(5^-5)*t^45 + O(5^-5)*t^46 + O(5^-6)*t^47 + O(5^-5)*t^48 +
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O(5^-6)*t^49 + O(5^-6)*t^50 + (46*5^-12 + O(5^-9))*t^51 +
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O(5^-6)*t^52
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> height_of_point(P,5,20);
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Time: 0.140
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s1 = -47683715820312 + O(5^20)
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Time: 0.000
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s2 = 3 + O(5^2)
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.069
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Reducing differentials modulo cohomology relations.
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Time: 0.200
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s2b = -28994174169036 + O(5^20)
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s2 - s2b = -11 + O(5^2)
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5 + O(5^2)
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> height_of_point(P,5,30);
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Time: 0.750
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s1 = -47683715820312 + O(5^20)
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Time: 0.010
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s2 = 3 + O(5^2)
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.07
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Reducing differentials modulo cohomology relations.
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Time: 0.200
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s2b = -28994174169036 + O(5^20)
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s2 - s2b = -11 + O(5^2)
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5 + O(5^2)
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
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Computing (y^Frobenius)^(-1)
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Expansion time: 0.009
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Reducing differentials modulo cohomology relations.
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r2 = -10
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d1 = -60
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s1 = 25/2
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s2 = 33543952 + O(5^12)
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> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
350
Computing (y^Frobenius)^(-1)
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Expansion time: 0
352
Reducing differentials modulo cohomology relations.
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5*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 + O(t^6)
354
5*t^-24 - 60*t^-23 + 335*t^-22 - 1215*t^-21 + 3382*t^-20 -
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7860*t^-19 + 16420*t^-18 - 33875*t^-17 + 68910*t^-16 -
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129969*t^-15 + 223520*t^-14 - 340515*t^-13 + 430510*t^-12 -
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413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 +
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604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 -
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99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t +
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O(t^2)
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r2 = -10
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d1 = -60
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s1 = 25/2
364
s2 = 33543952 + O(5^12)
365
> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec);
366
Computing (y^Frobenius)^(-1)
367
Expansion time: 0.009
368
Reducing differentials modulo cohomology relations.
369
5*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 - 620*t^6 - 56510*t^7 +
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92780*t^8 - 309860*t^9 + 6582199*t^10 - 9697985*t^11 +
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32499195*t^12 + O(t^13)
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5*t^-24 - 60*t^-23 + 335*t^-22 - 1215*t^-21 + 3382*t^-20 -
373
7860*t^-19 + 16420*t^-18 - 33875*t^-17 + 68910*t^-16 -
374
129969*t^-15 + 223520*t^-14 - 340515*t^-13 + 430510*t^-12 -
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413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 +
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604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 -
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99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t +
378
27635*t^2 + 164120*t^3 + 411630*t^4 + 805474*t^5 +
379
2058140*t^6 + 5084645*t^7 + 11266020*t^8 + 25126615*t^9 +
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57418854*t^10 + O(t^11)
381
r2 = -10
382
d1 = -60
383
s1 = 25/2
384
s2 = 33543952 + O(5^12)
385
> E;
386
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
387
Field
388
> 13 - 25/2;
389
1/2
390
> regulator(E,5, 20);
391
Computing (y^Frobenius)^(-1)
392
Expansion time: 0.03
393
Reducing differentials modulo cohomology relations.
394
5*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 - 620*t^6 - 56510*t^7 +
395
92780*t^8 - 309860*t^9 + 6582199*t^10 - 9697985*t^11 +
396
32499195*t^12 + O(t^13)
397
5*t^-24 - 60*t^-23 + 335*t^-22 - 1215*t^-21 + 3382*t^-20 -
398
7860*t^-19 + 16420*t^-18 - 33875*t^-17 + 68910*t^-16 -
399
129969*t^-15 + 223520*t^-14 - 340515*t^-13 + 430510*t^-12 -
400
413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 +
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604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 -
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99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t +
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27635*t^2 + 164120*t^3 + 411630*t^4 + 805474*t^5 +
404
2058140*t^6 + 5084645*t^7 + 11266020*t^8 + 25126615*t^9 +
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57418854*t^10 + O(t^11)
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r2 = -10
407
d1 = -60
408
s1 = 1/2
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s2 = 887301058934661 + O(5^22)
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-154932736694 + O(5^18)
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[1886073216441 + O(5^18) -39676830387*5^2 + O(5^18)]
412
[-39676830387*5^2 + O(5^18) 669458142766 + O(5^18)]
413
> F := WeierstrassModel(E);
414
> F;
415
Elliptic Curve defined by y^2 = x^3 + 621*x + 103086 over
416
Rational Field
417
> regulator(F, 5, 20);
418
Computing (y^Frobenius)^(-1)
419
Expansion time: 0.021
420
Reducing differentials modulo cohomology relations.
421
5*t - 775008*t^5 - 3451319280*t^7 + 99464526720*t^9 +
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740772243669600*t^11 + O(t^13)
423
5*t^-24 + 1242*t^-20 + 32987520*t^-18 - 175466655*t^-16 -
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212534467920*t^-14 - 36147307135050*t^-12 +
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444610840001184*t^-10 + 102435296676157440*t^-8 +
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6822123994586407680*t^-6 - 44185598971592148000*t^-4 +
427
295478433890435510843679 + 475055663305252414140720*t^2 +
428
652099700285420352098964210*t^4 +
429
65458490316153359172867977760*t^6 +
430
1938105976146212715913465569225*t^8 +
431
343820104471550964940560517257792*t^10 + O(t^11)
432
r2 = 0
433
d1 = 0
434
s1 = 1/2
435
s2 = 390595685806406 + O(5^22)
436
-15987114127344 + O(5^21)
437
[-661779322509 + O(5^21) -1294630351221*5^3 + O(5^21)]
438
[-1294630351221*5^3 + O(5^21) 60163026518941 + O(5^21)]
439
> E;
440
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
441
Field
442
> P;
443
(-1 : 1 : 1)
444
> Parent(P);
445
Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 +
446
2 over Rational Field with coordinates in Rational Field
447
> E := EC("1058C");
448
> E;
449
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
450
Field
451
> E := EC("1058C");
452
> height_of_point(P,5,30);
453
Time: 8.030
454
s1 = -47683715820312 + O(5^20)
455
Time: 0.010
456
s2 = 4*5 + O(5^3)
457
5 + O(5^2)
458
> height_of_point(P,5,30);
459
Time: 0.750
460
s1 = -47683715820312 + O(5^20)
461
Time: 0.010
462
s2 = 3 + O(5^2)
463
5 + O(5^2)
464
> 390595685806406 mod 5^2;
465
6
466
> height_of_point(P,5,10);
467
Time: 0.010
468
t + 1/2*t^2 + (s2 + 1/3)*t^3 + (3/2*s2 + 3/4)*t^4 + (1/2*s2^2 +
469
7/4*s2 + 811/576)*t^5 + (5/4*s2^2 + 27/8*s2 + 2519/1152)*t^6
470
+ (1/6*s2^3 + 25/12*s2^2 + 20279/2880*s2 + 246659/51840)*t^7
471
+ (7/12*s2^3 + 25/6*s2^2 + 74633/5760*s2 + 556159/51840)*t^8
472
+ (1/24*s2^4 + 91/72*s2^3 + 53083/5760*s2^2 +
473
3924689/145152*s2 + 1509906533/69672960)*t^9 + (3/16*s2^4 +
474
133/48*s2^3 + 73009/3840*s2^2 + 218445/3584*s2 +
475
2098466159/46448640)*t^10 + (1/2*s2^4 + 9317/1440*s2^3 +
476
38497/945*s2^2 + 212219863/1612800*s2 +
477
31777804447/319334400)*t^11 + (39/32*s2^4 + 41797/2880*s2^3 +
478
44785931/483840*s2^2 + 2746691069/9676800*s2 +
479
101091788431/464486400)*t^12 + O(t^13)
480
s1 = -47683715820312 + O(5^20)
481
Time: 0.000
482
s2 = -2 + O(0)
483
O(0)
484
> height_of_point(P,5,20);
485
Time: 0.010
486
t + 1/2*t^2 + (s2 + 1/3)*t^3 + (3/2*s2 + 3/4)*t^4 + (1/2*s2^2 +
487
7/4*s2 + 811/576)*t^5 + (5/4*s2^2 + 27/8*s2 + 2519/1152)*t^6
488
+ (1/6*s2^3 + 25/12*s2^2 + 20279/2880*s2 + 246659/51840)*t^7
489
+ (7/12*s2^3 + 25/6*s2^2 + 74633/5760*s2 + 556159/51840)*t^8
490
+ (1/24*s2^4 + 91/72*s2^3 + 53083/5760*s2^2 +
491
3924689/145152*s2 + 1509906533/69672960)*t^9 + (3/16*s2^4 +
492
133/48*s2^3 + 73009/3840*s2^2 + 218445/3584*s2 +
493
2098466159/46448640)*t^10 + (1/2*s2^4 + 9317/1440*s2^3 +
494
38497/945*s2^2 + 212219863/1612800*s2 +
495
31777804447/319334400)*t^11 + (39/32*s2^4 + 41797/2880*s2^3 +
496
44785931/483840*s2^2 + 2746691069/9676800*s2 +
497
101091788431/464486400)*t^12 + O(t^13)
498
s1 = -47683715820312 + O(5^20)
499
Time: 0.000
500
s2 = -2 + O(0)
501
Computing (y^Frobenius)^(-1)
502
Expansion time: 0.079
503
Reducing differentials modulo cohomology relations.
504
Time: 0.200
505
s2b = -28994174169036 + O(5^20)
506
s2 - s2b = -1 + O(0)
507
O(0)
508
> height_of_point(P,5,30);
509
Time: 0.140
510
t + 1/2*t^2 + (s2 + 1/3)*t^3 + (3/2*s2 + 3/4)*t^4 + (1/2*s2^2 +
511
7/4*s2 + 811/576)*t^5 + (5/4*s2^2 + 27/8*s2 + 2519/1152)*t^6
512
+ (1/6*s2^3 + 25/12*s2^2 + 20279/2880*s2 + 246659/51840)*t^7
513
+ (7/12*s2^3 + 25/6*s2^2 + 74633/5760*s2 + 556159/51840)*t^8
514
+ (1/24*s2^4 + 91/72*s2^3 + 53083/5760*s2^2 +
515
3924689/145152*s2 + 1509906533/69672960)*t^9 + (3/16*s2^4 +
516
133/48*s2^3 + 73009/3840*s2^2 + 218445/3584*s2 +
517
2098466159/46448640)*t^10 + (1/120*s2^5 + 1/2*s2^4 +
518
111827/17280*s2^3 + 14784757/362880*s2^2 +
519
15279827491/116121600*s2 + 52000039649/522547200)*t^11 +
520
(11/240*s2^5 + 39/32*s2^4 + 501817/34560*s2^3 +
521
134399791/1451520*s2^2 + 65920556561/232243200*s2 +
522
909825920251/4180377600)*t^12 + (1/720*s2^6 + 209/1440*s2^5 +
523
208751/69120*s2^4 + 141968539/4354560*s2^3 +
524
145243919153/696729600*s2^2 + 29049610444907/45984153600*s2 +
525
3160555396034153/6621718118400)*t^13 + (13/1440*s2^6 +
526
1133/2880*s2^5 + 1003043/138240*s2^4 + 659042959/8709120*s2^3
527
+ 649272490109/1393459200*s2^2 +
528
18570577832297/13138329600*s2 +
529
14164547432115413/13243436236800)*t^14 + (1/5040*s2^7 +
530
143/4320*s2^6 + 14359/13824*s2^5 + 149317151/8709120*s2^4 +
531
73583935421/418037760*s2^3 + 48606744813833/45984153600*s2^2
532
+ 1906941415564041251/602576348774400*s2 +
533
794218070502348929/328678008422400)*t^15 + (1/672*s2^7 +
534
143/1440*s2^6 + 122167/46080*s2^5 + 59521999/1451520*s2^4 +
535
113327417089/278691840*s2^3 + 37014051344279/15328051200*s2^\
536
2 + 2879145396107750743/401717565849600*s2 +
537
18723616992657277/3423729254400)*t^16 + (1/40320*s2^8 +
538
25/4032*s2^7 + 580499/2073600*s2^6 + 573901681/87091200*s2^5
539
+ 91326198881/928972800*s2^4 +
540
260517345451541/275904921600*s2^3 +
541
33269176318286881867/6025763487744000*s2^2 +
542
197236276007666020471/12051526975488000*s2 +
543
86521607370926689779361/6941679537881088000)*t^17 +
544
(17/80640*s2^8 + 55/2688*s2^7 + 3135523/4147200*s2^6 +
545
2842740077/174182400*s2^5 + 1305394367891/5573836800*s2^4 +
546
173648705020711/78829977600*s2^3 +
547
153001683131222060699/12051526975488000*s2^2 +
548
903573247002573294907/24103053950976000*s2 +
549
397013191910693546871857/13883359075762176000)*t^18 +
550
(1/362880*s2^9 + 17/17280*s2^8 + 894203/14515200*s2^7 +
551
25779853/13063680*s2^6 + 67435860449/1672151040*s2^5 +
552
153833929086527/275904921600*s2^4 +
553
18589728679913491271/3615458092646400*s2^3 +
554
132730481579535573749/4519322615808000*s2^2 +
555
1454174368731125752920551/16858364591996928000*s2 +
556
19073477253636348700489649/289657355262492672000)*t^19 +
557
(19/725760*s2^9 + 1717/483840*s2^8 + 5098697/29030400*s2^7 +
558
530134429/104509440*s2^6 + 66175829575/668860416*s2^5 +
559
1466923238256571/1103619686400*s2^4 +
560
86983331241368304989/7230916185292800*s2^3 +
561
448012491622797790163/6573560168448000*s2^2 +
562
6726128810135888376087749/33716729183993856000*s2 +
563
176368602966440710077202687/1158629421049970688000)*t^20 +
564
(589/4354560*s2^9 + 82739/7257600*s2^8 +
565
195302923/406425600*s2^7 + 101339416529/7838208000*s2^6 +
566
121211087867893/501645312000*s2^5 +
567
75724931923925957/23911759872000*s2^4 +
568
245054811254448307111/8677099422351360*s2^3 +
569
2925040101405720153670957/18438836272496640000*s2^2 +
570
2966049910573132782522965143/6406178544958832640000*s2 +
571
472933174310969748229722710861/1338216981312716144640000)*t^\
572
21 + (19/35840*s2^9 + 165733/4838400*s2^8 +
573
148987417/116121600*s2^7 + 56856056021/1741824000*s2^6 +
574
196962543543131/334430208000*s2^5 +
575
1082512600617623051/143470559232000*s2^4 +
576
128134203152937110147/1928244316078080*s2^3 +
577
650713476823318264635317/1756079644999680000*s2^2 +
578
96680075391894831728580743501/89686499629423656960000*s2 +
579
11658559886207441862632073989/14161026257277419520000)*t^22 +
580
O(t^23)
581
s1 = -47683715820312 + O(5^20)
582
Time: 0.000
583
s2 = 3 + O(5^2)
584
Computing (y^Frobenius)^(-1)
585
Expansion time: 0.071
586
Reducing differentials modulo cohomology relations.
