CoCalc Public Files
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1Total time: 213.469 seconds, Total memory usage: 5.24MB
2[email protected]:~/papers/padic_cyclotomic_height/e2heights$me 3Magma V2.11-1 Wed Sep 29 2004 23:46:36 on form [Seed = 135448808] 4Type ? for help. Type <Ctrl>-D to quit. 5 6Loading startup file "/home/was/magma/local/emacs.m" 7 8Loading "/home/was/magma/local/init.m" 9> E := EC("37A"); 10> prec := 30; sigma30 := sigma_alg3(E, p, prec : e2prec := prec); 11 12>> prec := 30; sigma30 := sigma_alg3(E, p, prec : e2prec := p 13 ^ 14User error: Identifier 'p' has not been declared or assigned 15> p := 5; 16> prec := 30; 17> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2; 18 19>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2; 20 ^ 21User error: Identifier 'E2' has not been declared or assigned 22 23>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2; 24 ^ 25User error: Identifier 'E2' has not been declared or assigned 26 27>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2; 28 ^ 29User error: Identifier 'f1' has not been declared or assigned 30> Attach("kedlaya.m"); 31> Attach("formal.m"); 32> >> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2; 33Computing (y^Frobenius)^(-1) 34Expansion time: 0.041 35Reducing differentials modulo cohomology relations. 36Computing (y^Frobenius)^(-1) 37Expansion time: 0.039 38Reducing differentials modulo cohomology relations. 39 40>> a:=25; b:=30; e1,f1:=E2(E,p,a); e2,f2:=E2(E,p,b); f1 - f2 41 ^ 42Runtime error in '-': Arguments are not compatible 43Argument types given: AlgMatElt, AlgMatElt 44> e1; 45-2767125139749567853 46> e2; 476650601347913228947772 48> f1; 49[2657944961933597168*5 + O(5^28) 437853581402776789*5 + O(5^28)] 50[-1600106773161651063 + O(5^27) 1611436384179670408 + O(5^27)] 51> f2; 52[5747055585190205081543*5 + O(5^33) -8396366479151751520086*5 + 53 O(5^33)] 54[10831544081154084442687 + O(5^32) -5452213560564062517092 + 55 O(5^32)] 56> f1[1,1] - f2[1,1]; 57 58>> f1[1,1] - f2[1,1]; 59 ^ 60Runtime error in '-': Arguments are not compatible 61Argument types given: FldPadElt, FldPadElt 62> Parent(f1); 63Full Matrix Algebra of degree 2 over pAdicField(5, 27) 64> Parent(f2); 65Full Matrix Algebra of degree 2 over pAdicField(5, 32) 66> Trace(f1); 67-2 + O(5^27) 68> Trace(f2); 69-2 + O(5^32) 70> prec := 10; s := sigma(E, p, prec : e2prec := prec); 71e2prec = 10 72Computing (y^Frobenius)^(-1) 73Expansion time: 0.001 74Reducing differentials modulo cohomology relations. 75e2 = -101130353 76K = 5-adic field mod 5^12 77r2 = 0 78d1 = 0 790 80-75421447 + O(5^12) 81> prec := 20; sigma20 := sigma_alg3(E, p, prec : e2prec := prec); 82e2prec = 20 83Computing (y^Frobenius)^(-1) 84Expansion time: 0.041 85Reducing differentials modulo cohomology relations. 86e2 = 914563619572772 87K = 5-adic field mod 5^22 88r2 = 0 89d1 = 0 900 91435471115984803 + O(5^22) 92> sigma - sigma20; 93O(5^12)*t + O(5^12)*t^2 + O(5^12)*t^3 + O(5^12)*t^4 + O(5^11)*t^5 94 + O(5^9)*t^6 + O(5^8)*t^7 + O(5^7)*t^8 + O(5^7)*t^9 + O(t^10) 95> h := height_function(E,5,30); 96e2prec = 30 97Computing (y^Frobenius)^(-1) 98Expansion time: 0.041 99Reducing differentials modulo cohomology relations. 100e2 = 6650601347913228947772 101K = 5-adic field mod 5^32 102r2 = 0 103d1 = 0 1040 105-11364423793197096905822 + O(5^32) 106> G, f := MordellWeilGroup(E); 107> P := f(G.1); 108> h(P); 109-1305176752965909410953*5 + O(5^32) 110> h40 := height_function(E,5,40); 111e2prec = 40 112Computing (y^Frobenius)^(-1) 113Expansion time: 0.079 114Reducing differentials modulo cohomology relations. 115e2 = -38243125061477700243509333478 116K = 5-adic field mod 5^42 117r2 = 0 118d1 = 0 1190 12055249955316579777359543719178 + O(5^42) 121> h40(P); 12228398557172280389803479672*5 + O(5^38) 123> a40 := h40(P); 124> a30 := h(P); 125> a30; 126-1305176752965909410953*5 + O(5^32) 127> a40; 12828398557172280389803479672*5 + O(5^38) 129> Parent(a30)!a40 - a30; 130-22*5^29 + O(5^32) 131> E := EC("1058C"); 132> Order(P); 1330 134> E := EC("1058C"); 135> regulator(E,5, 20); 136 137>> regulator(E,5); 138 ^ 139Runtime error in 'regulator': Bad argument types 140Argument types given: CrvEll[FldRat], RngIntElt 141> regulator(E,5, 20); 142Computing (y^Frobenius)^(-1) 143Expansion time: 0.02 144Reducing differentials modulo cohomology relations. 145 146regulator( 147 E: E, 148 p: 5, 149 prec: 20 150) 151In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\ 152al.m", line 584, column 17: 153>> B := [f(P) : P in G | Order(P) eq 0]; 154 ^ 155Runtime error in for: Iteration is not possible over this object 156> regulator(E,5, 20); 157 158In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\ 159al.m", line 584, column 22: 160>> B := [f(P) : P in Gens(G) | Order(P) eq 0]; 161 ^ 162Runtime error: Undefined reference 'Gens' in package 163"/home/was/papers/padic_cyclotomic_height/e2heights/formal.m" 164> regulator(E,5, 20); 165Computing (y^Frobenius)^(-1) 166Expansion time: 0.02 167Reducing differentials modulo cohomology relations. 168-736185286609*5^2 + O(5^20) 169[158952545746*5 + O(5^19) 54853902771*5^2 + O(5^19)] 170[54853902771*5^2 + O(5^19) 1264328610646*5 + O(5^19)] 171> regulator(E,5, 20); 172Computing (y^Frobenius)^(-1) 173Expansion time: 0.031 174Reducing differentials modulo cohomology relations. 175-736185286609 + O(5^18) 176[158952545746 + O(5^18) 54853902771*5 + O(5^18)] 177[54853902771*5 + O(5^18) 1264328610646 + O(5^18)] 178> E := EC("1058C"); 179> E; 180Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 181Field 182> regulator(E,5, 20); 183Computing (y^Frobenius)^(-1) 184Expansion time: 0.029 185Reducing differentials modulo cohomology relations. 186-736185286609 + O(5^18) 187[158952545746 + O(5^18) 54853902771*5 + O(5^18)] 188[54853902771*5 + O(5^18) 1264328610646 + O(5^18)] 189> P := f(G.1); Q := f(G.2); R := P+Q; 190 191>> P := f(G.1); Q := f(G.2); R := P+Q; 192 ^ 193Runtime error in '.': Argument 2 (2) should be in the range [-1 194.. 1] 195 196>> P := f(G.1); Q := f(G.2); R := P+Q; 197 ^ 198Runtime error in '+': Bad argument types 199Argument types given: PtEll[FldRat], FldRat 200> G, f := MordellWeilGroup(E); 201> >> P := f(G.1); Q := f(G.2); R := P+Q; 202> h(P); 203-321842139675355243103*5 + O(5^32) 204> P;l 205(-1 : 1 : 1) 206> ; 207 208>> P;l 209 ^ 210User error: Identifier 'l' has not been declared or assigned 211> height_of_point(P,5,30); 212Time: 0.760 213s1 = -47683715820312 + O(5^20) 214Time: 0.010 215s2 = 3 + O(5^2) 216Computing (y^Frobenius)^(-1) 217Expansion time: 0.079 218Reducing differentials modulo cohomology relations. 219Time: 0.180 220s2b = -28994174169036 + O(5^20) 221s2 - s2b = -11 + O(5^2) 2225 + O(5^2) 223> height_of_point(P,5,30)/5; 224Time: 0.750 225s1 = -47683715820312 + O(5^20) 226Time: 0.010 227s2 = 3 + O(5^2) 228Computing (y^Frobenius)^(-1) 229Expansion time: 0.07 230Reducing differentials modulo cohomology relations. 