587
Time: 0.200
588
s2b = -28994174169036 + O(5^20)
589
s2 - s2b = -11 + O(5^2)
590
5 + O(5^2)
591
> height_of_point(P,5,30);
592
Time: 0.750
593
s1 = -47683715820312 + O(5^20)
594
Time: 0.010
595
s2 = 3 + O(5^2)
596
Computing (y^Frobenius)^(-1)
597
Expansion time: 0.07
598
Reducing differentials modulo cohomology relations.
599
Time: 0.200
600
s2b = -28994174169036 + O(5^20)
601
s2 - s2b = -11 + O(5^2)
602
5 + O(5^2)
603
> E := EC("37A");
604
> height_of_point(P,5,30);
605
Time: 0.770
606
s1 = -47683715820312 + O(5^20)
607
Time: 0.000
608
s2 = 3 + O(5^2)
609
Computing (y^Frobenius)^(-1)
610
Expansion time: 0.079
611
Reducing differentials modulo cohomology relations.
612
Time: 0.200
613
s2b = -28994174169036 + O(5^20)
614
s2 - s2b = -11 + O(5^2)
615
5 + O(5^2)
616
> G, f := MordellWeilGroup(E);
617
> height_of_point(f(G.1),5,30);
618
Time: 0.540
619
s1 = O(5^20)
620
Time: 0.010
621
s2 = 3 + O(5^2)
622
Computing (y^Frobenius)^(-1)
623
Expansion time: 0.07
624
Reducing differentials modulo cohomology relations.
625
Time: 0.180
626
s2b = 41366042218322 + O(5^20)
627
s2 - s2b = 6 + O(5^2)
628
2*5 + O(5^2)
629
> height_of_point(f(G.1),5,30);
630
Time: 0.530
631
s1 = O(5^20)
632
Time: 0.010
633
s2 = 3 + O(5^2)
634
Computing (y^Frobenius)^(-1)
635
Expansion time: 0.07
636
Reducing differentials modulo cohomology relations.
637
Time: 0.180
638
s2b = -41366042218322 + O(5^20)
639
s2 - s2b = O(5^2)
640
2*5 + O(5^2)
641
> height_of_point(f(G.1),5,30);
642
Time: 0.530
643
s1 = O(5^20)
644
Time: 0.010
645
s2 = 3 + O(5^2)
646
Computing (y^Frobenius)^(-1)
647
Expansion time: 0.07
648
Reducing differentials modulo cohomology relations.
649
Time: 0.180
650
s2b = -41366042218322 + O(5^20)
651
s2 - s2b = O(5^2)
652
2*5 + O(5^2)
653
> height_of_point(P,5,30);
654
Time: 0.760
655
s1 = -47683715820312 + O(5^20)
656
Time: 0.000
657
s2 = 3 + O(5^2)
658
Computing (y^Frobenius)^(-1)
659
Expansion time: 0.07
660
Reducing differentials modulo cohomology relations.
661
Time: 0.190
662
s2b = -6916822718276*5 + O(5^20)
663
s2 - s2b = 8 + O(5^2)
664
5 + O(5^2)
665
> height_of_point(f(G.1),5,40);
666
Time: 1.850
667
s1 = O(5^20)
668
Time: 0.030
669
s2 = 53 + O(5^3)
670
Computing (y^Frobenius)^(-1)
671
Expansion time: 0.07
672
Reducing differentials modulo cohomology relations.
673
Time: 0.190
674
s2b = 41366042218322 + O(5^20)
675
s2 - s2b = -19 + O(5^3)
676
-3*5 + O(5^3)
677
> height_of_point(f(G.1),5,40);
678
Time: 1.860
679
s1 = O(5^20)
680
Time: 0.030
681
s2 = 53 + O(5^3)
682
Computing (y^Frobenius)^(-1)
683
Expansion time: 0.089
684
Reducing differentials modulo cohomology relations.
685
Time: 0.190
686
s2b = -41366042218322 + O(5^20)
687
s2 - s2b = O(5^3)
688
-3*5 + O(5^3)
689
> height_of_point(f(G.1),5,40);
690
Time: 1.850
691
s1 = O(5^20)
692
Time: 0.030
693
s2 = 53 + O(5^3)
694
Computing (y^Frobenius)^(-1)
695
Expansion time: 0.07
696
Reducing differentials modulo cohomology relations.
697
Time: 0.190
698
s2b = -41366042218322 + O(5^20)
699
s2 - s2b = O(5^3)
700
-3*5 + O(5^3)
701
> height_of_point(P,5,40);
702
Time: 2.680
703
s1 = -47683715820312 + O(5^20)
704
Time: 0.030
705
s2 = -47 + O(5^3)
706
Computing (y^Frobenius)^(-1)
707
Expansion time: 0.07
708
Reducing differentials modulo cohomology relations.
709
Time: 0.190
710
s2b = 28994174169036 + O(5^20)
711
s2 - s2b = 42 + O(5^3)
712
5 + O(5^3)
713
> ;
714
> E;
715
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
716
> aInvariants(E);
717
[ 0, 0, 1, -1, 0 ]
718
> height_function(E,5,10);
719
Computing (y^Frobenius)^(-1)
720
Expansion time: 0.009
721
Reducing differentials modulo cohomology relations.
722
wp_of_z = t^-2 + O(5^12)*t^-1 + O(5^12) + O(5^12)*t + (5^-1 +
723
O(5^11))*t^2 + O(5^10)*t^3 - (1046317 + O(5^10))*t^4 +
724
O(5^10)*t^5 - (81380208*5^-2 + O(5^10))*t^6 + O(5^10)*t^7 +
725
(11255834*5^-1 + O(5^10))*t^8 + O(5^9)*t^9 - (50391532*5^-3 +
726
O(5^9))*t^10 + O(5^9)*t^11 + O(t^12)
727
5*t - 310*t^4 + 1248*t^5 + 10540*t^7 - 113570*t^8 + 257920*t^9 -
728
311240*t^10 + 6359700*t^11 - 34275900*t^12 + O(t^13)
729
5*t^-24 - 60*t^-21 - 2*t^-20 + 365*t^-18 + 80*t^-17 - 455*t^-16 -
730
1415*t^-15 + 305*t^-14 + 3500*t^-13 + 925*t^-12 - 3710*t^-11
731
- 6279*t^-10 + 6660*t^-9 + 4090*t^-8 + 80*t^-7 - 7975*t^-6 +
732
2294*t^-5 + 1850*t^-4 + 905*t^-3 - 1115*t^-2 - 360*t^-1 + 308
733
+ 15*t - 50*t^2 + 75*t^3 - 30*t^4 + 69*t^5 - 890*t^6 +
734
3270*t^7 - 5660*t^8 + 940*t^9 + 21403*t^10 + O(t^11)
735
r2 = 0
736
d1 = 0
737
s1 = 1/2
738
s2 = -44903869 + O(5^12)
739
function(P) ... end function
740
> E := EC("1058C");
741
> height_function(E,5,10);
742
Computing (y^Frobenius)^(-1)
743
Expansion time: 0.01
744
Reducing differentials modulo cohomology relations.
745
wp_of_z = t^-2 + O(5^12)*t^-1 + O(5^12) + O(5^12)*t +
746
(116984049*5^-1 + O(5^11))*t^2 + O(5^10)*t^3 - (2691683 +
747
O(5^10))*t^4 + O(5^9)*t^5 - (1391658*5^-2 + O(5^9))*t^6 +
748
O(5^9)*t^7 - (4808491*5^-1 + O(5^9))*t^8 + O(5^9)*t^9 -
749
(95211518*5^-3 + O(5^9))*t^10 + O(5^8)*t^11 + O(t^12)
750
5*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 - 620*t^6 - 56510*t^7 +
751
92780*t^8 - 309860*t^9 + 6582199*t^10 - 9697985*t^11 +
752
32499195*t^12 + O(t^13)
753
5*t^-24 - 60*t^-23 + 335*t^-22 - 1215*t^-21 + 3382*t^-20 -
754
7860*t^-19 + 16420*t^-18 - 33875*t^-17 + 68910*t^-16 -
755
129969*t^-15 + 223520*t^-14 - 340515*t^-13 + 430510*t^-12 -
756
413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 +
757
604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 -
758
99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t +
759
27635*t^2 + 164120*t^3 + 411630*t^4 + 805474*t^5 +
760
2058140*t^6 + 5084645*t^7 + 11266020*t^8 + 25126615*t^9 +
761
57418854*t^10 + O(t^11)
762
r2 = -10
763
d1 = -60
764
s1 = 1/2
765
s2 = 33544036 + O(5^12)
766
function(P) ... end function
767
> aInvariants(E);
768
[ 1, 0, 1, 0, 2 ]
769
> E;
770
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
771
Field
772
> h := height_function(E,5,30);
773
Computing (y^Frobenius)^(-1)
774
Expansion time: 0.051
775
Reducing differentials modulo cohomology relations.
776
> >> P := f(G.1); Q := f(G.2); R := P+Q;
777
778
>> P := f(G.1); Q := f(G.2); R := P+Q;
779
^
780
Runtime error in '.': Argument 2 (2) should be in the range [-1
781
.. 1]
782
783
>> P := f(G.1); Q := f(G.2); R := P+Q;
784
^
785
Runtime error in '+': Arguments are not compatible
786
Argument types given: Pt[FldRat], Pt[FldRat]
787
> G, f := MordellWeilGroup(E);
788
> h := height_function(E,5,30);
789
Computing (y^Frobenius)^(-1)
790
Expansion time: 0.049
791
Reducing differentials modulo cohomology relations.
792
> >> P := f(G.1); Q := f(G.2); R := P+Q;
793
> P;
794
(-1 : 1 : 1)
795
> Q;
796
(1 : -3 : 1)
797
> P+Q;
798
(2 : 2 : 1)
799
> h(P);
800
616044959906266998871 + O(5^31)
801
> h(Q);
802
-1749390146105056154979 + O(5^31)
803
> h(2*P);
804
-749917183385185910766 + O(5^31)
805
> 4*h(P);
806
-2192433033452324582641 + O(5^31)
807
> 4*h(P) - h(2*P);
808
-121007*5^23 + O(5^31)
809
> 49*h(P) - h(7*P);
810
-102982*5^23 + O(5^31)
811
> height_of_point(P,5,40);
812
Time: 2.680
813
s1 = -47683715820312 + O(5^20)
814
Time: 0.030
815
s2 = -47 + O(5^3)
816
Computing (y^Frobenius)^(-1)
817
Expansion time: 0.08
818
Reducing differentials modulo cohomology relations.
819
Time: 0.190
820
s2b = -28994174169036 + O(5^20)
821
s2 - s2b = -11 + O(5^3)
822
5 + O(5^3)
823
> height_of_point(3*P,5,40);
824
Time: 2.640
825
s1 = -47683715820312 + O(5^20)
826
Time: 0.040
827
s2 = -47 + O(5^3)
828
Computing (y^Frobenius)^(-1)
829
Expansion time: 0.069
830
Reducing differentials modulo cohomology relations.
831
Time: 0.190
832
s2b = -28994174169036 + O(5^20)
833
s2 - s2b = -11 + O(5^3)
834
9*5 + O(5^3)
835
> h(5*P);
836
194840086849274083589 + O(5^31)
837
> 194840086849274083589 mod 5^3;
838
89
839
> 45 + 89;
840
134
841
> height_of_point(5*P,5,40);
842
Time: 2.680
843
s1 = -47683715820312 + O(5^20)
844
Time: 0.040
845
s2 = -47 + O(5^3)
846
Computing (y^Frobenius)^(-1)
847
Expansion time: 0.07
848
Reducing differentials modulo cohomology relations.
849
Time: 0.180
850
s2b = -28994174169036 + O(5^20)
851
s2 - s2b = -11 + O(5^3)
852
853
854
855
[Interrupted]
856
> > >
857
>
858
> h(5*P);
859
860
[Interrupted]
861
> h(5*P);
862
-83165207024885344879*5^2 + O(5^31)
863
> 25*h(P);
864
616044959906266998871*5^2 + O(5^33)
865
> 25*h(P) - h(5*P);
866
-3846*5^25 + O(5^31)
867
> height_of_point(2*P,5,33);
868
Time: 1.200
869
s1 = -47683715820312 + O(5^20)
870
Time: 0.020
871
s2 = -47 + O(5^3)
872
Computing (y^Frobenius)^(-1)
873
Expansion time: 0.069
874
Reducing differentials modulo cohomology relations.
875
Time: 0.190
876
s2b = -28994174169036 + O(5^20)
877
s2 - s2b = -11 + O(5^3)
878
4 + O(5^2)
879
> h(2*P);
880
-749917183385185910766 + O(5^31)
881
> -749917183385185910766 mod 5^2;
882
9
883
> h(Q);
884
-1749390146105056154979 + O(5^31)
885
> E := EC("37A");
886
> aInvariants(E);
887
[ 0, 0, 1, -1, 0 ]
888
> h := height_function(E,5,30);
889
Computing (y^Frobenius)^(-1)
890
Expansion time: 0.04
891
Reducing differentials modulo cohomology relations.
892
> h(E![0,0]);
893
-1305176752965909410953 + O(5^31)
894
> h := height_function(E,5,30);
895
Computing (y^Frobenius)^(-1)
896
Expansion time: 0.041
897
Reducing differentials modulo cohomology relations.
898
s1 = 0
899
s2 = -11364423793197096905822 + O(5^32)
900
> -11364423793197096905822 mod 5^3;
901
53
902
> -11364423793197096905822 mod 25;
903
3
904
> h := height_function(E,5,30);
905
Computing (y^Frobenius)^(-1)
906
Expansion time: 0.05
907
Reducing differentials modulo cohomology relations.
908
s1 = 0
909
s2 = 11364423793197096905822 + O(5^32)
910
> 11364423793197096905822 mod 25;
911
22
912
> height_of_point(P,5,25);
913
Time: 0.370
914
s1 = -47683715820312 + O(5^20)
915
Time: 0.010
916
s2 = 3 + O(5^2)
917
Computing (y^Frobenius)^(-1)
918
Expansion time: 0.079
919
Reducing differentials modulo cohomology relations.
920
Time: 0.200
921
s2b = -28994174169036 + O(5^20)
922
s2 - s2b = -11 + O(5^2)
923
1 + O(5)
924
> P;
925
(-1 : 1 : 1)
926
> Parent(P);
927
Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 +
928
2 over Rational Field with coordinates in Rational Field
929
> h := height_function(E,5,30);
930
Computing (y^Frobenius)^(-1)
931
Expansion time: 0.049
932
Reducing differentials modulo cohomology relations.
933
s1 = 0
934
s2 = -11364423793197096905822 + O(5^32)
935
> h(P);
936
937
h(
938
P: (-1 : 1 : 1)
939
)
940
In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\
941
al.m", line 569, column 24:
942
>> assert Parent(P) eq E;
943
^
944
Runtime error in 'eq': Bad argument types
945
Argument types given: SetPtEll[FldRat], CrvEll[FldRat]
946
> Parent(P);
947
Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 +
948
2 over Rational Field with coordinates in Rational Field
949
> Curve(P);
950
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
951
Field
952
> h := height_function(E,5,30);
953
Computing (y^Frobenius)^(-1)
954
Expansion time: 0.049
955
Reducing differentials modulo cohomology relations.
956
s1 = 0
957
s2 = -11364423793197096905822 + O(5^32)
958
> h(E![0,0]);
959
-1305176752965909410953 + O(5^31)
960
> h(E![0,0]);
961
-1305176752965909410953 + O(5^31)
962
> E;
963
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
964
> height_of_point(E![0,0],5,25);
965
Time: 0.230
966
s1 = O(5^20)
967
Time: 0.020
968
s2 = 3 + O(5^2)
969
Computing (y^Frobenius)^(-1)
970
Expansion time: 0.069
971
Reducing differentials modulo cohomology relations.
972
Time: 0.180
973
s2b = 41366042218322 + O(5^20)
974
s2 - s2b = 6 + O(5^2)
975
2 + O(5)
976
> -11364423793197096905822 mod 5^2;
977
3
978
> aInvariants(E);
979
[ 0, 0, 1, -1, 0 ]
980
> height_of_point(E![0,0],5,35);
981
Time: 1.090
982
s1 = O(5^20)
983
Time: 0.020
984
s2 = 53 + O(5^3)
985
Computing (y^Frobenius)^(-1)
986
Expansion time: 0.07
987
Reducing differentials modulo cohomology relations.
988
Time: 0.180
989
s2b = 41366042218322 + O(5^20)
990
s2 - s2b = -19 + O(5^3)
991
-3 + O(5^2)
992
> E := EC("1058C");
993
> E;
994
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
995
Field
996
> aInvariants(E);
997
[ 1, 0, 1, 0, 2 ]
998
> qEigenform(E,7);
999
q - q^2 - 2*q^3 + q^4 - 3*q^5 + 2*q^6 + O(q^7)
1000
> P;
1001
(-1 : 1 : 1)
1002
> h := height_function(E,5,30);
1003
Computing (y^Frobenius)^(-1)
1004
Expansion time: 0.051
1005
Reducing differentials modulo cohomology relations.
1006
s1 = 25/2
1007
s2 = 1174596243288607762702 + O(5^32)
1008
> h(P);
1009
616044959906266998871 + O(5^31)
1010
> height_of_point(P,5,35);
1011
Time: 1.590
1012
s1 = -47683715820312 + O(5^20)
1013
Time: 0.020
1014
s2 = -47 + O(5^3)
1015
Computing (y^Frobenius)^(-1)
1016
Expansion time: 0.069
1017
Reducing differentials modulo cohomology relations.
1018
Time: 0.190
1019
s2b = -28994174169036 + O(5^20)
1020
s2 - s2b = -11 + O(5^3)
1021
1 + O(5^2)
1022
> E;
1023
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
1024
Field
1025
> sigma_using_e2(E,5,20);
1026
Computing (y^Frobenius)^(-1)
1027
Expansion time: 0.111
1028
Reducing differentials modulo cohomology relations.
1029
s1 = 25/2
1030
s2 = 110294297236694266286783602328465827 + O(5^52)
1031
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 -
1032
(629854385846743427328970843065414381 + O(5^52))*t^3 -
1033
(77934013291507374156328086110542283*5 + O(5^52))*t^4 -
1034
(220172699283231708595841105601471946 + O(5^51))*t^5 +
1035
(1673672812530827450451130460784403*5 + O(5^49))*t^6 +
1036
(4417868661595977484621003705993611*5^-1 + O(5^48))*t^7 +
1037
(1661938937009558479946781305492786*5^-1 + O(5^47))*t^8 -
1038
(4574286464576783649504865839794*5 + O(5^46))*t^9 -
1039
(12772721231859411292031233322273 + O(5^45))*t^10 -
1040
(335389858314801008160952928479*5^-2 + O(5^41))*t^11 -
1041
(1734566437614434002025218517*5^-2 + O(5^38))*t^12 -
1042
(142625041296925740237371068*5^-2 + O(5^37))*t^13 +
1043
(22491253790292392461780617*5^-2 + O(5^36))*t^14 +
1044
(164912848985610385137397089*5^-2 + O(5^36))*t^15 +
1045
(54956008876191474687462988*5^-2 + O(5^36))*t^16 +
1046
(178946475950880797069233306*5^-3 + O(5^35))*t^17 -
1047
(180414110732202736576745224*5^-3 + O(5^35))*t^18 -
1048
(16550329369713898663200319*5^-3 + O(5^34))*t^19 + O(t^20)
1049
> h30 := height_function(E,5,30);
1050
Computing (y^Frobenius)^(-1)
1051
Expansion time: 0.04
1052
Reducing differentials modulo cohomology relations.
1053
s1 = 25/2
1054
s2 = 1174596243288607762702 + O(5^32)
1055
> - (1110223024625156540423631668090820312 + O(5^52) + 2
1056
> sigma_using_e2(E,5,20);
1057
Computing (y^Frobenius)^(-1)
1058
Expansion time: 0.109
1059
Reducing differentials modulo cohomology relations.
1060
s1 = 1/2
1061
s2 = 110294297236694266286783602328465911 + O(5^52)
1062
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 -
1063
(629854385846743427328970843065414297 + O(5^52))*t^3 -
1064
(389670066457536870781640430552711289 + O(5^52))*t^4 +
1065
(162764071597834336104928145263098697 + O(5^51))*t^5 -
1066
(2404186207817254245228035109846774 + O(5^49))*t^6 -
1067
(4990897897258098001816325444151238*5^-1 + O(5^48))*t^7 +
1068
(359437676698161044649203153266187*5^-1 + O(5^47))*t^8 -
1069
(95761481299689786373060143704122*5^-1 + O(5^46))*t^9 +
1070
(28447373515228957242145926166521*5^-1 + O(5^45))*t^10 -
1071
(182898434999223008342809499728*5^-2 + O(5^41))*t^11 -
1072
(800135779858798980231071314*5^-2 + O(5^38))*t^12 -
1073
(445019511166745653307297972*5^-2 + O(5^37))*t^13 -
1074
(32156961753867404468164021*5^-1 + O(5^36))*t^14 -
1075
(2981332774116526138122263*5^-2 + O(5^36))*t^15 +
1076
(109467613771521244191981292*5^-2 + O(5^36))*t^16 +
1077
(141748983545992937954322264*5^-3 + O(5^35))*t^17 +
1078
(141733191779444003946819274*5^-3 + O(5^35))*t^18 -
1079
(18239611789423131658498001*5^-3 + O(5^34))*t^19 + O(t^20)
1080
> sigma_using_e2(E,5,20);
1081
Computing (y^Frobenius)^(-1)
1082
Expansion time: 0.09
1083
Reducing differentials modulo cohomology relations.
1084
s1 = 1/2
1085
s2 = 120
1086
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 -
1087
(740148683083437693615754445393880088 + O(5^52))*t^3 +
1088
(555111512312578270211815834045410337 + O(5^52))*t^4 -
1089
(54740163019712579423665172523914979 + O(5^51))*t^5 +
1090
(8218734335072339389524939987412952 + O(5^49))*t^6 -
1091
(1081096803300113853406345033228757*5^-1 + O(5^48))*t^7 -
1092
(676756319023050668931071767639632*5^-1 + O(5^47))*t^8 +
1093
(14493303594954845892175119681424*5^-1 + O(5^46))*t^9 -
1094
(51223340065950321746557208488097*5^-1 + O(5^45))*t^10 -
1095
(99476298472490002199852592122*5^-2 + O(5^41))*t^11 +
1096
(4496994857405744881922738434*5^-2 + O(5^38))*t^12 -
1097
(784468168041799361668545092*5^-2 + O(5^37))*t^13 -
1098
(40987269503926289081432511*5^-2 + O(5^36))*t^14 -
1099
(71120346115436094002164256*5^-2 + O(5^36))*t^15 +
1100
(126948302361700956519542867*5^-2 + O(5^36))*t^16 -
1101
(92486829963149380562133236*5^-3 + O(5^35))*t^17 +
1102
(165687869369338465358518459*5^-3 + O(5^35))*t^18 +
1103
(32617839794063525807019929*5^-3 + O(5^34))*t^19 + O(t^20)
1104
> sigma_using_e2(E,5,20);
1105
Computing (y^Frobenius)^(-1)
1106
Expansion time: 0.099
1107
Reducing differentials modulo cohomology relations.
1108
s1 = 1/2
1109
s2 = 120
1110
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 -
1111
(740148683083437693615754445393880203 + O(5^52))*t^3 -
1112
(555111512312578270211815834045410148 + O(5^52))*t^4 -
1113
(165762465482228233466028339333004399 + O(5^51))*t^5 -
1114
(2883495911179226014711376693513608 + O(5^49))*t^6 -
1115
(4010852007172054723968706381173217*5^-1 + O(5^48))*t^7 +
1116
(837464528451815612591159196548783*5^-1 + O(5^47))*t^8 -
1117
(283715146304251112159267693441976*5^-1 + O(5^46))*t^9 +
1118
(68508926363565072297462563215903*5^-1 + O(5^45))*t^10 -
1119
(519971555742131274676490275682*5^-2 + O(5^41))*t^11 -
1120
(1958821819447146127648838321*5^-2 + O(5^38))*t^12 -
1121
(751595175910825090322565812*5^-2 + O(5^37))*t^13 +
1122
(171133102080767567523117694*5^-2 + O(5^36))*t^14 +
1123
(100155792583878677549529004*5^-2 + O(5^36))*t^15 -
1124
(139345755439051861589380113*5^-2 + O(5^36))*t^16 -
1125
(107770027030979093569029996*5^-3 + O(5^35))*t^17 -
1126
(144521567240994259353648501*5^-3 + O(5^35))*t^18 -
1127
(9872688500406788965437206*5^-3 + O(5^34))*t^19 + O(t^20)
1128
> sigma_using_e2(E,5,20);
1129
Computing (y^Frobenius)^(-1)
1130
Expansion time: 0.1
1131
Reducing differentials modulo cohomology relations.
1132
s1 = 1/2
1133
s2 = 120
1134
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 -
1135
(740148683083437693615754445393880208 + O(5^52))*t^3 +
1136
(555111512312578270211815834045410157 + O(5^52))*t^4 -
1137
(54740163019712579423665172523922389 + O(5^51))*t^5 +
1138
(8218734335072339389524939987394547 + O(5^49))*t^6 -
1139
(1821245486383551547022099480216862*5^-1 + O(5^48))*t^7 -
1140
(2*5^-1 + O(1))*t^8 + O(5^-1)*t^9 + O(5^-53)*t^10 +
1141
O(5^-55)*t^11 + O(5^-109)*t^12 + O(5^-110)*t^13 +
1142
O(5^-162)*t^14 + O(5^-163)*t^15 + O(5^-215)*t^16 +
1143
O(5^-216)*t^17 + O(5^-268)*t^18 + O(5^-269)*t^19 + O(t^20)
1144
> sigma_using_e2(EC("37A"),5,20);
1145
Computing (y^Frobenius)^(-1)
1146
Expansion time: 0.099
1147
Reducing differentials modulo cohomology relations.
1148
s1 = 1/2
1149
s2 = 120
1150
t + O(5^52)*t^2 + O(5^52)*t^3 - (1110223024625156540423631668090\
1151
820312 + O(5^52))*t^4 + (7401486830834376936157544453938802*\
1152
5 + O(5^51))*t^5 + O(5^49)*t^6 -
1153
(8141635513917814629773298899332678*5^-1 + O(5^48))*t^7 +
1154
O(1)*t^8 + O(5^-1)*t^9 + O(5^-52)*t^10 + O(5^-53)*t^11 +
1155
O(5^-106)*t^12 + O(5^-107)*t^13 + O(5^-159)*t^14 +
1156
O(5^-160)*t^15 + O(5^-212)*t^16 + O(5^-213)*t^17 +
1157
O(5^-265)*t^18 + O(5^-266)*t^19 + O(t^20)
1158
> sigma_using_e2(EC("37A"),5,20);
1159
Computing (y^Frobenius)^(-1)
1160
Expansion time: 0.1
1161
Reducing differentials modulo cohomology relations.
1162
s1 = 1/2
1163
s2 = 120
1164
t + O(5^52)*t^2 + t^3 - (1110223024625156540423631668090820312 +
1165
O(5^52))*t^4 - (185037170770859423403938611348470052 +
1166
O(5^51))*t^5 - (8881784197001252323389053344726561 +
1167
O(5^49))*t^6 - (3700743415417188468078772226969402*5^-1 +
1168
O(5^48))*t^7 + (325665420556712585190931955973307 +
1169
O(5^47))*t^8 - (992151687085655765489689887516067*5^-1 +
1170
O(5^47))*t^9 + (192438657601693800340096155802393*5^-1 +
1171
O(5^46))*t^10 + (5639228061588096713262891012522*5^-2 +
1172
O(5^45))*t^11 + (53184969655852737126960640861916*5^-1 +
1173
O(5^45))*t^12 + (32281020533866103622373698211084*5^-2 +
1174
O(5^44))*t^13 + (57661106929738288893113060603256*5^-2 +
1175
O(5^44))*t^14 + (3180647822492471018173308499538*5^-2 +
1176
O(5^43))*t^15 + (449144578440627702869473691621*5^-2 +
1177
O(5^42))*t^16 + (2086347241407767586787117146951*5^-3 +
1178
O(5^41))*t^17 - (512245785428633261709138321746*5^-2 +
1179
O(5^41))*t^18 - (330076174900100845183154138172*5^-3 +
1180
O(5^40))*t^19 + O(t^20)
1181
> sigma_using_e2(EC("37A"),5,20);
1182
Computing (y^Frobenius)^(-1)
1183
Expansion time: 0.099
1184
Reducing differentials modulo cohomology relations.