231Time: 0.190 232s2b = -28994174169036 + O(5^20) 233s2 - s2b = -11 + O(5^2) 2341 + O(5) 235> prec := 10; s := sigma(E, p, prec : e2prec := prec); 236 237>> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 238 ^ 239Runtime error: Attempting to call something that is not callable 240> Dettach("formal.m"); 241 242>> Dettach("formal.m"); 243 ^ 244User error: Identifier 'Dettach' has not been declared or 245assigned 246> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 247Computing (y^Frobenius)^(-1) 248Expansion time: 0 249Reducing differentials modulo cohomology relations. 250> s; 251t - (122070312 + O(5^12))*t^2 - (47836256 + O(5^12))*t^3 + 252 (22270217*5 + O(5^12))*t^4 + (13762429 + O(5^11))*t^5 - 253 (153097*5 + O(5^9))*t^6 + (915486*5^-1 + O(5^8))*t^7 + 254 (24036*5^-1 + O(5^7))*t^8 + (3956*5 + O(5^7))*t^9 + O(t^10) 255> s1 := sigma_alg1(E,5,50); 256Time: 7.250 257Time: 0.080 258> s - s1; 259O(5^3)*t + O(5^3)*t^2 - (1 + O(5^3))*t^3 + (61 + O(5^3))*t^4 - 260 (48 + O(5^3))*t^5 + (6 + O(5^3))*t^6 + (26*5^-1 + O(5^2))*t^7 261 + (5^-1 + O(5^2))*t^8 + (2 + O(5^2))*t^9 + O(t^10) 262> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 263Computing (y^Frobenius)^(-1) 264Expansion time: 0.01 265Reducing differentials modulo cohomology relations. 266s1 = 25/2 267s2 = 33543952 + O(5^12) 268> E := EC("37A"); 269> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 270Computing (y^Frobenius)^(-1) 271Expansion time: 0 272Reducing differentials modulo cohomology relations. 273s1 = 0 274s2 = -75421447 + O(5^12) 275> s1 := sigma_alg1(E,5,50); 276Time: 5.210 277Time: 0.070 278> s1 - s; 279O(5^3)*t + O(5^3)*t^2 + O(5^3)*t^3 + O(5^3)*t^4 + O(5^3)*t^5 + 280 O(5^3)*t^6 + O(5^2)*t^7 + O(5^3)*t^8 + O(5^2)*t^9 + O(t^10) 281> E; 282Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field 283> E := EC("1058C"); 284Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 285Field 286> E := EC("1058C"); 287> E; 288Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 289Field 290> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 291Computing (y^Frobenius)^(-1) 292Expansion time: 0.01 293Reducing differentials modulo cohomology relations. 294s1 = 25/2 295s2 = 33543952 + O(5^12) 296> s2; 297 298>> s2; 299 ^ 300User error: Identifier 's2' has not been declared or assigned 301> s1; 302(1 + O(5^3))*t + O(5^3)*t^2 + (53 + O(5^3))*t^3 - (62 + 303 O(5^3))*t^4 - (23 + O(5^3))*t^5 + (17 + O(5^3))*t^6 - (6 + 304 O(5^2))*t^7 - (58 + O(5^3))*t^8 - (5 + O(5^2))*t^9 + (11 + 305 O(5^2))*t^10 - (2 + O(5))*t^11 + (11 + O(5^2))*t^12 - (1 + 306 O(5))*t^13 + O(5)*t^14 - (1 + O(5))*t^15 + (1 + O(5))*t^16 + 307 O(1)*t^17 + (2 + O(5))*t^18 + O(1)*t^19 + O(1)*t^20 + 308 O(5^-1)*t^21 + O(1)*t^22 + O(5^-1)*t^23 + O(5^-1)*t^24 + 309 O(5^-1)*t^25 + O(5^-1)*t^26 + O(5^-2)*t^27 + O(5^-1)*t^28 + 310 O(5^-2)*t^29 + O(5^-2)*t^30 + O(5^-3)*t^31 + O(5^-2)*t^32 + 311 O(5^-3)*t^33 + O(5^-3)*t^34 + O(5^-3)*t^35 + O(5^-3)*t^36 + 312 O(5^-4)*t^37 + O(5^-3)*t^38 + O(5^-4)*t^39 + O(5^-4)*t^40 + 313 O(5^-5)*t^41 + O(5^-4)*t^42 + O(5^-5)*t^43 + O(5^-5)*t^44 + 314 O(5^-5)*t^45 + O(5^-5)*t^46 + O(5^-6)*t^47 + O(5^-5)*t^48 + 315 O(5^-6)*t^49 + O(5^-6)*t^50 + (46*5^-12 + O(5^-9))*t^51 + 316 O(5^-6)*t^52 317> height_of_point(P,5,20); 318Time: 0.140 319s1 = -47683715820312 + O(5^20) 320Time: 0.000 321s2 = 3 + O(5^2) 322Computing (y^Frobenius)^(-1) 323Expansion time: 0.069 324Reducing differentials modulo cohomology relations. 325Time: 0.200 326s2b = -28994174169036 + O(5^20) 327s2 - s2b = -11 + O(5^2) 3285 + O(5^2) 329> height_of_point(P,5,30); 330Time: 0.750 331s1 = -47683715820312 + O(5^20) 332Time: 0.010 333s2 = 3 + O(5^2) 334Computing (y^Frobenius)^(-1) 335Expansion time: 0.07 336Reducing differentials modulo cohomology relations. 337Time: 0.200 338s2b = -28994174169036 + O(5^20) 339s2 - s2b = -11 + O(5^2) 3405 + O(5^2) 341> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 342Computing (y^Frobenius)^(-1) 343Expansion time: 0.009 344Reducing differentials modulo cohomology relations. 345r2 = -10 346d1 = -60 347s1 = 25/2 348s2 = 33543952 + O(5^12) 349> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 350Computing (y^Frobenius)^(-1) 351Expansion time: 0 352Reducing differentials modulo cohomology relations. 3535*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 + O(t^6) 3545*t^-24 - 60*t^-23 + 335*t^-22 - 1215*t^-21 + 3382*t^-20 - 355 7860*t^-19 + 16420*t^-18 - 33875*t^-17 + 68910*t^-16 - 356 129969*t^-15 + 223520*t^-14 - 340515*t^-13 + 430510*t^-12 - 357 413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 + 358 604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 - 359 99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t + 360 O(t^2) 361r2 = -10 362d1 = -60 363s1 = 25/2 364s2 = 33543952 + O(5^12) 365> >> prec := 10; s := sigma_using_e2(E, p, prec : e2prec := prec); 366Computing (y^Frobenius)^(-1) 367Expansion time: 0.009 368Reducing differentials modulo cohomology relations. 3695*t - 10*t^2 + 10*t^3 - 315*t^4 + 307*t^5 - 620*t^6 - 56510*t^7 + 370 92780*t^8 - 309860*t^9 + 6582199*t^10 - 9697985*t^11 + 371 32499195*t^12 + O(t^13) 3725*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 - 375 413530*t^-11 + 248018*t^-10 + 44970*t^-9 - 393495*t^-8 + 376 604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 - 377 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 + 380 57418854*t^10 + O(t^11) 381r2 = -10 382d1 = -60 383s1 = 25/2 384s2 = 33543952 + O(5^12) 385> E; 386Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 387Field 388> 13 - 25/2; 3891/2 390> regulator(E,5, 20); 391Computing (y^Frobenius)^(-1) 392Expansion time: 0.03 393Reducing differentials modulo cohomology relations. 3945*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) 3975*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 + 401 604105*t^-7 - 546235*t^-6 + 296600*t^-5 - 44900*t^-4 - 402 99460*t^-3 + 123050*t^-2 - 63245*t^-1 + 73723 + 3735*t + 403 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 + 405 57418854*t^10 + O(t^11) 406r2 = -10 407d1 = -60 408s1 = 1/2 409s2 = 887301058934661 + O(5^22) 410-154932736694 + O(5^18) 411[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; 415Elliptic Curve defined by y^2 = x^3 + 621*x + 103086 over 416Rational Field 417> regulator(F, 5, 20); 418Computing (y^Frobenius)^(-1) 419Expansion time: 0.021 420Reducing differentials modulo cohomology relations. 4215*t - 775008*t^5 - 3451319280*t^7 + 99464526720*t^9 + 422 740772243669600*t^11 + O(t^13) 4235*t^-24 + 1242*t^-20 + 32987520*t^-18 - 175466655*t^-16 - 424 212534467920*t^-14 - 36147307135050*t^-12 + 425 444610840001184*t^-10 + 102435296676157440*t^-8 + 426 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) 432r2 = 0 433d1 = 0 434s1 = 1/2 435s2 = 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; 440Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 441Field 442> P; 443(-1 : 1 : 1) 444> Parent(P); 445Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 + 4462 over Rational Field with coordinates in Rational Field 447> E := EC("1058C"); 448> E; 449Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 450Field 451> E := EC("1058C"); 452> height_of_point(P,5,30); 453Time: 8.030 454s1 = -47683715820312 + O(5^20) 455Time: 0.010 456s2 = 4*5 + O(5^3) 4575 + O(5^2) 458> height_of_point(P,5,30); 459Time: 0.750 460s1 = -47683715820312 + O(5^20) 461Time: 0.010 462s2 = 3 + O(5^2) 4635 + O(5^2) 464> 390595685806406 mod 5^2; 4656 466> height_of_point(P,5,10); 467Time: 0.010 468t + 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) 480s1 = -47683715820312 + O(5^20) 481Time: 0.000 482s2 = -2 + O(0) 483O(0) 484> height_of_point(P,5,20); 485Time: 0.010 486t + 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) 498s1 = -47683715820312 + O(5^20) 499Time: 0.000 500s2 = -2 + O(0) 501Computing (y^Frobenius)^(-1) 502Expansion time: 0.079 503Reducing differentials modulo cohomology relations. 