1185
t + O(5^52)*t^2 + (331883312126563673413235306321062928 +
1186
O(5^52))*t^3 - (1110223024625156540423631668090820312 +
1187
O(5^52))*t^4 - (114496958818289158695517187452183148 +
1188
O(5^51))*t^5 + (445053157775380010065972176906892 +
1189
O(5^49))*t^6 - (246425046817409275170929836993681 +
1190
O(5^48))*t^7 - (248945902282571925665450930262558 +
1191
O(5^47))*t^8 - (9730291437202192499870687559126*5 +
1192
O(5^47))*t^9 + (42490376307877879990909502013661 +
1193
O(5^46))*t^10 - (7675221160712331595301225730587 +
1194
O(5^45))*t^11 - (10121360477771639598919647047014 +
1195
O(5^45))*t^12 + (508742572750785974838087065064 +
1196
O(5^44))*t^13 - (47223357661752843893077363668*5 +
1197
O(5^44))*t^14 - (238907170363878162509258107286 +
1198
O(5^43))*t^15 + (85145841166214130496466860646 +
1199
O(5^42))*t^16 + (13592483819664419928276730898 +
1200
O(5^41))*t^17 + (9451463052915136304055110597 + O(5^41))*t^18
1201
- (1228398393563077324992870341 + O(5^40))*t^19 + O(t^20)
1202
> P;
1203
(-1 : 1 : 1)
1204
> Parent(P);
1205
Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 +
1206
2 over Rational Field with coordinates in Rational Field
1207
> E := EC("37A");
1208
> G, f := MordellWeilGroup(E);
1209
> P := f(G.1);
1210
> P;
1211
(0 : 0 : 1)
1212
> height_of_point(P,5,35);
1213
Time: 1.080
1214
s1 = O(5^20)
1215
Time: 0.010
1216
s2 = 53 + O(5^3)
1217
Computing (y^Frobenius)^(-1)
1218
Expansion time: 0.069
1219
Reducing differentials modulo cohomology relations.
1220
Time: 0.190
1221
s2b = 41366042218322 + O(5^20)
1222
s2 - s2b = -19 + O(5^3)
1223
-3 + O(5^2)
1224
> h := height_function(E,5,30);
1225
Computing (y^Frobenius)^(-1)
1226
Expansion time: 0.051
1227
Reducing differentials modulo cohomology relations.
1228
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1229
Computing (y^Frobenius)^(-1)
1230
Expansion time: 0.09
1231
Reducing differentials modulo cohomology relations.
1232
t + O(5^52)*t^2 + (331883312126563673413235306321062928 +
1233
O(5^52))*t^3 - (1110223024625156540423631668090820312 +
1234
O(5^52))*t^4 - (114496958818289158695517187452183148 +
1235
O(5^51))*t^5 + (445053157775380010065972176906892 +
1236
O(5^49))*t^6 - (246425046817409275170929836993681 +
1237
O(5^48))*t^7 - (248945902282571925665450930262558 +
1238
O(5^47))*t^8 - (9730291437202192499870687559126*5 +
1239
O(5^47))*t^9 + (42490376307877879990909502013661 +
1240
O(5^46))*t^10 - (7675221160712331595301225730587 +
1241
O(5^45))*t^11 - (10121360477771639598919647047014 +
1242
O(5^45))*t^12 + (508742572750785974838087065064 +
1243
O(5^44))*t^13 - (47223357661752843893077363668*5 +
1244
O(5^44))*t^14 - (238907170363878162509258107286 +
1245
O(5^43))*t^15 + (85145841166214130496466860646 +
1246
O(5^42))*t^16 + (13592483819664419928276730898 +
1247
O(5^41))*t^17 + (9451463052915136304055110597 + O(5^41))*t^18
1248
- (1228398393563077324992870341 + O(5^40))*t^19 + O(t^20)
1249
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1250
Computing (y^Frobenius)^(-1)
1251
Expansion time: 0.099
1252
Reducing differentials modulo cohomology relations.
1253
331883312126563673413235306321062928 + O(5^52)
1254
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1255
Computing (y^Frobenius)^(-1)
1256
Expansion time: 0.1
1257
Reducing differentials modulo cohomology relations.
1258
-331883312126563673413235306321062928 + O(5^52)
1259
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1260
Computing (y^Frobenius)^(-1)
1261
Expansion time: 0.099
1262
Reducing differentials modulo cohomology relations.
1263
sigma_of_z = z + O(5^52)*z^2 + (3318833121265636734132353063210\
1264
62928 + O(5^52))*z^3 + O(5^52)*z^4 -
1265
(572484794091445793477585937260915738*5^-1 + O(5^51))*z^5 +
1266
O(5^49)*z^6 - (1989565230700169856559061432977002*5^-1 +
1267
O(5^49))*z^7 + O(5^48)*z^8 - (295205514675397750004341499150\
1268
1261*5^-1 + O(5^48))*z^9 + O(5^48)*z^10 +
1269
(6383056611836077441016060024375336*5^-2 + O(5^47))*z^11 +
1270
O(5^46)*z^12 - (970086973722876449310372081716669*5^-2 +
1271
O(5^46))*z^13 + O(5^46)*z^14 -
1272
(1525252462976469179547332754163016*5^-3 + O(5^45))*z^15 +
1273
O(5^45)*z^16 - (187386734196935414796599570828493*5^-3 +
1274
O(5^44))*z^17 + O(5^44)*z^18 +
1275
(242720203645724787456505786084508*5^-3 + O(5^44))*z^19 +
1276
O(z^20)
1277
sigma_of_t = z + O(5^52)*z^2 + (3318833121265636734132353063210\
1278
62928 + O(5^52))*z^3 - (111022302462515654042363166809082031\
1279
2 + O(5^52))*z^4 - (114496958818289158695517187452183148 +
1280
O(5^51))*z^5 + (445053157775380010065972176906892 +
1281
O(5^49))*z^6 - (246425046817409275170929836993681 +
1282
O(5^48))*z^7 - (248945902282571925665450930262558 +
1283
O(5^47))*z^8 - (9730291437202192499870687559126*5 +
1284
O(5^47))*z^9 + (42490376307877879990909502013661 +
1285
O(5^46))*z^10 - (7675221160712331595301225730587 +
1286
O(5^45))*z^11 - (10121360477771639598919647047014 +
1287
O(5^45))*z^12 + (508742572750785974838087065064 +
1288
O(5^44))*z^13 - (47223357661752843893077363668*5 +
1289
O(5^44))*z^14 - (238907170363878162509258107286 +
1290
O(5^43))*z^15 + (85145841166214130496466860646 +
1291
O(5^42))*z^16 + (13592483819664419928276730898 +
1292
O(5^41))*z^17 + (9451463052915136304055110597 + O(5^41))*z^18
1293
- (1228398393563077324992870341 + O(5^40))*z^19 +
1294
(4210721468439696872157677218 + O(5^40))*z^20 +
1295
(488040730614903301156568960493*5^-4 + O(5^39))*z^21 -
1296
(366090865413409916507680948 + O(5^39))*z^22 +
1297
(27352436193426794540060570549*5^-4 + O(5^38))*z^23 -
1298
(7242727124186526331718572011*5^-4 + O(5^38))*z^24 + O(z^25)
1299
331883312126563673413235306321062928 + O(5^52)
1300
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1301
Computing (y^Frobenius)^(-1)
1302
Expansion time: 0.101
1303
Reducing differentials modulo cohomology relations.
1304
sigma_of_z = z + O(5^52)*z^2 + (3318833121265636734132353063210\
1305
62928 + O(5^52))*z^3 + O(5^52)*z^4 -
1306
(572484794091445793477585937260915738*5^-1 + O(5^51))*z^5 +
1307
O(5^49)*z^6 - (1989565230700169856559061432977002*5^-1 +
1308
O(5^49))*z^7 + O(5^48)*z^8 - (295205514675397750004341499150\
1309
1261*5^-1 + O(5^48))*z^9 + O(5^48)*z^10 +
1310
(6383056611836077441016060024375336*5^-2 + O(5^47))*z^11 +
1311
O(5^46)*z^12 - (970086973722876449310372081716669*5^-2 +
1312
O(5^46))*z^13 + O(5^46)*z^14 -
1313
(1525252462976469179547332754163016*5^-3 + O(5^45))*z^15 +
1314
O(5^45)*z^16 - (187386734196935414796599570828493*5^-3 +
1315
O(5^44))*z^17 + O(5^44)*z^18 +
1316
(242720203645724787456505786084508*5^-3 + O(5^44))*z^19 +
1317
O(z^20)
1318
sigma_of_t = t + O(5^52)*t^2 + (3318833121265636734132353063210\
1319
62928 + O(5^52))*t^3 - (111022302462515654042363166809082031\
1320
2 + O(5^52))*t^4 - (114496958818289158695517187452183148 +
1321
O(5^51))*t^5 + (445053157775380010065972176906892 +
1322
O(5^49))*t^6 - (246425046817409275170929836993681 +
1323
O(5^48))*t^7 - (248945902282571925665450930262558 +
1324
O(5^47))*t^8 - (9730291437202192499870687559126*5 +
1325
O(5^47))*t^9 + (42490376307877879990909502013661 +
1326
O(5^46))*t^10 - (7675221160712331595301225730587 +
1327
O(5^45))*t^11 - (10121360477771639598919647047014 +
1328
O(5^45))*t^12 + (508742572750785974838087065064 +
1329
O(5^44))*t^13 - (47223357661752843893077363668*5 +
1330
O(5^44))*t^14 - (238907170363878162509258107286 +
1331
O(5^43))*t^15 + (85145841166214130496466860646 +
1332
O(5^42))*t^16 + (13592483819664419928276730898 +
1333
O(5^41))*t^17 + (9451463052915136304055110597 + O(5^41))*t^18
1334
- (1228398393563077324992870341 + O(5^40))*t^19 +
1335
(4210721468439696872157677218 + O(5^40))*t^20 +
1336
(488040730614903301156568960493*5^-4 + O(5^39))*t^21 -
1337
(366090865413409916507680948 + O(5^39))*t^22 +
1338
(27352436193426794540060570549*5^-4 + O(5^38))*t^23 -
1339
(7242727124186526331718572011*5^-4 + O(5^38))*t^24 + O(t^25)
1340
331883312126563673413235306321062928 + O(5^52)
1341
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1342
Computing (y^Frobenius)^(-1)
1343
Expansion time: 0.09
1344
Reducing differentials modulo cohomology relations.
1345
sigma_of_z = z + O(5^52)*z^2 - (3318833121265636734132353063210\
1346
62928 + O(5^52))*z^3 + O(5^52)*z^4 -
1347
(572484794091445793477585937260915738*5^-1 + O(5^51))*z^5 +
1348
O(5^49)*z^6 - (37132579446567251091702244966413809*5^-1 +
1349
O(5^49))*z^7 + O(5^48)*z^8 - (351951012083163939367181498304\
1350
6653*5^-1 + O(5^48))*z^9 + O(5^48)*z^10 +
1351
(1508498676482117717568145936189131*5^-2 + O(5^47))*z^11 +
1352
O(5^46)*z^12 + (81800697895818887737947097052042*5^-1 +
1353
O(5^46))*z^13 + O(5^46)*z^14 -
1354
(452361705555103897582665660102321*5^-3 + O(5^45))*z^15 +
1355
O(5^45)*z^16 + (251713869361766805733035104823793*5^-3 +
1356
O(5^44))*z^17 + O(5^44)*z^18 +
1357
(137074885431587834358238230892353*5^-3 + O(5^44))*z^19 +
1358
O(z^20)
1359
sigma_of_t = t + O(5^52)*t^2 - (3318833121265636734132353063210\
1360
62928 + O(5^52))*t^3 - (111022302462515654042363166809082031\
1361
2 + O(5^52))*t^4 - (114496958818289158695517187452183148 +
1362
O(5^51))*t^5 - (445053157775380010065972176906892 +
1363
O(5^49))*t^6 + (2712422600253921763086158075756174*5^-1 +
1364
O(5^48))*t^7 - (248945902282571925665450930262558 +
1365
O(5^47))*t^8 + (500210808339014563245973107051528*5^-1 +
1366
O(5^47))*t^9 + (350061117604023247193425868431064*5^-1 +
1367
O(5^46))*t^10 - (102547378314196000448652776310529*5^-2 +
1368
O(5^45))*t^11 + (1534450102375366490687977784606*5^-1 +
1369
O(5^45))*t^12 + (11287252120088070907312416171067*5^-1 +
1370
O(5^44))*t^13 - (2124209910855379565780183900342*5^-2 +
1371
O(5^44))*t^14 - (7169219439052372633549416052522*5^-2 +
1372
O(5^43))*t^15 + (69167910730606868564893818148 +
1373
O(5^42))*t^16 + (1201290126587875861046621005874*5^-3 +
1374
O(5^41))*t^17 - (248166440348117994852359297968*5^-2 +
1375
O(5^41))*t^18 - (384727456478966449426400512883*5^-3 +
1376
O(5^40))*t^19 - (11437230645978973371010417101*5^-3 +
1377
O(5^40))*t^20 - (251547777777881746439008047509*5^-4 +
1378
O(5^39))*t^21 + (78980927752525059272201248054*5^-3 +
1379
O(5^39))*t^22 + (9621538632222121039093315306*5^-3 +
1380
O(5^38))*t^23 - (37462151861312383777694576527*5^-4 +
1381
O(5^38))*t^24 + O(t^25)
1382
-331883312126563673413235306321062928 + O(5^52)
1383
> aInvariants(E);
1384
[ 0, 0, 1, -1, 0 ]
1385
> Coefficient(sigma_using_e2(EC("37A"),5,20),3);
1386
Computing (y^Frobenius)^(-1)
1387
Expansion time: 0.089
1388
Reducing differentials modulo cohomology relations.