504Time: 0.200 505s2b = -28994174169036 + O(5^20) 506s2 - s2b = -1 + O(0) 507O(0) 508> height_of_point(P,5,30); 509Time: 0.140 510t + 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) 581s1 = -47683715820312 + O(5^20) 582Time: 0.000 583s2 = 3 + O(5^2) 584Computing (y^Frobenius)^(-1) 585Expansion time: 0.071 586Reducing differentials modulo cohomology relations. 587Time: 0.200 588s2b = -28994174169036 + O(5^20) 589s2 - s2b = -11 + O(5^2) 5905 + O(5^2) 591> height_of_point(P,5,30); 592Time: 0.750 593s1 = -47683715820312 + O(5^20) 594Time: 0.010 595s2 = 3 + O(5^2) 596Computing (y^Frobenius)^(-1) 597Expansion time: 0.07 598Reducing differentials modulo cohomology relations. 599Time: 0.200 600s2b = -28994174169036 + O(5^20) 601s2 - s2b = -11 + O(5^2) 6025 + O(5^2) 603> E := EC("37A"); 604> height_of_point(P,5,30); 605Time: 0.770 606s1 = -47683715820312 + O(5^20) 607Time: 0.000 608s2 = 3 + O(5^2) 609Computing (y^Frobenius)^(-1) 610Expansion time: 0.079 611Reducing differentials modulo cohomology relations. 612Time: 0.200 613s2b = -28994174169036 + O(5^20) 614s2 - s2b = -11 + O(5^2) 6155 + O(5^2) 616> G, f := MordellWeilGroup(E); 617> height_of_point(f(G.1),5,30); 618Time: 0.540 619s1 = O(5^20) 620Time: 0.010 621s2 = 3 + O(5^2) 622Computing (y^Frobenius)^(-1) 623Expansion time: 0.07 624Reducing differentials modulo cohomology relations. 625Time: 0.180 626s2b = 41366042218322 + O(5^20) 627s2 - s2b = 6 + O(5^2) 6282*5 + O(5^2) 629> height_of_point(f(G.1),5,30); 630Time: 0.530 631s1 = O(5^20) 632Time: 0.010 633s2 = 3 + O(5^2) 634Computing (y^Frobenius)^(-1) 635Expansion time: 0.07 636Reducing differentials modulo cohomology relations. 637Time: 0.180 638s2b = -41366042218322 + O(5^20) 639s2 - s2b = O(5^2) 6402*5 + O(5^2) 641> height_of_point(f(G.1),5,30); 642Time: 0.530 643s1 = O(5^20) 644Time: 0.010 645s2 = 3 + O(5^2) 646Computing (y^Frobenius)^(-1) 647Expansion time: 0.07 648Reducing differentials modulo cohomology relations. 649Time: 0.180 650s2b = -41366042218322 + O(5^20) 651s2 - s2b = O(5^2) 6522*5 + O(5^2) 653> height_of_point(P,5,30); 654Time: 0.760 655s1 = -47683715820312 + O(5^20) 656Time: 0.000 657s2 = 3 + O(5^2) 658Computing (y^Frobenius)^(-1) 659Expansion time: 0.07 660Reducing differentials modulo cohomology relations. 661Time: 0.190 662s2b = -6916822718276*5 + O(5^20) 663s2 - s2b = 8 + O(5^2) 6645 + O(5^2) 665> height_of_point(f(G.1),5,40); 666Time: 1.850 667s1 = O(5^20) 668Time: 0.030 669s2 = 53 + O(5^3) 670Computing (y^Frobenius)^(-1) 671Expansion time: 0.07 672Reducing differentials modulo cohomology relations. 673Time: 0.190 674s2b = 41366042218322 + O(5^20) 675s2 - s2b = -19 + O(5^3) 676-3*5 + O(5^3) 677> height_of_point(f(G.1),5,40); 678Time: 1.860 679s1 = O(5^20) 680Time: 0.030 681s2 = 53 + O(5^3) 682Computing (y^Frobenius)^(-1) 683Expansion time: 0.089 684Reducing differentials modulo cohomology relations. 685Time: 0.190 686s2b = -41366042218322 + O(5^20) 687s2 - s2b = O(5^3) 688-3*5 + O(5^3) 689> height_of_point(f(G.1),5,40); 690Time: 1.850 691s1 = O(5^20) 692Time: 0.030 693s2 = 53 + O(5^3) 694Computing (y^Frobenius)^(-1) 695Expansion time: 0.07 696Reducing differentials modulo cohomology relations. 697Time: 0.190 698s2b = -41366042218322 + O(5^20) 699s2 - s2b = O(5^3) 700-3*5 + O(5^3) 701> height_of_point(P,5,40); 702Time: 2.680 703s1 = -47683715820312 + O(5^20) 704Time: 0.030 705s2 = -47 + O(5^3) 706Computing (y^Frobenius)^(-1) 707Expansion time: 0.07 708Reducing differentials modulo cohomology relations. 709Time: 0.190 710s2b = 28994174169036 + O(5^20) 711s2 - s2b = 42 + O(5^3) 7125 + O(5^3) 713> ; 714> E; 715Elliptic 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); 719Computing (y^Frobenius)^(-1) 720Expansion time: 0.009 721Reducing differentials modulo cohomology relations. 722wp_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) 7275*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) 7295*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) 735r2 = 0 736d1 = 0 737s1 = 1/2 738s2 = -44903869 + O(5^12) 739function(P) ... end function 740> E := EC("1058C"); 741> height_function(E,5,10); 742Computing (y^Frobenius)^(-1) 743Expansion time: 0.01 744Reducing differentials modulo cohomology relations. 745wp_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) 7505*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) 7535*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) 762r2 = -10 763d1 = -60 764s1 = 1/2 765s2 = 33544036 + O(5^12) 766function(P) ... end function 767> aInvariants(E); 768[ 1, 0, 1, 0, 2 ] 769> E; 770Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 771Field 772> h := height_function(E,5,30); 773Computing (y^Frobenius)^(-1) 774Expansion time: 0.051 775Reducing 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 ^ 780Runtime 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 ^ 785Runtime error in '+': Arguments are not compatible 786Argument types given: Pt[FldRat], Pt[FldRat] 787> G, f := MordellWeilGroup(E); 788> h := height_function(E,5,30); 789Computing (y^Frobenius)^(-1) 790Expansion time: 0.049 791Reducing 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); 800616044959906266998871 + 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); 812Time: 2.680 813s1 = -47683715820312 + O(5^20) 814Time: 0.030 815s2 = -47 + O(5^3) 816Computing (y^Frobenius)^(-1) 817Expansion time: 0.08 818Reducing differentials modulo cohomology relations. 819Time: 0.190 820s2b = -28994174169036 + O(5^20) 821s2 - s2b = -11 + O(5^3) 8225 + O(5^3) 823> height_of_point(3*P,5,40); 824Time: 2.640 825s1 = -47683715820312 + O(5^20) 826Time: 0.040 827s2 = -47 + O(5^3) 828Computing (y^Frobenius)^(-1) 829Expansion time: 0.069 830Reducing differentials modulo cohomology relations. 831Time: 0.190 832s2b = -28994174169036 + O(5^20) 833s2 - s2b = -11 + O(5^3) 8349*5 + O(5^3) 835> h(5*P); 836194840086849274083589 + O(5^31) 837> 194840086849274083589 mod 5^3; 83889 839> 45 + 89; 840134 841> height_of_point(5*P,5,40); 842Time: 2.680 843s1 = -47683715820312 + O(5^20) 844Time: 0.040 845s2 = -47 + O(5^3) 846Computing (y^Frobenius)^(-1) 847Expansion time: 0.07 848Reducing differentials modulo cohomology relations. 849Time: 0.180 850s2b = -28994174169036 + O(5^20) 851s2 - 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); 864616044959906266998871*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); 868Time: 1.200 869s1 = -47683715820312 + O(5^20) 870Time: 0.020 871s2 = -47 + O(5^3) 872Computing (y^Frobenius)^(-1) 873Expansion time: 0.069 874Reducing differentials modulo cohomology relations. 875Time: 0.190 876s2b = -28994174169036 + O(5^20) 877s2 - s2b = -11 + O(5^3) 8784 + O(5^2) 879> h(2*P); 880-749917183385185910766 + O(5^31) 881> -749917183385185910766 mod 5^2; 8829 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); 889Computing (y^Frobenius)^(-1) 890Expansion time: 0.04 891Reducing differentials modulo cohomology relations. 892> h(E![0,0]); 893-1305176752965909410953 + O(5^31) 894> h := height_function(E,5,30); 895Computing (y^Frobenius)^(-1) 896Expansion time: 0.041 897Reducing differentials modulo cohomology relations. 898s1 = 0 899s2 = -11364423793197096905822 + O(5^32) 900> -11364423793197096905822 mod 5^3; 90153 902> -11364423793197096905822 mod 25; 9033 904> h := height_function(E,5,30); 905Computing (y^Frobenius)^(-1) 906Expansion time: 0.05 907Reducing differentials modulo cohomology relations. 908s1 = 0 909s2 = 11364423793197096905822 + O(5^32) 910> 11364423793197096905822 mod 25; 91122 912> height_of_point(P,5,25); 913Time: 0.370 914s1 = -47683715820312 + O(5^20) 915Time: 0.010 916s2 = 3 + O(5^2) 917Computing (y^Frobenius)^(-1) 918Expansion time: 0.079 919Reducing differentials modulo cohomology relations. 920Time: 0.200 921s2b = -28994174169036 + O(5^20) 922s2 - s2b = -11 + O(5^2) 9231 + O(5) 924> P; 925(-1 : 1 : 1) 926> Parent(P); 927Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 + 9282 over Rational Field with coordinates in Rational Field 929> h := height_function(E,5,30); 930Computing (y^Frobenius)^(-1) 931Expansion time: 0.049 932Reducing differentials modulo cohomology relations. 933s1 = 0 934s2 = -11364423793197096905822 + O(5^32) 935> h(P); 936 937h( 938 P: (-1 : 1 : 1) 939) 940In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\ 941al.m", line 569, column 24: 942>> assert Parent(P) eq E; 943 ^ 944Runtime error in 'eq': Bad argument types 945Argument types given: SetPtEll[FldRat], CrvEll[FldRat] 946> Parent(P); 947Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 + 9482 over Rational Field with coordinates in Rational Field 949> Curve(P); 950Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 951Field 952> h := height_function(E,5,30); 953Computing (y^Frobenius)^(-1) 954Expansion time: 0.049 955Reducing differentials modulo cohomology relations. 956s1 = 0 957s2 = -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; 963Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field 964> height_of_point(E![0,0],5,25); 965Time: 0.230 966s1 = O(5^20) 967Time: 0.020 968s2 = 3 + O(5^2) 969Computing (y^Frobenius)^(-1) 970Expansion time: 0.069 971Reducing differentials modulo cohomology relations. 972Time: 0.180 973s2b = 41366042218322 + O(5^20) 974s2 - s2b = 6 + O(5^2) 9752 + O(5) 976> -11364423793197096905822 mod 5^2; 9773 978> aInvariants(E); 979[ 0, 0, 1, -1, 0 ] 980> height_of_point(E![0,0],5,35); 981Time: 1.090 982s1 = O(5^20) 983Time: 0.020 984s2 = 53 + O(5^3) 985Computing (y^Frobenius)^(-1) 986Expansion time: 0.07 987Reducing differentials modulo cohomology relations. 988Time: 0.180 989s2b = 41366042218322 + O(5^20) 990s2 - s2b = -19 + O(5^3) 991-3 + O(5^2) 992> E := EC("1058C"); 993> E; 994Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 995Field 996> aInvariants(E); 997[ 1, 0, 1, 0, 2 ] 998> qEigenform(E,7); 999q - 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); 1003Computing (y^Frobenius)^(-1) 1004Expansion time: 0.051 1005Reducing differentials modulo cohomology relations. 1006s1 = 25/2 1007s2 = 1174596243288607762702 + O(5^32) 1008> h(P); 1009616044959906266998871 + O(5^31) 1010> height_of_point(P,5,35); 1011Time: 1.590 1012s1 = -47683715820312 + O(5^20) 1013Time: 0.020 1014s2 = -47 + O(5^3) 1015Computing (y^Frobenius)^(-1) 1016Expansion time: 0.069 1017Reducing differentials modulo cohomology relations. 1018Time: 0.190 1019s2b = -28994174169036 + O(5^20) 1020s2 - s2b = -11 + O(5^3) 10211 + O(5^2) 1022> E; 1023Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 1024Field 1025> sigma_using_e2(E,5,20); 1026Computing (y^Frobenius)^(-1) 1027Expansion time: 0.111 1028Reducing differentials modulo cohomology relations. 1029s1 = 25/2 1030s2 = 110294297236694266286783602328465827 + O(5^52) 1031t - (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); 1050Computing (y^Frobenius)^(-1) 1051Expansion time: 0.04 1052Reducing differentials modulo cohomology relations. 1053s1 = 25/2 1054s2 = 1174596243288607762702 + O(5^32) 1055> - (1110223024625156540423631668090820312 + O(5^52) + 2 1056> sigma_using_e2(E,5,20); 1057Computing (y^Frobenius)^(-1) 1058Expansion time: 0.109 1059Reducing differentials modulo cohomology relations. 1060s1 = 1/2 1061s2 = 110294297236694266286783602328465911 + O(5^52) 1062t - (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); 1081Computing (y^Frobenius)^(-1) 1082Expansion time: 0.09 1083Reducing differentials modulo cohomology relations. 1084s1 = 1/2 1085s2 = 120 1086t - (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); 1105Computing (y^Frobenius)^(-1) 1106Expansion time: 0.099 1107Reducing differentials modulo cohomology relations. 1108s1 = 1/2 1109s2 = 120 1110t - (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); 1129Computing (y^Frobenius)^(-1) 1130Expansion time: 0.1 1131Reducing differentials modulo cohomology relations. 1132s1 = 1/2 1133s2 = 120 1134t - (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); 1145Computing (y^Frobenius)^(-1) 1146Expansion time: 0.099 1147Reducing differentials modulo cohomology relations. 1148s1 = 1/2 1149s2 = 120 1150t + 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); 1159Computing (y^Frobenius)^(-1) 1160Expansion time: 0.1 1161Reducing differentials modulo cohomology relations. 1162s1 = 1/2 1163s2 = 120 1164t + 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); 1182Computing (y^Frobenius)^(-1) 1183Expansion time: 0.099 1184Reducing differentials modulo cohomology relations. 1185t + 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); 1205Set of points of Elliptic Curve defined by y^2 + x*y + y = x^3 + 12062 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); 1213Time: 1.080 1214s1 = O(5^20) 1215Time: 0.010 1216s2 = 53 + O(5^3) 1217Computing (y^Frobenius)^(-1) 1218Expansion time: 0.069 1219Reducing differentials modulo cohomology relations. 1220Time: 0.190 1221s2b = 41366042218322 + O(5^20) 1222s2 - s2b = -19 + O(5^3) 1223-3 + O(5^2) 1224> h := height_function(E,5,30); 1225Computing (y^Frobenius)^(-1) 1226Expansion time: 0.051 1227Reducing differentials modulo cohomology relations. 1228> Coefficient(sigma_using_e2(EC("37A"),5,20),3); 1229Computing (y^Frobenius)^(-1) 1230Expansion time: 0.09 1231Reducing differentials modulo cohomology relations. 1232t + 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); 1250Computing (y^Frobenius)^(-1) 1251Expansion time: 0.099 1252Reducing differentials modulo cohomology relations. 1253331883312126563673413235306321062928 + O(5^52) 1254> Coefficient(sigma_using_e2(EC("37A"),5,20),3); 1255Computing (y^Frobenius)^(-1) 1256Expansion time: 0.1 1257Reducing differentials modulo cohomology relations. 1258-331883312126563673413235306321062928 + O(5^52) 1259> Coefficient(sigma_using_e2(EC("37A"),5,20),3); 1260Computing (y^Frobenius)^(-1) 1261Expansion time: 0.099 1262Reducing differentials modulo cohomology relations. 