1389
sigma_of_z = z + O(5^52)*z^2 + (3318833121265636734132353063210\
1390
62928 + O(5^52))*z^3 + O(5^52)*z^4 -
1391
(572484794091445793477585937260915738*5^-1 + O(5^51))*z^5 +
1392
O(5^49)*z^6 - (1989565230700169856559061432977002*5^-1 +
1393
O(5^49))*z^7 + O(5^48)*z^8 - (295205514675397750004341499150\
1394
1261*5^-1 + O(5^48))*z^9 + O(5^48)*z^10 +
1395
(6383056611836077441016060024375336*5^-2 + O(5^47))*z^11 +
1396
O(5^46)*z^12 - (970086973722876449310372081716669*5^-2 +
1397
O(5^46))*z^13 + O(5^46)*z^14 -
1398
(1525252462976469179547332754163016*5^-3 + O(5^45))*z^15 +
1399
O(5^45)*z^16 - (187386734196935414796599570828493*5^-3 +
1400
O(5^44))*z^17 + O(5^44)*z^18 +
1401
(242720203645724787456505786084508*5^-3 + O(5^44))*z^19 +
1402
O(z^20)
1403
sigma_of_t = t + O(5^52)*t^2 + (3318833121265636734132353063210\
1404
62928 + O(5^52))*t^3 - (111022302462515654042363166809082031\
1405
2 + O(5^52))*t^4 - (114496958818289158695517187452183148 +
1406
O(5^51))*t^5 + (445053157775380010065972176906892 +
1407
O(5^49))*t^6 - (246425046817409275170929836993681 +
1408
O(5^48))*t^7 - (248945902282571925665450930262558 +
1409
O(5^47))*t^8 - (9730291437202192499870687559126*5 +
1410
O(5^47))*t^9 + (42490376307877879990909502013661 +
1411
O(5^46))*t^10 - (7675221160712331595301225730587 +
1412
O(5^45))*t^11 - (10121360477771639598919647047014 +
1413
O(5^45))*t^12 + (508742572750785974838087065064 +
1414
O(5^44))*t^13 - (47223357661752843893077363668*5 +
1415
O(5^44))*t^14 - (238907170363878162509258107286 +
1416
O(5^43))*t^15 + (85145841166214130496466860646 +
1417
O(5^42))*t^16 + (13592483819664419928276730898 +
1418
O(5^41))*t^17 + (9451463052915136304055110597 + O(5^41))*t^18
1419
- (1228398393563077324992870341 + O(5^40))*t^19 +
1420
(4210721468439696872157677218 + O(5^40))*t^20 +
1421
(488040730614903301156568960493*5^-4 + O(5^39))*t^21 -
1422
(366090865413409916507680948 + O(5^39))*t^22 +
1423
(27352436193426794540060570549*5^-4 + O(5^38))*t^23 -
1424
(7242727124186526331718572011*5^-4 + O(5^38))*t^24 + O(t^25)
1425
331883312126563673413235306321062928 + O(5^52)
1426
> height_of_point(P,5,35);
1427
Time: 1.070
1428
s1 = O(5^20)
1429
Time: 0.020
1430
s2 = 53 + O(5^3)
1431
Computing (y^Frobenius)^(-1)
1432
Expansion time: 0.069
1433
Reducing differentials modulo cohomology relations.
1434
Time: 0.180
1435
s2b = 41366042218322 + O(5^20)
1436
s2 - s2b = -19 + O(5^3)
1437
-3 + O(5^2)
1438
> height_of_point(P,5,45);
1439
Time: 3.330
1440
s1 = O(5^20)
1441
Time: 0.040
1442
s2 = 53 + O(5^3)
1443
Computing (y^Frobenius)^(-1)
1444
Expansion time: 0.069
1445
Reducing differentials modulo cohomology relations.
1446
Time: 0.200
1447
s2b = 41366042218322 + O(5^20)
1448
s2 - s2b = -19 + O(5^3)
1449
-3 + O(5^2)
1450
> E := EC("1058C");
1451
> aInvariants(E);
1452
[ 1, 0, 1, 0, 2 ]
1453
> E;
1454
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
1455
Field
1456
> G, f := MordellWeilGroup(E);
1457
> P := f(G.1);
1458
> P;
1459
(-1 : 1 : 1)
1460
> height_of_point(P,5,45);
1461
Time: 4.790
1462
s1 = -47683715820312 + O(5^20)
1463
Time: 0.050
1464
s2 = -47 + O(5^3)
1465
Computing (y^Frobenius)^(-1)
1466
Expansion time: 0.07
1467
Reducing differentials modulo cohomology relations.
1468
Time: 0.190
1469
s2b = -28994174169036 + O(5^20)
1470
s2 - s2b = -11 + O(5^3)
1471
1 + O(5^2)
1472
> h := height_function(E,5,30);
1473
Computing (y^Frobenius)^(-1)
1474
Expansion time: 0.051
1475
Reducing differentials modulo cohomology relations.
1476
sigma_of_z = z + O(5^32)*z^2 - (6586425211840379867422 +
1477
O(5^32))*z^3 + O(5^32)*z^4 - (2054070730961408546742*5^-1 +
1478
O(5^31))*z^5 + O(5^30)*z^6 - (227755506296130888078*5^-1 +
1479
O(5^29))*z^7 + O(5^29)*z^8 - (417414431500402178496*5^-1 +
1480
O(5^29))*z^9 + O(5^28)*z^10 + (1906843524742089631*5^-2 +
1481
O(5^26))*z^11 + O(5^26)*z^12 + (2100638315129136364*5^-2 +
1482
O(5^25))*z^13 + O(5^24)*z^14 - (67031490260633954*5^-3 +
1483
O(5^22))*z^15 + O(5^21)*z^16 + (808268455808457*5^-3 +
1484
O(5^19))*z^17 + O(5^19)*z^18 - (958454004995672*5^-3 +
1485
O(5^19))*z^19 + O(5^19)*z^20 + (935579698359078*5^-4 +
1486
O(5^18))*z^21 + O(5^17)*z^22 + (51123741962324*5^-4 +
1487
O(5^17))*z^23 + O(5^16)*z^24 + (157229664188094*5^-6 +
1488
O(5^15))*z^25 + O(5^13)*z^26 - (4127700788859*5^-6 +
1489
O(5^13))*z^27 + O(5^13)*z^28 - (8594787547078*5^-6 +
1490
O(5^13))*z^29 + O(z^30)
1491
sigma_of_t = t - (11641532182693481445312 + O(5^32))*t^2 +
1492
(1787123539683519078599*5 + O(5^32))*t^3 -
1493
(4058871726413829078476 + O(5^32))*t^4 -
1494
(295526084219465032023 + O(5^31))*t^5 +
1495
(104454124496266441009 + O(5^30))*t^6 - (7816372567323560883
1496
+ O(5^29))*t^7 + (30694029669476612232 + O(5^29))*t^8 -
1497
(7078732682188443772 + O(5^28))*t^9 - (5473317214354319621 +
1498
O(5^28))*t^10 + (613727091278416833 + O(5^26))*t^11 +
1499
(33885851644877526 + O(5^25))*t^12 - (21774254701697454 +
1500
O(5^24))*t^13 + (29799514491687162 + O(5^24))*t^14 -
1501
(333931262365241 + O(5^22))*t^15 + (123150656551243 +
1502
O(5^21))*t^16 - (5861255724438 + O(5^19))*t^17 +
1503
(1087647337866 + O(5^18))*t^18 - (50623617284*5 +
1504
O(5^17))*t^19 + (2656628927 + O(5^16))*t^20 + (73239002*5^3 +
1505
O(5^15))*t^21 + (132025756*5 + O(5^14))*t^22 - (42338568 +
1506
O(5^12))*t^23 - (20527317 + O(5^11))*t^24 - (555369 +
1507
O(5^9))*t^25 - (313437 + O(5^9))*t^26 - (2846*5 +
1508
O(5^8))*t^27 + (1604*5 + O(5^7))*t^28 + (3761 + O(5^6))*t^29
1509
+ (52*5^2 + O(5^5))*t^30 + (92773484*5^-7 + O(5^5))*t^31 +
1510
(2520252*5^-7 + O(5^4))*t^32 - (3609196*5^-7 + O(5^4))*t^33 +
1511
(3061342*5^-7 + O(5^3))*t^34 + O(t^35)
1512
> E;
1513
Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational
1514
Field
1515
> sigma_using_e2(E,5,20);
1516
Computing (y^Frobenius)^(-1)
1517
Expansion time: 0.1
1518
Reducing differentials modulo cohomology relations.
1519
sigma_of_z = z + O(5^52)*z^2 - (6298543858467434273289708430654\
1520
14297 + O(5^52))*z^3 + O(5^52)*z^4 -
1521
(769258654111136873757143772687843617*5^-1 + O(5^51))*z^5 +
1522
O(5^49)*z^6 + (2667623452623147004644613292940047*5^-1 +
1523
O(5^48))*z^7 + O(5^48)*z^8 + (166626391604314726280696395329\
1524
0254*5^-1 + O(5^47))*z^9 + O(5^46)*z^10 -
1525
(8796517002357125787395668066619*5^-2 + O(5^43))*z^11 +
1526
O(5^41)*z^12 - (118527727825329133108210707386*5^-2 +
1527
O(5^42))*z^13 + O(5^41)*z^14 -
1528
(2544432097662158286678248915204*5^-3 + O(5^41))*z^15 +
1529
O(5^40)*z^16 + (148065011957841727763084714707*5^-3 +
1530
O(5^40))*z^17 + O(5^40)*z^18 -
1531
(315505256803804402362696401922*5^-3 + O(5^40))*z^19 +
1532
O(z^20)
1533
sigma_of_t = t - (1110223024625156540423631668090820312 +
1534
O(5^52))*t^2 + (170088596064026391980507609544469224*5 +
1535
O(5^52))*t^3 + (720552958167619669641991237538109024 +
1536
O(5^52))*t^4 - (34851578966356178111132895002141398 +
1537
O(5^51))*t^5 + (3897197146110341098869369313316009 +
1538
O(5^49))*t^6 - (937958051051020988045388368482758 +
1539
O(5^48))*t^7 + (101932791998268065474371625049732 +
1540
O(5^47))*t^8 + (17654698179021849547194520540603 +
1541
O(5^46))*t^9 + (12380146449798832713059083180379 +
1542
O(5^45))*t^10 - (3674179353696308614287989417 + O(5^41))*t^11
1543
+ (97688337002307787680033776 + O(5^38))*t^12 +
1544
(23546698310058631040490046 + O(5^37))*t^13 +
1545
(6538955539462142665515287 + O(5^36))*t^14 -
1546
(4953133890962745959630866 + O(5^36))*t^15 -
1547
(4211223062868945290714382 + O(5^36))*t^16 -
1548
(931540141569276783068188 + O(5^35))*t^17 -
1549
(368393368136271971802759 + O(5^35))*t^18 -
1550
(40241289524739344320409*5 + O(5^34))*t^19 +
1551
(10517833771212373425802 + O(5^33))*t^20 -
1552
(13699160248637224844687203*5^-4 + O(5^33))*t^21 -
1553
(25905043052708000912439069*5^-4 + O(5^33))*t^22 -
1554
(4022427664407666537123993*5^-3 + O(5^33))*t^23 -
1555
(1205067592978531567377507*5^-4 + O(5^32))*t^24 + O(t^25)
1556
t - (1110223024625156540423631668090820312 + O(5^52))*t^2 +
1557
(170088596064026391980507609544469224*5 + O(5^52))*t^3 +
1558
(720552958167619669641991237538109024 + O(5^52))*t^4 -
1559
(34851578966356178111132895002141398 + O(5^51))*t^5 +
1560
(3897197146110341098869369313316009 + O(5^49))*t^6 -
1561
(937958051051020988045388368482758 + O(5^48))*t^7 +
1562
(101932791998268065474371625049732 + O(5^47))*t^8 +
1563
(17654698179021849547194520540603 + O(5^46))*t^9 +
1564
(12380146449798832713059083180379 + O(5^45))*t^10 -
1565
(3674179353696308614287989417 + O(5^41))*t^11 +
1566
(97688337002307787680033776 + O(5^38))*t^12 +
1567
(23546698310058631040490046 + O(5^37))*t^13 +
1568
(6538955539462142665515287 + O(5^36))*t^14 -
1569
(4953133890962745959630866 + O(5^36))*t^15 -
1570
(4211223062868945290714382 + O(5^36))*t^16 -
1571
(931540141569276783068188 + O(5^35))*t^17 -
1572
(368393368136271971802759 + O(5^35))*t^18 -
1573
(40241289524739344320409*5 + O(5^34))*t^19 + O(t^20)
1574
> regulator(E,5,30)
1575
> ;
1576
Computing (y^Frobenius)^(-1)
1577
Expansion time: 0.049
1578
Reducing differentials modulo cohomology relations.
1579
sigma_of_z = z + O(5^32)*z^2 - (6586425211840379867422 +
1580
O(5^32))*z^3 + O(5^32)*z^4 - (2054070730961408546742*5^-1 +
1581
O(5^31))*z^5 + O(5^30)*z^6 - (227755506296130888078*5^-1 +
1582
O(5^29))*z^7 + O(5^29)*z^8 - (417414431500402178496*5^-1 +
1583
O(5^29))*z^9 + O(5^28)*z^10 + (1906843524742089631*5^-2 +
1584
O(5^26))*z^11 + O(5^26)*z^12 + (2100638315129136364*5^-2 +
1585
O(5^25))*z^13 + O(5^24)*z^14 - (67031490260633954*5^-3 +
1586
O(5^22))*z^15 + O(5^21)*z^16 + (808268455808457*5^-3 +
1587
O(5^19))*z^17 + O(5^19)*z^18 - (958454004995672*5^-3 +
1588
O(5^19))*z^19 + O(5^19)*z^20 + (935579698359078*5^-4 +
1589
O(5^18))*z^21 + O(5^17)*z^22 + (51123741962324*5^-4 +
1590
O(5^17))*z^23 + O(5^16)*z^24 + (157229664188094*5^-6 +
1591
O(5^15))*z^25 + O(5^13)*z^26 - (4127700788859*5^-6 +
1592
O(5^13))*z^27 + O(5^13)*z^28 - (8594787547078*5^-6 +
1593
O(5^13))*z^29 + O(z^30)
1594
sigma_of_t = t - (11641532182693481445312 + O(5^32))*t^2 +
1595
(1787123539683519078599*5 + O(5^32))*t^3 -
1596
(4058871726413829078476 + O(5^32))*t^4 -
1597
(295526084219465032023 + O(5^31))*t^5 +
1598
(104454124496266441009 + O(5^30))*t^6 - (7816372567323560883
1599
+ O(5^29))*t^7 + (30694029669476612232 + O(5^29))*t^8 -
1600
(7078732682188443772 + O(5^28))*t^9 - (5473317214354319621 +
1601
O(5^28))*t^10 + (613727091278416833 + O(5^26))*t^11 +
1602
(33885851644877526 + O(5^25))*t^12 - (21774254701697454 +
1603
O(5^24))*t^13 + (29799514491687162 + O(5^24))*t^14 -
1604
(333931262365241 + O(5^22))*t^15 + (123150656551243 +
1605
O(5^21))*t^16 - (5861255724438 + O(5^19))*t^17 +
1606
(1087647337866 + O(5^18))*t^18 - (50623617284*5 +
1607
O(5^17))*t^19 + (2656628927 + O(5^16))*t^20 + (73239002*5^3 +
1608
O(5^15))*t^21 + (132025756*5 + O(5^14))*t^22 - (42338568 +
1609
O(5^12))*t^23 - (20527317 + O(5^11))*t^24 - (555369 +
1610
O(5^9))*t^25 - (313437 + O(5^9))*t^26 - (2846*5 +
1611
O(5^8))*t^27 + (1604*5 + O(5^7))*t^28 + (3761 + O(5^6))*t^29
1612
+ (52*5^2 + O(5^5))*t^30 + (92773484*5^-7 + O(5^5))*t^31 +
1613
(2520252*5^-7 + O(5^4))*t^32 - (3609196*5^-7 + O(5^4))*t^33 +
1614
(3061342*5^-7 + O(5^3))*t^34 + O(t^35)
1615
1438935235176290340451 + O(5^31)
1616
[-636005667032888288524 + O(5^31) -373959132520556057038*5 +
1617
O(5^31)]
1618
[-373959132520556057038*5 + O(5^31) -808975102078321911174 +
1619
O(5^31)]
1620
> P;
1621
(-1 : 1 : 1)
1622
> Q;
1623
(1 : -3 : 1)
1624
> aInvariants(E);
1625
[ 1, 0, 1, 0, 2 ]
1626
> regulator(E,5,30)
1627
> regulator(E,5,30)
1628
1629
>> regulator(E,5,30);
1630
^
1631
User error: bad syntax
1632
> ;
1633
> >> regulator(E,5,30);
1634
Computing (y^Frobenius)^(-1)
1635
Expansion time: 0.05
1636
Reducing differentials modulo cohomology relations.