1263sigma_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) 1277sigma_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) 1299331883312126563673413235306321062928 + O(5^52) 1300> Coefficient(sigma_using_e2(EC("37A"),5,20),3); 1301Computing (y^Frobenius)^(-1) 1302Expansion time: 0.101 1303Reducing differentials modulo cohomology relations. 1304sigma_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) 1318sigma_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) 1340331883312126563673413235306321062928 + O(5^52) 1341> Coefficient(sigma_using_e2(EC("37A"),5,20),3); 1342Computing (y^Frobenius)^(-1) 1343Expansion time: 0.09 1344Reducing differentials modulo cohomology relations. 1345sigma_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) 1359sigma_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); 1386Computing (y^Frobenius)^(-1) 1387Expansion time: 0.089 1388Reducing differentials modulo cohomology relations. 1389sigma_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) 1403sigma_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) 1425331883312126563673413235306321062928 + O(5^52) 1426> height_of_point(P,5,35); 1427Time: 1.070 1428s1 = O(5^20) 1429Time: 0.020 1430s2 = 53 + O(5^3) 1431Computing (y^Frobenius)^(-1) 1432Expansion time: 0.069 1433Reducing differentials modulo cohomology relations. 1434Time: 0.180 1435s2b = 41366042218322 + O(5^20) 1436s2 - s2b = -19 + O(5^3) 1437-3 + O(5^2) 1438> height_of_point(P,5,45); 1439Time: 3.330 1440s1 = O(5^20) 1441Time: 0.040 1442s2 = 53 + O(5^3) 1443Computing (y^Frobenius)^(-1) 1444Expansion time: 0.069 1445Reducing differentials modulo cohomology relations. 1446Time: 0.200 1447s2b = 41366042218322 + O(5^20) 1448s2 - 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; 1454Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 1455Field 1456> G, f := MordellWeilGroup(E); 1457> P := f(G.1); 1458> P; 1459(-1 : 1 : 1) 1460> height_of_point(P,5,45); 1461Time: 4.790 1462s1 = -47683715820312 + O(5^20) 1463Time: 0.050 1464s2 = -47 + O(5^3) 1465Computing (y^Frobenius)^(-1) 1466Expansion time: 0.07 1467Reducing differentials modulo cohomology relations. 1468Time: 0.190 1469s2b = -28994174169036 + O(5^20) 1470s2 - s2b = -11 + O(5^3) 14711 + O(5^2) 1472> h := height_function(E,5,30); 1473Computing (y^Frobenius)^(-1) 1474Expansion time: 0.051 1475Reducing differentials modulo cohomology relations. 1476sigma_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) 1491sigma_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; 1513Elliptic Curve defined by y^2 + x*y + y = x^3 + 2 over Rational 1514Field 1515> sigma_using_e2(E,5,20); 1516Computing (y^Frobenius)^(-1) 1517Expansion time: 0.1 1518Reducing differentials modulo cohomology relations. 1519sigma_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) 1533sigma_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) 1556t - (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> ; 1576Computing (y^Frobenius)^(-1) 1577Expansion time: 0.049 1578Reducing differentials modulo cohomology relations. 1579sigma_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) 1594sigma_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) 16151438935235176290340451 + 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 ^ 1631User error: bad syntax 1632> ; 1633> >> regulator(E,5,30); 1634Computing (y^Frobenius)^(-1) 1635Expansion time: 0.05 1636Reducing differentials modulo cohomology relations. 16371438935235176290340451 + 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); 1643Computing (y^Frobenius)^(-1) 1644Expansion time: 0.04 1645Reducing 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); 1652Computing (y^Frobenius)^(-1) 1653Expansion time: 0.04 1654Reducing differentials modulo cohomology relations. 16551438935235176290340451 + 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 1662In file "/home/was/papers/padic_cyclotomic_height/e2heights/form\ 1663al.m", line 550, column 28: 1664>> pr := AbsolutePrecision(f); 1665 ^ 1666Runtime error: Undefined reference 'f' in package 1667"/home/was/papers/padic_cyclotomic_height/e2heights/formal.m" 1668> regulator(E,5,30); 1669Computing (y^Frobenius)^(-1) 1670Expansion time: 0.041 1671Reducing differentials modulo cohomology relations. 16721438935235176290340451 + 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, 168067, 71, 73, 79, 89, 97 ] 1681> r := regulator(E,7); 1682Computing (y^Frobenius)^(-1) 1683Expansion time: 0.041 1684Reducing differentials modulo cohomology relations. 168534330260522044597 + 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); 1689Computing (y^Frobenius)^(-1) 1690Expansion time: 0.039 1691Reducing differentials modulo cohomology relations. 1692> r; 169334330260522044597 + O(7^20) 1694> [<p,regulator(E,p)> : p in G | p ge 5 and p le 29]; 1695Computing (y^Frobenius)^(-1) 1696Expansion time: 0.031 1697Reducing differentials modulo cohomology relations. 1698Computing (y^Frobenius)^(-1) 1699Expansion time: 0.041 1700Reducing differentials modulo cohomology relations. 1701 1702>> [* <p,regulator(E,p)> : p in G | p ge 5 and p le 29 *]; 1703 ^ 1704Runtime error in sequence construction: Could not find a valid 1705universe 1706> regulator(E,11); 1707Computing (y^Frobenius)^(-1) 1708Expansion time: 0.081 1709Reducing differentials modulo cohomology relations. 17101664557466859596710369 + 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 *]; 1716Computing (y^Frobenius)^(-1) 1717Expansion time: 0.021 1718Reducing differentials modulo cohomology relations. 1719Computing (y^Frobenius)^(-1) 1720Expansion time: 0.039 1721Reducing differentials modulo cohomology relations. 1722Computing (y^Frobenius)^(-1) 1723Expansion time: 0.079 1724Reducing differentials modulo cohomology relations. 1725Computing (y^Frobenius)^(-1) 1726Expansion time: 0.1 1727Reducing differentials modulo cohomology relations. 1728Computing (y^Frobenius)^(-1) 1729Expansion time: 0.131 1730Reducing differentials modulo cohomology relations. 1731Computing (y^Frobenius)^(-1) 1732Expansion time: 0.221 1733Reducing differentials modulo cohomology relations. 1734Computing (y^Frobenius)^(-1) 1735Expansion time: 0.359 1736Reducing differentials modulo cohomology relations. 1737[* <5, 1106466121701 + O(5^18)>, <7, 34330260522044597 + 1738O(7^20)>, <11, 1664557466859596710369 + O(11^21)>, <13, 1739103609576221026126766923 + O(13^21)>, <17, 1740-979424087926189051470349 + O(17^20)>, <19, 1741-16905965663076553616324627 + O(19^20)>, <29, 174272071173330516195538881320295 + O(29^20)> *] 1743> x :=$1;
1744> x;
1745[* <5, 1106466121701 + O(5^18)>, <7, 34330260522044597 +
1746O(7^20)>, <11, 1664557466859596710369 + O(11^21)>, <13,
1747103609576221026126766923 + O(13^21)>, <17,
1748-979424087926189051470349 + O(17^20)>, <19,
1749-16905965663076553616324627 + O(19^20)>, <29,
175072071173330516195538881320295 + 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;
1753for>
1754Computing (y^Frobenius)^(-1)
1755Expansion time: 0.03
1756Reducing differentials modulo cohomology relations.
1757Computing (y^Frobenius)^(-1)
1758Expansion time: 0.039
1759Reducing differentials modulo cohomology relations.
1760Computing (y^Frobenius)^(-1)
1761Expansion time: 0.079
1762Reducing differentials modulo cohomology relations.
1763Computing (y^Frobenius)^(-1)
1764Expansion time: 0.12
1765Reducing differentials modulo cohomology relations.
1766Computing (y^Frobenius)^(-1)
1767Expansion time: 0.14
1768Reducing differentials modulo cohomology relations.
1769Computing (y^Frobenius)^(-1)
1770Expansion time: 0.231
1771Reducing differentials modulo cohomology relations.
1772Computing (y^Frobenius)^(-1)
1773Expansion time: 0.36
1774Reducing differentials modulo cohomology relations.
1775Computing (y^Frobenius)^(-1)
1776Expansion time: 0.409
1777Reducing differentials modulo cohomology relations.
1778Computing (y^Frobenius)^(-1)
1779Expansion time: 0.681
1780Reducing differentials modulo cohomology relations.
1781Computing (y^Frobenius)^(-1)
1782Expansion time: 0.72
1783Reducing differentials modulo cohomology relations.
1784Computing (y^Frobenius)^(-1)
1785Expansion time: 0.689
1786Reducing differentials modulo cohomology relations.
1787Computing (y^Frobenius)^(-1)
1788Expansion time: 0.979
1789Reducing differentials modulo cohomology relations.
1790Computing (y^Frobenius)^(-1)
1791Expansion time: 1.16
1792Reducing differentials modulo cohomology relations.
1793Computing (y^Frobenius)^(-1)
1794Expansion time: 1.26
1795Reducing differentials modulo cohomology relations.
1796Computing (y^Frobenius)^(-1)
1797Expansion time: 1.329
1798Reducing differentials modulo cohomology relations.
1799Computing (y^Frobenius)^(-1)
1800Expansion time: 1.4
1801Reducing differentials modulo cohomology relations.
1802Computing (y^Frobenius)^(-1)
1803Expansion time: 1.82
1804Reducing differentials modulo cohomology relations.
1805Computing (y^Frobenius)^(-1)
1806Expansion time: 1.829
1807Reducing differentials modulo cohomology relations.
1808Computing (y^Frobenius)^(-1)
1809Expansion time: 1.891
1810Reducing differentials modulo cohomology relations.
1811Computing (y^Frobenius)^(-1)
1812Expansion time: 2.029
1813Reducing differentials modulo cohomology relations.
1814Computing (y^Frobenius)^(-1)
1815Expansion time: 2.93
1816Reducing differentials modulo cohomology relations.
1817> A;
1818[* <5, 1106466121701 + O(5^18), 0>, <7, 34330260522044597 +
1819O(7^20), 0>, <11, 1664557466859596710369 + O(11^21), 0>, <13,
1820103609576221026126766923 + O(13^21), 0>, <17,
1821-979424087926189051470349 + O(17^20), 0>, <19,
1822-16905965663076553616324627 + O(19^20), 0>, <29,
182372071173330516195538881320295 + O(29^20), 0>, <31,
1824-173944231991525079009482582188 + O(31^20), 0>, <37,
182510390854614217075712273759793878 + O(37^20), 0>, <41,
182664521799079830548248787468756513 + 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,
183053186567475602244093889022258956772 + O(59^20), 0>, <61,
18313076131498368429672977101883433*61^-2 + O(61^16), -2>, <67,
1832167936089130563072609672366369770203 + O(67^20), 0>, <71,
18332645825244440902073298471121935090827 + O(71^20), 0>, <73,
1834-895075117177840654650337174738339688 + O(73^20), 0>, <79,
18358896576480895465746291587395824330359 + O(79^20), 0>, <89,
1836-463071254062057122311079391324128760494 + O(89^20), 0>, <97,
1837269501708627173241996085597295066750*97^-2 + O(97^16), -2> *]
1838> regulator(E,61,40);
1839Computing (y^Frobenius)^(-1)
1840Expansion time: 5.349
1841Reducing differentials modulo cohomology relations.
1842-370438686758431742575952482434711859118701081729752071017439432\
18436910*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);
1851Computing (y^Frobenius)^(-1)
1852Expansion time: 5.37
1853Reducing differentials modulo cohomology relations.
1854>P;
1855(-1 : 1 : 1)
1856> Q;
1857(1 : -3 : 1)
1858> h(P);
18597235590364451759412233487342066596711624481343161361427996724504\
18608924769*61^-2 + O(61^38)
1861> h(Q);
1862-673922145771191356308640328270344812001327531147182728021623000\
186308957039*61^-2 + O(61^38)
1864> h(2*P);
1865-426498439984155838211045243792382254102362859887852505754773483\
186643235456*61^-2 + O(61^38)
1867> 4*h(P);
18683048703885235699461273383751997784765406621448076984849992269174\
18693608675*61^-2 + O(61^38)
1870> 4*h(P) - h(2*P);
18711051*61^36 + O(61^38)
1872> h(P);
18737235590364451759412233487342066596711624481343161361427996724504\
18748924769*61^-2 + O(61^38)
1875> h(Q);
1876-673922145771191356308640328270344812001327531147182728021623000\
187708957039*61^-2 + O(61^38)
1878> Coefficient(qEigenform(E,65),61);
18791
1880> Coefficient(qEigenform(E,100),97);
18811
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,
18846, 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,
188973, 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,
189379, 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;
1896Computing (y^Frobenius)^(-1)
1897Expansion time: 0.03
1898Reducing differentials modulo cohomology relations.
1899Computing (y^Frobenius)^(-1)
1900Expansion time: 0.03
1901Reducing differentials modulo cohomology relations.
1902Computing (y^Frobenius)^(-1)
1903Expansion time: 0.079
1904Reducing differentials modulo cohomology relations.
1905Computing (y^Frobenius)^(-1)
1906Expansion time: 0.111
1907Reducing differentials modulo cohomology relations.