1637
1438935235176290340451 + O(5^31)
1638
[-636005667032888288524 + O(5^31) -373959132520556057038*5 +
1639
O(5^31)]
1640
[-373959132520556057038*5 + O(5^31) -808975102078321911174 +
1641
O(5^31)]
1642
> >> regulator(E,5,30);
1643
Computing (y^Frobenius)^(-1)
1644
Expansion time: 0.04
1645
Reducing differentials modulo cohomology relations.
1646
-372767215421039257159 + O(5^31)
1647
[239545855354013131971 + O(5^31) -26615011923010630709*5 +
1648
O(5^31)]
1649
[-26615011923010630709*5 + O(5^31) -1465315445362506007879 +
1650
O(5^31)]
1651
> regulator(E,5,30);
1652
Computing (y^Frobenius)^(-1)
1653
Expansion time: 0.04
1654
Reducing differentials modulo cohomology relations.
1655
1438935235176290340451 + O(5^31)
1656
[-636005667032888288524 + O(5^31) -373959132520556057038*5 +
1657
O(5^31)]
1658
[-373959132520556057038*5 + O(5^31) -808975102078321911174 +
1659
O(5^31)]
1660
> regulator(E,5,30);
1661
1662
In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\
1663
al.m", line 550, column 28:
1664
>> pr := AbsolutePrecision(f);
1665
^
1666
Runtime error: Undefined reference 'f' in package
1667
"/home/was/papers/padic_cyclotomic_height/e2heights/formal.m"
1668
> regulator(E,5,30);
1669
Computing (y^Frobenius)^(-1)
1670
Expansion time: 0.041
1671
Reducing differentials modulo cohomology relations.
1672
1438935235176290340451 + O(5^31)
1673
[-636005667032888288524 + O(5^31) -373959132520556057038*5 +
1674
O(5^31)]
1675
[-373959132520556057038*5 + O(5^31) -808975102078321911174 +
1676
O(5^31)]
1677
> G := [p : p in PrimeSeq(2,100) | IsGoodOrdinary(E,p)];
1678
> G;
1679
[ 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61,
1680
67, 71, 73, 79, 89, 97 ]
1681
> r := regulator(E,7);
1682
Computing (y^Frobenius)^(-1)
1683
Expansion time: 0.041
1684
Reducing differentials modulo cohomology relations.
1685
34330260522044597 + O(7^20)
1686
[28393329287189844 + O(7^20) 8692966313868747 + O(7^20)]
1687
[8692966313868747 + O(7^20) -14606141452113312 + O(7^20)]
1688
> r := regulator(E,7);
1689
Computing (y^Frobenius)^(-1)
1690
Expansion time: 0.039
1691
Reducing differentials modulo cohomology relations.
1692
> r;
1693
34330260522044597 + O(7^20)
1694
> [<p,regulator(E,p)> : p in G | p ge 5 and p le 29];
1695
Computing (y^Frobenius)^(-1)
1696
Expansion time: 0.031
1697
Reducing differentials modulo cohomology relations.
1698
Computing (y^Frobenius)^(-1)
1699
Expansion time: 0.041
1700
Reducing differentials modulo cohomology relations.
1701
1702
>> [* <p,regulator(E,p)> : p in G | p ge 5 and p le 29 *];
1703
^
1704
Runtime error in sequence construction: Could not find a valid
1705
universe
1706
> regulator(E,11);
1707
Computing (y^Frobenius)^(-1)
1708
Expansion time: 0.081
1709
Reducing differentials modulo cohomology relations.
1710
1664557466859596710369 + O(11^21)
1711
[-2203571954321795386286 + O(11^21) -1922017207079415409985 +
1712
O(11^21)]
1713
[-1922017207079415409985 + O(11^21) 876258682167493485144 +
1714
O(11^21)]
1715
> >> [* <p,regulator(E,p)> : p in G | p ge 5 and p le 29 *];
1716
Computing (y^Frobenius)^(-1)
1717
Expansion time: 0.021
1718
Reducing differentials modulo cohomology relations.
1719
Computing (y^Frobenius)^(-1)
1720
Expansion time: 0.039
1721
Reducing differentials modulo cohomology relations.
1722
Computing (y^Frobenius)^(-1)
1723
Expansion time: 0.079
1724
Reducing differentials modulo cohomology relations.
1725
Computing (y^Frobenius)^(-1)
1726
Expansion time: 0.1
1727
Reducing differentials modulo cohomology relations.
1728
Computing (y^Frobenius)^(-1)
1729
Expansion time: 0.131
1730
Reducing differentials modulo cohomology relations.
1731
Computing (y^Frobenius)^(-1)
1732
Expansion time: 0.221
1733
Reducing differentials modulo cohomology relations.
1734
Computing (y^Frobenius)^(-1)
1735
Expansion time: 0.359
1736
Reducing differentials modulo cohomology relations.
1737
[* <5, 1106466121701 + O(5^18)>, <7, 34330260522044597 +
1738
O(7^20)>, <11, 1664557466859596710369 + O(11^21)>, <13,
1739
103609576221026126766923 + O(13^21)>, <17,
1740
-979424087926189051470349 + O(17^20)>, <19,
1741
-16905965663076553616324627 + O(19^20)>, <29,
1742
72071173330516195538881320295 + O(29^20)> *]
1743
> x := $1;
1744
> x;
1745
[* <5, 1106466121701 + O(5^18)>, <7, 34330260522044597 +
1746
O(7^20)>, <11, 1664557466859596710369 + O(11^21)>, <13,
1747
103609576221026126766923 + O(13^21)>, <17,
1748
-979424087926189051470349 + O(17^20)>, <19,
1749
-16905965663076553616324627 + O(19^20)>, <29,
1750
72071173330516195538881320295 + O(29^20)> *]
1751
> A := [* *];
1752
> time for p in G do if p gt 3 then r := regulator(E,p); Append(~A,<p,r,Valuation(r)>); end if; end for;
1753
for>
1754
Computing (y^Frobenius)^(-1)
1755
Expansion time: 0.03
1756
Reducing differentials modulo cohomology relations.
1757
Computing (y^Frobenius)^(-1)
1758
Expansion time: 0.039
1759
Reducing differentials modulo cohomology relations.
1760
Computing (y^Frobenius)^(-1)
1761
Expansion time: 0.079
1762
Reducing differentials modulo cohomology relations.
1763
Computing (y^Frobenius)^(-1)
1764
Expansion time: 0.12
1765
Reducing differentials modulo cohomology relations.
1766
Computing (y^Frobenius)^(-1)
1767
Expansion time: 0.14
1768
Reducing differentials modulo cohomology relations.
1769
Computing (y^Frobenius)^(-1)
1770
Expansion time: 0.231
1771
Reducing differentials modulo cohomology relations.
1772
Computing (y^Frobenius)^(-1)
1773
Expansion time: 0.36
1774
Reducing differentials modulo cohomology relations.
1775
Computing (y^Frobenius)^(-1)
1776
Expansion time: 0.409
1777
Reducing differentials modulo cohomology relations.
1778
Computing (y^Frobenius)^(-1)
1779
Expansion time: 0.681
1780
Reducing differentials modulo cohomology relations.
1781
Computing (y^Frobenius)^(-1)
1782
Expansion time: 0.72
1783
Reducing differentials modulo cohomology relations.
1784
Computing (y^Frobenius)^(-1)
1785
Expansion time: 0.689
1786
Reducing differentials modulo cohomology relations.
1787
Computing (y^Frobenius)^(-1)
1788
Expansion time: 0.979
1789
Reducing differentials modulo cohomology relations.
1790
Computing (y^Frobenius)^(-1)
1791
Expansion time: 1.16
1792
Reducing differentials modulo cohomology relations.
1793
Computing (y^Frobenius)^(-1)
1794
Expansion time: 1.26
1795
Reducing differentials modulo cohomology relations.
1796
Computing (y^Frobenius)^(-1)
1797
Expansion time: 1.329
1798
Reducing differentials modulo cohomology relations.
1799
Computing (y^Frobenius)^(-1)
1800
Expansion time: 1.4
1801
Reducing differentials modulo cohomology relations.
1802
Computing (y^Frobenius)^(-1)
1803
Expansion time: 1.82
1804
Reducing differentials modulo cohomology relations.
1805
Computing (y^Frobenius)^(-1)
1806
Expansion time: 1.829
1807
Reducing differentials modulo cohomology relations.
1808
Computing (y^Frobenius)^(-1)
1809
Expansion time: 1.891
1810
Reducing differentials modulo cohomology relations.
1811
Computing (y^Frobenius)^(-1)
1812
Expansion time: 2.029
1813
Reducing differentials modulo cohomology relations.
1814
Computing (y^Frobenius)^(-1)
1815
Expansion time: 2.93
1816
Reducing differentials modulo cohomology relations.
1817
> A;
1818
[* <5, 1106466121701 + O(5^18), 0>, <7, 34330260522044597 +
1819
O(7^20), 0>, <11, 1664557466859596710369 + O(11^21), 0>, <13,
1820
103609576221026126766923 + O(13^21), 0>, <17,
1821
-979424087926189051470349 + O(17^20), 0>, <19,
1822
-16905965663076553616324627 + O(19^20), 0>, <29,
1823
72071173330516195538881320295 + O(29^20), 0>, <31,
1824
-173944231991525079009482582188 + O(31^20), 0>, <37,
1825
10390854614217075712273759793878 + O(37^20), 0>, <41,
1826
64521799079830548248787468756513 + O(41^20), 0>, <43,
1827
-179327636570955691741962748603937 + O(43^20), 0>, <47,
1828
-271587216348857870741024380860904 + O(47^20), 0>, <53,
1829
-12162564753722638963498226479559914 + O(53^20), 0>, <59,
1830
53186567475602244093889022258956772 + O(59^20), 0>, <61,
1831
3076131498368429672977101883433*61^-2 + O(61^16), -2>, <67,
1832
167936089130563072609672366369770203 + O(67^20), 0>, <71,
1833
2645825244440902073298471121935090827 + O(71^20), 0>, <73,
1834
-895075117177840654650337174738339688 + O(73^20), 0>, <79,
1835
8896576480895465746291587395824330359 + O(79^20), 0>, <89,
1836
-463071254062057122311079391324128760494 + O(89^20), 0>, <97,
1837
269501708627173241996085597295066750*97^-2 + O(97^16), -2> *]
1838
> regulator(E,61,40);
1839
Computing (y^Frobenius)^(-1)
1840
Expansion time: 5.349
1841
Reducing differentials modulo cohomology relations.
1842
-370438686758431742575952482434711859118701081729752071017439432\
1843
6910*61^-2 + O(61^36)
1844
[-93680825643725386032887135610285079821362956426006239567705919\
1845
220542497*61^-2 + O(61^38) -15768916285877803870082359007005\
1846
398879798330586794869806672872686060419*61^-2 + O(61^38)]
1847
[-15768916285877803870082359007005398879798330586794869806672872\
1848
686060419*61^-2 + O(61^38) -46585432491123892099258257177305\
1849
075066710823866005642332567045978541820*61^-2 + O(61^38)]
1850
> h := height_function(E,61,40);
1851
Computing (y^Frobenius)^(-1)
1852
Expansion time: 5.37
1853
Reducing differentials modulo cohomology relations.
1854
>P;
1855
(-1 : 1 : 1)
1856
> Q;
1857
(1 : -3 : 1)
1858
> h(P);
1859
7235590364451759412233487342066596711624481343161361427996724504\
1860
8924769*61^-2 + O(61^38)
1861
> h(Q);
1862
-673922145771191356308640328270344812001327531147182728021623000\
1863
08957039*61^-2 + O(61^38)
1864
> h(2*P);
1865
-426498439984155838211045243792382254102362859887852505754773483\
1866
43235456*61^-2 + O(61^38)
1867
> 4*h(P);
1868
3048703885235699461273383751997784765406621448076984849992269174\
1869
3608675*61^-2 + O(61^38)
1870
> 4*h(P) - h(2*P);
1871
1051*61^36 + O(61^38)
1872
> h(P);
1873
7235590364451759412233487342066596711624481343161361427996724504\
1874
8924769*61^-2 + O(61^38)
1875
> h(Q);
1876
-673922145771191356308640328270344812001327531147182728021623000\
1877
08957039*61^-2 + O(61^38)
1878
> Coefficient(qEigenform(E,65),61);
1879
1
1880
> Coefficient(qEigenform(E,100),97);
1881
1
1882
> [Coefficient(qEigenform(E,100),p) : p in G];
1883
[ -2, -3, -2, -6, -1, -6, -2, -9, -4, -2, -9, 4, 6, -3, 6, 1, -8,
1884
6, 11, -8, 9, 1 ]
1885
> E := EC("37A");
1886
> G := [p : p in PrimeSeq(2,100) | IsGoodOrdinary(E,p)];
1887
> G;
1888
[ 2, 3, 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71,
1889
73, 79, 83, 89, 97 ]
1890
> G := [p : p in PrimeSeq(2,100) | IsGoodOrdinary(E,p) and p ge 5];
1891
> G;
1892
[ 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73,
1893
79, 83, 89, 97 ]
1894
> A := [* *];
1895
> time for p in G do if p gt 3 then r := regulator(E,p); Append(~A,<p,r,Valuation(r)>); end if; end for;
1896
Computing (y^Frobenius)^(-1)
1897
Expansion time: 0.03
1898
Reducing differentials modulo cohomology relations.