1908Computing (y^Frobenius)^(-1)
1909Expansion time: 0.27
1910Reducing differentials modulo cohomology relations.
1911Computing (y^Frobenius)^(-1)
1912Expansion time: 0.359
1913Reducing differentials modulo cohomology relations.
1914Computing (y^Frobenius)^(-1)
1915Expansion time: 0.419
1916Reducing differentials modulo cohomology relations.
1917Computing (y^Frobenius)^(-1)
1918Expansion time: 0.699
1919Reducing differentials modulo cohomology relations.
1920Computing (y^Frobenius)^(-1)
1921Expansion time: 0.671
1922Reducing differentials modulo cohomology relations.
1923Computing (y^Frobenius)^(-1)
1924Expansion time: 0.969
1925Reducing differentials modulo cohomology relations.
1926Computing (y^Frobenius)^(-1)
1927Expansion time: 1.15
1928Reducing differentials modulo cohomology relations.
1929Computing (y^Frobenius)^(-1)
1930Expansion time: 1.239
1931Reducing differentials modulo cohomology relations.
1932Computing (y^Frobenius)^(-1)
1933Expansion time: 1.31
1934Reducing differentials modulo cohomology relations.
1935Computing (y^Frobenius)^(-1)
1936Expansion time: 1.371
1937Reducing differentials modulo cohomology relations.
1938Computing (y^Frobenius)^(-1)
1939Expansion time: 1.799
1940Reducing differentials modulo cohomology relations.
1941Computing (y^Frobenius)^(-1)
1942Expansion time: 1.839
1943Reducing differentials modulo cohomology relations.
1944Computing (y^Frobenius)^(-1)
1945Expansion time: 1.901
1946Reducing differentials modulo cohomology relations.
1947Computing (y^Frobenius)^(-1)
1948Expansion time: 1.93
1949Reducing differentials modulo cohomology relations.
1950Computing (y^Frobenius)^(-1)
1951Expansion time: 2.039
1952Reducing differentials modulo cohomology relations.
1953Computing (y^Frobenius)^(-1)
1954Expansion time: 2.909
1955Reducing differentials modulo cohomology relations.
1956Time: 35.600
1957> A;
1958[* <5, 138360994885922 + O(5^21), 0>, <7, 4709403600911866 +
1959O(7^20), 0>, <11, 3502722142035391199047 + O(11^21), 0>, <13,
19603328786448953657679826*13 + O(13^21), 1>, <23,
1961-18507705578301047964577596 + O(23^20), 0>, <29,
196216742788144500360438362532376 + O(29^20), 0>, <31,
1963214585630396079048220257036022 + O(31^20), 0>, <41,
196462872643243806364912447375677980 + O(41^20), 0>, <43,
196598201982475772371322430732508192 + O(43^20), 0>, <47,
1966-950963360517191754231629728419340 + O(47^20), 0>, <53,
19673013481933339232802567988878307847*53^-2 + O(53^18), -2>, <59,
1968-36750556362766803167564839340913550 + O(59^20), 0>, <61,
1969139607664981793518838513655007581617 + O(61^20), 0>, <67,
1970-12619683101217607828398831962484117*67 + O(67^20), 1>, <71,
19712376688066444732847472979255838549466 + 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,
197527000563061297979788158574760212991695 + O(89^20), 0>, <97,
19762257574446473128489077697545138873920970 + O(97^20), 0> *]
1977> SetColumns(100);
1978> A;
1979[* <5, 138360994885922 + O(5^21), 0>, <7, 4709403600911866 + O(7^20), 0>, <11,
19803502722142035391199047 + 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,
1982214585630396079048220257036022 + O(31^20), 0>, <41, 62872643243806364912447375677980 + O(41^20), 0>,
1983<43, 98201982475772371322430732508192 + O(43^20), 0>, <47, -950963360517191754231629728419340 +
1984O(47^20), 0>, <53, 3013481933339232802567988878307847*53^-2 + O(53^18), -2>, <59,
1985-36750556362766803167564839340913550 + O(59^20), 0>, <61, 139607664981793518838513655007581617 +
1986O(61^20), 0>, <67, -12619683101217607828398831962484117*67 + O(67^20), 1>, <71,
19872376688066444732847472979255838549466 + O(71^20), 0>, <73, -4793861347633889739942985981494036464 +
1988O(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),
20010>
2002<43, 98201982475772371322430732508192 + O(43^20),
20030>
2004<47, -950963360517191754231629728419340 +
2005O(47^20), 0>
2006<53, 3013481933339232802567988878307847*53^-2 +
2007O(53^18), -2>
2008<59, -36750556362766803167564839340913550 +
2009O(59^20), 0>
2010<61, 139607664981793518838513655007581617 +
2011O(61^20), 0>
2012<67, -12619683101217607828398831962484117*67 +
2013O(67^20), 1>
2014<71, 2376688066444732847472979255838549466 +
2015O(71^20), 0>
2016<73, -4793861347633889739942985981494036464 +
2017O(73^20), 0>
2018<79, -39751046706757915792433195709308099930 +
2019O(79^20), 0>
2020<83, -118006589717737960880381609746899564294 +
2021O(83^20), 0>
2022<89, 27000563061297979788158574760212991695 +
2023O(89^20), 0>
2024<97, 2257574446473128489077697545138873920970 +
2025O(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);
20491
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);
20538
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;
2058Computing (y^Frobenius)^(-1)
2059Expansion time: 0.03
2060Reducing differentials modulo cohomology relations.
2061Computing (y^Frobenius)^(-1)
2062Expansion time: 0.039
2063Reducing differentials modulo cohomology relations.
2064Computing (y^Frobenius)^(-1)
2065Expansion time: 0.09
2066Reducing differentials modulo cohomology relations.
2067Computing (y^Frobenius)^(-1)
2068Expansion time: 0.099
2069Reducing differentials modulo cohomology relations.
2070Computing (y^Frobenius)^(-1)
2071Expansion time: 0.14
2072Reducing differentials modulo cohomology relations.
2073Computing (y^Frobenius)^(-1)
2074Expansion time: 0.21
2075Reducing differentials modulo cohomology relations.
2076Computing (y^Frobenius)^(-1)
2077Expansion time: 0.259
2078Reducing differentials modulo cohomology relations.
2079Computing (y^Frobenius)^(-1)
2080Expansion time: 0.37
2081Reducing differentials modulo cohomology relations.
2082Computing (y^Frobenius)^(-1)
2083Expansion time: 0.409
2084Reducing differentials modulo cohomology relations.
2085Computing (y^Frobenius)^(-1)
2086Expansion time: 0.679
2087Reducing differentials modulo cohomology relations.
2088Computing (y^Frobenius)^(-1)
2089Expansion time: 0.701
2090Reducing differentials modulo cohomology relations.
2091Computing (y^Frobenius)^(-1)
2092Expansion time: 0.689
2093Reducing differentials modulo cohomology relations.
2094Computing (y^Frobenius)^(-1)
2095Expansion time: 0.981
2096Reducing differentials modulo cohomology relations.
2097Computing (y^Frobenius)^(-1)
2098Expansion time: 1.169
2099Reducing differentials modulo cohomology relations.
2100Computing (y^Frobenius)^(-1)
2101Expansion time: 1.25
2102Reducing differentials modulo cohomology relations.
2103Computing (y^Frobenius)^(-1)
2104Expansion time: 1.319
2105Reducing differentials modulo cohomology relations.
2106Computing (y^Frobenius)^(-1)
2107Expansion time: 1.389
2108Reducing differentials modulo cohomology relations.
2109Computing (y^Frobenius)^(-1)
2110Expansion time: 1.819
2111Reducing differentials modulo cohomology relations.
2112Computing (y^Frobenius)^(-1)
2113Expansion time: 1.869
2114Reducing differentials modulo cohomology relations.
2115Computing (y^Frobenius)^(-1)
2116Expansion time: 1.889
2117Reducing differentials modulo cohomology relations.
2118Computing (y^Frobenius)^(-1)
2119Expansion time: 1.961
2120Reducing differentials modulo cohomology relations.