1899
Computing (y^Frobenius)^(-1)
1900
Expansion time: 0.03
1901
Reducing differentials modulo cohomology relations.
1902
Computing (y^Frobenius)^(-1)
1903
Expansion time: 0.079
1904
Reducing differentials modulo cohomology relations.
1905
Computing (y^Frobenius)^(-1)
1906
Expansion time: 0.111
1907
Reducing differentials modulo cohomology relations.
1908
Computing (y^Frobenius)^(-1)
1909
Expansion time: 0.27
1910
Reducing differentials modulo cohomology relations.
1911
Computing (y^Frobenius)^(-1)
1912
Expansion time: 0.359
1913
Reducing differentials modulo cohomology relations.
1914
Computing (y^Frobenius)^(-1)
1915
Expansion time: 0.419
1916
Reducing differentials modulo cohomology relations.
1917
Computing (y^Frobenius)^(-1)
1918
Expansion time: 0.699
1919
Reducing differentials modulo cohomology relations.
1920
Computing (y^Frobenius)^(-1)
1921
Expansion time: 0.671
1922
Reducing differentials modulo cohomology relations.
1923
Computing (y^Frobenius)^(-1)
1924
Expansion time: 0.969
1925
Reducing differentials modulo cohomology relations.
1926
Computing (y^Frobenius)^(-1)
1927
Expansion time: 1.15
1928
Reducing differentials modulo cohomology relations.
1929
Computing (y^Frobenius)^(-1)
1930
Expansion time: 1.239
1931
Reducing differentials modulo cohomology relations.
1932
Computing (y^Frobenius)^(-1)
1933
Expansion time: 1.31
1934
Reducing differentials modulo cohomology relations.
1935
Computing (y^Frobenius)^(-1)
1936
Expansion time: 1.371
1937
Reducing differentials modulo cohomology relations.
1938
Computing (y^Frobenius)^(-1)
1939
Expansion time: 1.799
1940
Reducing differentials modulo cohomology relations.
1941
Computing (y^Frobenius)^(-1)
1942
Expansion time: 1.839
1943
Reducing differentials modulo cohomology relations.
1944
Computing (y^Frobenius)^(-1)
1945
Expansion time: 1.901
1946
Reducing differentials modulo cohomology relations.
1947
Computing (y^Frobenius)^(-1)
1948
Expansion time: 1.93
1949
Reducing differentials modulo cohomology relations.
1950
Computing (y^Frobenius)^(-1)
1951
Expansion time: 2.039
1952
Reducing differentials modulo cohomology relations.
1953
Computing (y^Frobenius)^(-1)
1954
Expansion time: 2.909
1955
Reducing differentials modulo cohomology relations.
1956
Time: 35.600
1957
> A;
1958
[* <5, 138360994885922 + O(5^21), 0>, <7, 4709403600911866 +
1959
O(7^20), 0>, <11, 3502722142035391199047 + O(11^21), 0>, <13,
1960
3328786448953657679826*13 + O(13^21), 1>, <23,
1961
-18507705578301047964577596 + O(23^20), 0>, <29,
1962
16742788144500360438362532376 + O(29^20), 0>, <31,
1963
214585630396079048220257036022 + O(31^20), 0>, <41,
1964
62872643243806364912447375677980 + O(41^20), 0>, <43,
1965
98201982475772371322430732508192 + O(43^20), 0>, <47,
1966
-950963360517191754231629728419340 + O(47^20), 0>, <53,
1967
3013481933339232802567988878307847*53^-2 + O(53^18), -2>, <59,
1968
-36750556362766803167564839340913550 + O(59^20), 0>, <61,
1969
139607664981793518838513655007581617 + O(61^20), 0>, <67,
1970
-12619683101217607828398831962484117*67 + O(67^20), 1>, <71,
1971
2376688066444732847472979255838549466 + O(71^20), 0>, <73,
1972
-4793861347633889739942985981494036464 + O(73^20), 0>, <79,
1973
-39751046706757915792433195709308099930 + O(79^20), 0>, <83,
1974
-118006589717737960880381609746899564294 + O(83^20), 0>, <89,
1975
27000563061297979788158574760212991695 + O(89^20), 0>, <97,
1976
2257574446473128489077697545138873920970 + O(97^20), 0> *]
1977
> SetColumns(100);
1978
> A;
1979
[* <5, 138360994885922 + O(5^21), 0>, <7, 4709403600911866 + O(7^20), 0>, <11,
1980
3502722142035391199047 + O(11^21), 0>, <13, 3328786448953657679826*13 + O(13^21), 1>, <23,
1981
-18507705578301047964577596 + O(23^20), 0>, <29, 16742788144500360438362532376 + O(29^20), 0>, <31,
1982
214585630396079048220257036022 + O(31^20), 0>, <41, 62872643243806364912447375677980 + O(41^20), 0>,
1983
<43, 98201982475772371322430732508192 + O(43^20), 0>, <47, -950963360517191754231629728419340 +
1984
O(47^20), 0>, <53, 3013481933339232802567988878307847*53^-2 + O(53^18), -2>, <59,
1985
-36750556362766803167564839340913550 + O(59^20), 0>, <61, 139607664981793518838513655007581617 +
1986
O(61^20), 0>, <67, -12619683101217607828398831962484117*67 + O(67^20), 1>, <71,
1987
2376688066444732847472979255838549466 + O(71^20), 0>, <73, -4793861347633889739942985981494036464 +
1988
O(73^20), 0>, <79, -39751046706757915792433195709308099930 + O(79^20), 0>, <83,
1989
-118006589717737960880381609746899564294 + O(83^20), 0>, <89, 27000563061297979788158574760212991695
1990
+ O(89^20), 0>, <97, 2257574446473128489077697545138873920970 + O(97^20), 0> *]
1991
> for x in A do print x; end for;
1992
> for x in A do print x; end for;
1993
<5, 138360994885922 + O(5^21), 0>
1994
<7, 4709403600911866 + O(7^20), 0>
1995
<11, 3502722142035391199047 + O(11^21), 0>
1996
<13, 3328786448953657679826*13 + O(13^21), 1>
1997
<23, -18507705578301047964577596 + O(23^20), 0>
1998
<29, 16742788144500360438362532376 + O(29^20), 0>
1999
<31, 214585630396079048220257036022 + O(31^20), 0>
2000
<41, 62872643243806364912447375677980 + O(41^20),
2001
0>
2002
<43, 98201982475772371322430732508192 + O(43^20),
2003
0>
2004
<47, -950963360517191754231629728419340 +
2005
O(47^20), 0>
2006
<53, 3013481933339232802567988878307847*53^-2 +
2007
O(53^18), -2>
2008
<59, -36750556362766803167564839340913550 +
2009
O(59^20), 0>
2010
<61, 139607664981793518838513655007581617 +
2011
O(61^20), 0>
2012
<67, -12619683101217607828398831962484117*67 +
2013
O(67^20), 1>
2014
<71, 2376688066444732847472979255838549466 +
2015
O(71^20), 0>
2016
<73, -4793861347633889739942985981494036464 +
2017
O(73^20), 0>
2018
<79, -39751046706757915792433195709308099930 +
2019
O(79^20), 0>
2020
<83, -118006589717737960880381609746899564294 +
2021
O(83^20), 0>
2022
<89, 27000563061297979788158574760212991695 +
2023
O(89^20), 0>
2024
<97, 2257574446473128489077697545138873920970 +
2025
O(97^20), 0>
2026
> SetColumns(0);
2027
> for x in A do print x; end for;
2028
<5, 138360994885922 + O(5^21), 0>
2029
<7, 4709403600911866 + O(7^20), 0>
2030
<11, 3502722142035391199047 + O(11^21), 0>
2031
<13, 3328786448953657679826*13 + O(13^21), 1>
2032
<23, -18507705578301047964577596 + O(23^20), 0>
2033
<29, 16742788144500360438362532376 + O(29^20), 0>
2034
<31, 214585630396079048220257036022 + O(31^20), 0>
2035
<41, 62872643243806364912447375677980 + O(41^20), 0>
2036
<43, 98201982475772371322430732508192 + O(43^20), 0>
2037
<47, -950963360517191754231629728419340 + O(47^20), 0>
2038
<53, 3013481933339232802567988878307847*53^-2 + O(53^18), -2>
2039
<59, -36750556362766803167564839340913550 + O(59^20), 0>
2040
<61, 139607664981793518838513655007581617 + O(61^20), 0>
2041
<67, -12619683101217607828398831962484117*67 + O(67^20), 1>
2042
<71, 2376688066444732847472979255838549466 + O(71^20), 0>
2043
<73, -4793861347633889739942985981494036464 + O(73^20), 0>
2044
<79, -39751046706757915792433195709308099930 + O(79^20), 0>
2045
<83, -118006589717737960880381609746899564294 + O(83^20), 0>
2046
<89, 27000563061297979788158574760212991695 + O(89^20), 0>
2047
<97, 2257574446473128489077697545138873920970 + O(97^20), 0>
2048
> Coefficient(qEigenform(E,54),53);
2049
1
2050
> [Coefficient(qEigenform(E,100),p) : p in G];
2051
[ -2, -1, -5, -2, 2, 6, -4, -9, 2, -9, 1, 8, -8, 8, 9, -1, 4, -15, 4, 4 ]
2052
> Coefficient(qEigenform(E,68),67);
2053
8
2054
> E := EC("389A");
2055
> A := [* *];
2056
> G := [p : p in PrimeSeq(2,100) | IsGoodOrdinary(E,p) and p ge 5];
2057
> time for p in G do if p gt 3 then r := regulator(E,p); Append(~A,<p,r,Valuation(r)>); end if; end for;
2058
Computing (y^Frobenius)^(-1)
2059
Expansion time: 0.03
2060
Reducing differentials modulo cohomology relations.
2061
Computing (y^Frobenius)^(-1)
2062
Expansion time: 0.039
2063
Reducing differentials modulo cohomology relations.
2064
Computing (y^Frobenius)^(-1)
2065
Expansion time: 0.09
2066
Reducing differentials modulo cohomology relations.
2067
Computing (y^Frobenius)^(-1)
2068
Expansion time: 0.099
2069
Reducing differentials modulo cohomology relations.
2070
Computing (y^Frobenius)^(-1)
2071
Expansion time: 0.14
2072
Reducing differentials modulo cohomology relations.
2073
Computing (y^Frobenius)^(-1)
2074
Expansion time: 0.21
2075
Reducing differentials modulo cohomology relations.
2076
Computing (y^Frobenius)^(-1)
2077
Expansion time: 0.259
2078
Reducing differentials modulo cohomology relations.
2079
Computing (y^Frobenius)^(-1)
2080
Expansion time: 0.37
2081
Reducing differentials modulo cohomology relations.
2082
Computing (y^Frobenius)^(-1)
2083
Expansion time: 0.409
2084
Reducing differentials modulo cohomology relations.
2085
Computing (y^Frobenius)^(-1)
2086
Expansion time: 0.679
2087
Reducing differentials modulo cohomology relations.
2088
Computing (y^Frobenius)^(-1)
2089
Expansion time: 0.701
2090
Reducing differentials modulo cohomology relations.
2091
Computing (y^Frobenius)^(-1)
2092
Expansion time: 0.689
2093
Reducing differentials modulo cohomology relations.
2094
Computing (y^Frobenius)^(-1)
2095
Expansion time: 0.981
2096
Reducing differentials modulo cohomology relations.
2097
Computing (y^Frobenius)^(-1)
2098
Expansion time: 1.169
2099
Reducing differentials modulo cohomology relations.
2100
Computing (y^Frobenius)^(-1)
2101
Expansion time: 1.25
2102
Reducing differentials modulo cohomology relations.
2103
Computing (y^Frobenius)^(-1)
2104
Expansion time: 1.319
2105
Reducing differentials modulo cohomology relations.
2106
Computing (y^Frobenius)^(-1)
2107
Expansion time: 1.389
2108
Reducing differentials modulo cohomology relations.
2109
Computing (y^Frobenius)^(-1)
2110
Expansion time: 1.819
2111
Reducing differentials modulo cohomology relations.
2112
Computing (y^Frobenius)^(-1)
2113
Expansion time: 1.869
2114
Reducing differentials modulo cohomology relations.
2115
Computing (y^Frobenius)^(-1)
2116
Expansion time: 1.889
2117
Reducing differentials modulo cohomology relations.
2118
Computing (y^Frobenius)^(-1)
2119
Expansion time: 1.961
2120
Reducing differentials modulo cohomology relations.
2121
Computing (y^Frobenius)^(-1)
2122
Expansion time: 2.05
2123
Reducing differentials modulo cohomology relations.
2124
Computing (y^Frobenius)^(-1)
2125
Expansion time: 2.971
2126
Reducing differentials modulo cohomology relations.
2127
Time: 40.050
2128
> for x in A do print x; end for;
2129
<5, 216689873081859 + O(5^21), 0>
2130
<7, 22975764581280320 + O(7^20), 0>
2131
<11, -2680231549475987377782 + O(11^21), 0>
2132
<13, 63403295801973034516257 + O(13^21), 0>
2133
<17, -1645488482806115361636866 + O(17^20), 0>
2134
<19, -7361284703823783446020710 + O(19^20), 0>
2135
<23, 4106360481404920381961161 + O(23^20), 0>
2136
<29, 17964006617098479194918517716 + O(29^20), 0>
2137
<31, 237769376451635960529874242927 + O(31^20), 0>
2138
<37, -6222330979469162993492786173376 + O(37^20), 0>
2139
<41, 68919996614654231147559338062161 + O(41^20), 0>
2140
<43, -4388984629363251702793918481448 + O(43^20), 0>
2141
<47, 96545625489717893697514257058316 + O(47^20), 0>
2142
<53, 8860077704993049937775387646524621 + O(53^20), 0>
2143
<59, -127728440963158589575687903800085513 + O(59^20), 0>
2144
<61, 205526659703992435223306981822800091 + O(61^20), 0>
2145
<67, -265706304644454114655343127828111580 + O(67^20), 0>
2146
<71, -836510465695579033880382317842141217 + O(71^20), 0>
2147
<73, 29596118639212386334923490945319667 + O(73^20), 0>
2148
<79, -4462739915965556249266498025090944298 + O(79^20), 0>
2149
<83, -54671538990404116030547810106443627720 + O(83^20), 0>
2150
<89, -167549488405712967064976645166530686723 + O(89^20), 0>
2151
<97, -2211820102333727027675650895720665441402 + O(97^20), 0>
2152
> E := EC("1058C");
2153
> G := [p : p in PrimeSeq(101,500) | IsGoodOrdinary(E,p) and p ge 5];
2154
> A := [* *];
2155
> for p in G do if p gt 3 then r := regulator(E,p); print p, r, Valuation(r); Append(~A,<p,r,Valuation(r)>); end if; end for;
2156
Computing (y^Frobenius)^(-1)
2157
Expansion time: 3.35
2158
Reducing differentials modulo cohomology relations.
2159
101 476480469707036332646161303910245319379 + O(101^20) 0
2160
Computing (y^Frobenius)^(-1)
2161
Expansion time: 3.64
2162
Reducing differentials modulo cohomology relations.
2163
103 6401358360394788081773821530465566398861 + O(103^20) 0
2164
Computing (y^Frobenius)^(-1)
2165
Expansion time: 3.73
2166
Reducing differentials modulo cohomology relations.
2167
107 4922767642974165115674022301621135949010 + O(107^20) 0
2168
Computing (y^Frobenius)^(-1)
2169
Expansion time: 3.719
2170
Reducing differentials modulo cohomology relations.
2171
109 -2448717765918646375218544243833474256546 + O(109^20) 0
2172
Computing (y^Frobenius)^(-1)
2173
Expansion time: 3.779
2174
Reducing differentials modulo cohomology relations.
2175
113 -5991211946727979721541061121026578552436 + O(113^20) 0
2176
Computing (y^Frobenius)^(-1)
2177
Expansion time: 4.069
2178
Reducing differentials modulo cohomology relations.
2179
127 -266954596482088375341789280269115541482483 + O(127^20) 0
2180
Computing (y^Frobenius)^(-1)
2181
Expansion time: 4.12
2182
Reducing differentials modulo cohomology relations.
2183
131 885928978952035516753483804506698950208963 + O(131^20) 0
2184
Computing (y^Frobenius)^(-1)
2185
Expansion time: 5.2
2186
Reducing differentials modulo cohomology relations.
2187
137 862355935221473255587603984788742846836475 + O(137^20) 0
2188
Computing (y^Frobenius)^(-1)
2189
Expansion time: 5.219
2190
Reducing differentials modulo cohomology relations.
2191
139 893498523927167174829901480037367363781685 + O(139^20) 0
2192
Computing (y^Frobenius)^(-1)
2193
Expansion time: 5.36
2194
Reducing differentials modulo cohomology relations.
2195
149 -1963835319577170752807743270103875287090958 + O(149^20) 0
2196
Computing (y^Frobenius)^(-1)
2197
Expansion time: 5.23
2198
Reducing differentials modulo cohomology relations.
2199
151 16455424611592423659744030523534896659535280 + O(151^20) 0
2200
Computing (y^Frobenius)^(-1)
2201
Expansion time: 5.519
2202
Reducing differentials modulo cohomology relations.
2203
157 1304985225261657442384029871097880458103944 + O(157^20) 0
2204
Computing (y^Frobenius)^(-1)
2205
Expansion time: 5.581
2206
Reducing differentials modulo cohomology relations.
2207
163 -74941293240715728290524282127729257855982741 + O(163^20) 0
2208
Computing (y^Frobenius)^(-1)
2209
Expansion time: 5.79
2210
Reducing differentials modulo cohomology relations.
2211
173 -31742200745061737323031668621113028647258865 + O(173^20) 0
2212
Computing (y^Frobenius)^(-1)
2213
Expansion time: 5.82
2214
Reducing differentials modulo cohomology relations.
2215
181 -301059950417872607277161663862120583299702134 + O(181^20) 0
2216
Computing (y^Frobenius)^(-1)
2217
Expansion time: 7.12
2218
Reducing differentials modulo cohomology relations.
2219
191 -1455648087029648260445874502808106259621227117 + O(191^20) 0
2220
Computing (y^Frobenius)^(-1)
2221
Expansion time: 7.12
2222
Reducing differentials modulo cohomology relations.
2223
193 -1629590413978391579327532226453194166132346094 + O(193^20) 0
2224
Computing (y^Frobenius)^(-1)
2225
Expansion time: 7.18
2226
Reducing differentials modulo cohomology relations.
2227
197 1077468644282969836512862412801672173327406615 + O(197^20) 0
2228
Computing (y^Frobenius)^(-1)
2229
Expansion time: 7.24
2230
Reducing differentials modulo cohomology relations.
2231
199 1886144760299789908083666104019078586322981159 + O(199^20) 0
2232
Computing (y^Frobenius)^(-1)
2233
Expansion time: 9.409
2234
Reducing differentials modulo cohomology relations.
2235
211 -6094803705500786294536521433382052159437991849 + O(211^20) 0
2236
Computing (y^Frobenius)^(-1)
2237
Expansion time: 9.561
2238
Reducing differentials modulo cohomology relations.
2239
223 37415158734533778175044858726928248342273569392 + O(223^20) 0
2240
Computing (y^Frobenius)^(-1)
2241
Expansion time: 9.649
2242
Reducing differentials modulo cohomology relations.
2243
227 19606670814333954015853846026123361999186774801 + O(227^20) 0
2244
Computing (y^Frobenius)^(-1)
2245
Expansion time: 9.789
2246
Reducing differentials modulo cohomology relations.
2247
229 -64214558386218506627020154712722573995815576492 + O(229^20) 0
2248
Computing (y^Frobenius)^(-1)
2249
Expansion time: 9.719
2250
Reducing differentials modulo cohomology relations.
2251
233 77248690651481504297812725415750130294760334531 + O(233^20) 0
2252
Computing (y^Frobenius)^(-1)
2253
Expansion time: 10
2254
Reducing differentials modulo cohomology relations.
2255
241 3252317888874066245912857299289743895693945*241^-2 + O(241^16) -2
2256
Computing (y^Frobenius)^(-1)
2257
Expansion time: 10.189
2258
Reducing differentials modulo cohomology relations.
2259
251 173581002964881140281151080833285796828100400284 + O(251^20) 0
2260
Computing (y^Frobenius)^(-1)
2261
Expansion time: 10.529
2262
Reducing differentials modulo cohomology relations.
2263
257 -426905762455721247734906601556431432396879136369 + O(257^20) 0
2264
Computing (y^Frobenius)^(-1)
2265
Expansion time: 10.461
2266
Reducing differentials modulo cohomology relations.
2267
263 -233733854342114140446544882182634535598861917469 + O(263^20) 0
2268
Computing (y^Frobenius)^(-1)
2269
Expansion time: 10.59
2270
Reducing differentials modulo cohomology relations.
2271
269 1177971198454019204373622054275629621785669528394 + O(269^20) 0
2272
Computing (y^Frobenius)^(-1)
2273
Expansion time: 10.52
2274
Reducing differentials modulo cohomology relations.
2275
271 -1357771339230472431076104469179938359011825717340 + O(271^20) 0
2276
Computing (y^Frobenius)^(-1)
2277
Expansion time: 13.23
2278
Reducing differentials modulo cohomology relations.
2279
277 1902113850890970472471844883013311805608608538071 + O(277^20) 0
2280
Computing (y^Frobenius)^(-1)
2281
Expansion time: 13.23
2282
Reducing differentials modulo cohomology relations.
2283
281 -771426215266877657517824927117197317064053922031 + O(281^20) 0
2284
Computing (y^Frobenius)^(-1)
2285
Expansion time: 13.23
2286
Reducing differentials modulo cohomology relations.
2287
283 -3909759751225677755196122951554393508916782635068 + O(283^20) 0
2288
Computing (y^Frobenius)^(-1)
2289
Expansion time: 15
2290
Reducing differentials modulo cohomology relations.
2291
293 9650766525262982951395332384521495636939673842618 + O(293^20) 0
2292
Computing (y^Frobenius)^(-1)
2293
Expansion time: 15.05
2294
Reducing differentials modulo cohomology relations.
2295
307 16215076042046210153447355333778959437456886944269 + O(307^20) 0
2296
Computing (y^Frobenius)^(-1)
2297
Expansion time: 15.03
2298
Reducing differentials modulo cohomology relations.
2299
311 9924568793992273436656018879412285712710587501738 + O(311^20) 0
2300
Computing (y^Frobenius)^(-1)
2301
Expansion time: 15.321
2302
Reducing differentials modulo cohomology relations.
2303
313 16338997345749621985343710006034679174320839*313^-2 + O(313^16) -2
2304
Computing (y^Frobenius)^(-1)
2305
Expansion time: 15.141
2306
Reducing differentials modulo cohomology relations.
2307
317 -8180959439659097002455664090975829418728238319179 + O(317^20) 0
2308
Computing (y^Frobenius)^(-1)
2309
Expansion time: 15.5
2310
Reducing differentials modulo cohomology relations.
2311
331 106324002766280001349902268973298443532764328404373 + O(331^20) 0
2312
Computing (y^Frobenius)^(-1)
2313
Expansion time: 15.559
2314
Reducing differentials modulo cohomology relations.
2315
337 1487641797148146783174749484529981177684649280*337^-2 + O(337^16) -2
2316
Computing (y^Frobenius)^(-1)
2317
Expansion time: 15.01
2318
Reducing differentials modulo cohomology relations.
2319
347 288881056768437771513194445718327926423466873535164 + O(347^20) 0
2320
Computing (y^Frobenius)^(-1)
2321
Expansion time: 14.851
2322
Reducing differentials modulo cohomology relations.
2323
349 -324627174783822167409446159532805108501548546483875 + O(349^20) 0
2324
Computing (y^Frobenius)^(-1)
2325
Expansion time: 15.359
2326
Reducing differentials modulo cohomology relations.
2327
353 -150041228084538689303273049361188586637161356864518 + O(353^20) 0
2328
Computing (y^Frobenius)^(-1)
2329
Expansion time: 15.1
2330
Reducing differentials modulo cohomology relations.
2331
359 508244171233401628697951496404505641462642110066758 + O(359^20) 0
2332
Computing (y^Frobenius)^(-1)
2333
Expansion time: 15.769
2334
Reducing differentials modulo cohomology relations.
2335
367 -416527758588057821093991552213449352177065535454474 + O(367^20) 0
2336
Computing (y^Frobenius)^(-1)
2337
Expansion time: 19.52
2338
Reducing differentials modulo cohomology relations.
2339
373 -1185597183209223899081873937523134791093301772229805 + O(373^20) 0
2340
Computing (y^Frobenius)^(-1)
2341
Expansion time: 19.569
2342
Reducing differentials modulo cohomology relations.
2343
379 642664726719383534578067014671603665079200147802845 + O(379^20) 0
2344
Computing (y^Frobenius)^(-1)
2345
Expansion time: 19.611
2346
Reducing differentials modulo cohomology relations.
2347
383 -1980679639877875794142899451152825399982395507331373 + O(383^20) 0
2348
Computing (y^Frobenius)^(-1)
2349
Expansion time: 19.711
2350
Reducing differentials modulo cohomology relations.
2351
389 912910025622472388115214247783732215543517477434958 + O(389^20) 0
2352
Computing (y^Frobenius)^(-1)
2353
Expansion time: 19.819
2354
Reducing differentials modulo cohomology relations.
2355
397 -2289985440897127570150400503318597019362572185376739 + O(397^20) 0
2356
Computing (y^Frobenius)^(-1)
2357
Expansion time: 19.92
2358
Reducing differentials modulo cohomology relations.
2359
401 2662460691462671991566057040285291569696404461033180 + O(401^20) 0
2360
Computing (y^Frobenius)^(-1)
2361
Expansion time: 22.52
2362
Reducing differentials modulo cohomology relations.
2363
409 -6035761433262382557092768771901984647658378844632414 + O(409^20) 0
2364
Computing (y^Frobenius)^(-1)
2365
Expansion time: 24.969
2366
Reducing differentials modulo cohomology relations.
2367
419 6966665477845050730634394930098503632468159092330232 + O(419^20) 0
2368
Computing (y^Frobenius)^(-1)
2369
Expansion time: 24.98
2370
Reducing differentials modulo cohomology relations.
2371
421 -8722498142436839434340378205352120473632018692238306 + O(421^20) 0
2372
Computing (y^Frobenius)^(-1)
2373
Expansion time: 25.289
2374
Reducing differentials modulo cohomology relations.
2375
431 -23154404799097883257764550108478971805586495350456898 + O(431^20) 0
2376
Computing (y^Frobenius)^(-1)
2377
Expansion time: 25.36
2378
Reducing differentials modulo cohomology relations.
2379
433 100101112887074728257664136622452394504319579883*433^-2 + O(433^16) -2
2380
Computing (y^Frobenius)^(-1)
2381
Expansion time: 25.449
2382
Reducing differentials modulo cohomology relations.
2383
439 27100444100108260647618106076004738147820891403030283 + O(439^20) 0
2384
Computing (y^Frobenius)^(-1)
2385
Expansion time: 25.471
2386
Reducing differentials modulo cohomology relations.
2387
443 6219765964400288347185296345919434522982680131332921 + O(443^20) 0
2388
Computing (y^Frobenius)^(-1)
2389
Expansion time: 25.609
2390
Reducing differentials modulo cohomology relations.
2391
449 2367703250647633532500453909151780788048681517906996 + O(449^20) 0
2392
Computing (y^Frobenius)^(-1)
2393
Expansion time: 25.731
2394
Reducing differentials modulo cohomology relations.
2395
457 -48835807557429023353520353913148898222094238891889268 + O(457^20) 0
2396
Computing (y^Frobenius)^(-1)
2397
Expansion time: 25.8
2398
Reducing differentials modulo cohomology relations.
2399
461 -26453166166427471420028949259756585259124118028066374 + O(461^20) 0
2400
Computing (y^Frobenius)^(-1)
2401
Expansion time: 25.879
2402
Reducing differentials modulo cohomology relations.
2403
463 57576177554821815956381222050012596877468570136176904 + O(463^20) 0
2404
Computing (y^Frobenius)^(-1)
2405
Expansion time: 31.471
2406
Reducing differentials modulo cohomology relations.
2407
467 113311978683408496093899189133874063724678183358071625 + O(467^20) 0
2408
Computing (y^Frobenius)^(-1)
2409
Expansion time: 31.721
2410
Reducing differentials modulo cohomology relations.
2411
487 5978459439427873076881614773786500598115284065894479 + O(487^20) 0
2412
Computing (y^Frobenius)^(-1)
2413
Expansion time: 31.929
2414
Reducing differentials modulo cohomology relations.
2415
499 -337366512483466327430173577004337987723710229937060602 + O(499^20) 0
2416
>