2121Computing (y^Frobenius)^(-1)
2122Expansion time: 2.05
2123Reducing differentials modulo cohomology relations.
2124Computing (y^Frobenius)^(-1)
2125Expansion time: 2.971
2126Reducing differentials modulo cohomology relations.
2127Time: 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;
2156Computing (y^Frobenius)^(-1)
2157Expansion time: 3.35
2158Reducing differentials modulo cohomology relations.
2159101 476480469707036332646161303910245319379 + O(101^20) 0
2160Computing (y^Frobenius)^(-1)
2161Expansion time: 3.64
2162Reducing differentials modulo cohomology relations.
2163103 6401358360394788081773821530465566398861 + O(103^20) 0
2164Computing (y^Frobenius)^(-1)
2165Expansion time: 3.73
2166Reducing differentials modulo cohomology relations.
2167107 4922767642974165115674022301621135949010 + O(107^20) 0
2168Computing (y^Frobenius)^(-1)
2169Expansion time: 3.719
2170Reducing differentials modulo cohomology relations.
2171109 -2448717765918646375218544243833474256546 + O(109^20) 0
2172Computing (y^Frobenius)^(-1)
2173Expansion time: 3.779
2174Reducing differentials modulo cohomology relations.
2175113 -5991211946727979721541061121026578552436 + O(113^20) 0
2176Computing (y^Frobenius)^(-1)
2177Expansion time: 4.069
2178Reducing differentials modulo cohomology relations.
2179127 -266954596482088375341789280269115541482483 + O(127^20) 0
2180Computing (y^Frobenius)^(-1)
2181Expansion time: 4.12
2182Reducing differentials modulo cohomology relations.
2183131 885928978952035516753483804506698950208963 + O(131^20) 0
2184Computing (y^Frobenius)^(-1)
2185Expansion time: 5.2
2186Reducing differentials modulo cohomology relations.
2187137 862355935221473255587603984788742846836475 + O(137^20) 0
2188Computing (y^Frobenius)^(-1)
2189Expansion time: 5.219
2190Reducing differentials modulo cohomology relations.
2191139 893498523927167174829901480037367363781685 + O(139^20) 0
2192Computing (y^Frobenius)^(-1)
2193Expansion time: 5.36
2194Reducing differentials modulo cohomology relations.
2195149 -1963835319577170752807743270103875287090958 + O(149^20) 0
2196Computing (y^Frobenius)^(-1)
2197Expansion time: 5.23
2198Reducing differentials modulo cohomology relations.
2199151 16455424611592423659744030523534896659535280 + O(151^20) 0
2200Computing (y^Frobenius)^(-1)
2201Expansion time: 5.519
2202Reducing differentials modulo cohomology relations.
2203157 1304985225261657442384029871097880458103944 + O(157^20) 0
2204Computing (y^Frobenius)^(-1)
2205Expansion time: 5.581
2206Reducing differentials modulo cohomology relations.
2207163 -74941293240715728290524282127729257855982741 + O(163^20) 0
2208Computing (y^Frobenius)^(-1)
2209Expansion time: 5.79
2210Reducing differentials modulo cohomology relations.
2211173 -31742200745061737323031668621113028647258865 + O(173^20) 0
2212Computing (y^Frobenius)^(-1)
2213Expansion time: 5.82
2214Reducing differentials modulo cohomology relations.
2215181 -301059950417872607277161663862120583299702134 + O(181^20) 0
2216Computing (y^Frobenius)^(-1)
2217Expansion time: 7.12
2218Reducing differentials modulo cohomology relations.
2219191 -1455648087029648260445874502808106259621227117 + O(191^20) 0
2220Computing (y^Frobenius)^(-1)
2221Expansion time: 7.12
2222Reducing differentials modulo cohomology relations.
2223193 -1629590413978391579327532226453194166132346094 + O(193^20) 0
2224Computing (y^Frobenius)^(-1)
2225Expansion time: 7.18
2226Reducing differentials modulo cohomology relations.
2227197 1077468644282969836512862412801672173327406615 + O(197^20) 0
2228Computing (y^Frobenius)^(-1)
2229Expansion time: 7.24
2230Reducing differentials modulo cohomology relations.
2231199 1886144760299789908083666104019078586322981159 + O(199^20) 0
2232Computing (y^Frobenius)^(-1)
2233Expansion time: 9.409
2234Reducing differentials modulo cohomology relations.
2235211 -6094803705500786294536521433382052159437991849 + O(211^20) 0
2236Computing (y^Frobenius)^(-1)
2237Expansion time: 9.561
2238Reducing differentials modulo cohomology relations.
2239223 37415158734533778175044858726928248342273569392 + O(223^20) 0
2240Computing (y^Frobenius)^(-1)
2241Expansion time: 9.649
2242Reducing differentials modulo cohomology relations.
2243227 19606670814333954015853846026123361999186774801 + O(227^20) 0
2244Computing (y^Frobenius)^(-1)
2245Expansion time: 9.789
2246Reducing differentials modulo cohomology relations.
2247229 -64214558386218506627020154712722573995815576492 + O(229^20) 0
2248Computing (y^Frobenius)^(-1)
2249Expansion time: 9.719
2250Reducing differentials modulo cohomology relations.
2251233 77248690651481504297812725415750130294760334531 + O(233^20) 0
2252Computing (y^Frobenius)^(-1)
2253Expansion time: 10
2254Reducing differentials modulo cohomology relations.
2255241 3252317888874066245912857299289743895693945*241^-2 + O(241^16) -2
2256Computing (y^Frobenius)^(-1)
2257Expansion time: 10.189
2258Reducing differentials modulo cohomology relations.
2259251 173581002964881140281151080833285796828100400284 + O(251^20) 0
2260Computing (y^Frobenius)^(-1)
2261Expansion time: 10.529
2262Reducing differentials modulo cohomology relations.
2263257 -426905762455721247734906601556431432396879136369 + O(257^20) 0
2264Computing (y^Frobenius)^(-1)
2265Expansion time: 10.461
2266Reducing differentials modulo cohomology relations.
2267263 -233733854342114140446544882182634535598861917469 + O(263^20) 0
2268Computing (y^Frobenius)^(-1)
2269Expansion time: 10.59
2270Reducing differentials modulo cohomology relations.
2271269 1177971198454019204373622054275629621785669528394 + O(269^20) 0
2272Computing (y^Frobenius)^(-1)
2273Expansion time: 10.52
2274Reducing differentials modulo cohomology relations.
2275271 -1357771339230472431076104469179938359011825717340 + O(271^20) 0
2276Computing (y^Frobenius)^(-1)
2277Expansion time: 13.23
2278Reducing differentials modulo cohomology relations.
2279277 1902113850890970472471844883013311805608608538071 + O(277^20) 0
2280Computing (y^Frobenius)^(-1)
2281Expansion time: 13.23
2282Reducing differentials modulo cohomology relations.
2283281 -771426215266877657517824927117197317064053922031 + O(281^20) 0
2284Computing (y^Frobenius)^(-1)
2285Expansion time: 13.23
2286Reducing differentials modulo cohomology relations.
2287283 -3909759751225677755196122951554393508916782635068 + O(283^20) 0
2288Computing (y^Frobenius)^(-1)
2289Expansion time: 15
2290Reducing differentials modulo cohomology relations.
2291293 9650766525262982951395332384521495636939673842618 + O(293^20) 0
2292Computing (y^Frobenius)^(-1)
2293Expansion time: 15.05
2294Reducing differentials modulo cohomology relations.
2295307 16215076042046210153447355333778959437456886944269 + O(307^20) 0
2296Computing (y^Frobenius)^(-1)
2297Expansion time: 15.03
2298Reducing differentials modulo cohomology relations.
2299311 9924568793992273436656018879412285712710587501738 + O(311^20) 0
2300Computing (y^Frobenius)^(-1)
2301Expansion time: 15.321
2302Reducing differentials modulo cohomology relations.
2303313 16338997345749621985343710006034679174320839*313^-2 + O(313^16) -2
2304Computing (y^Frobenius)^(-1)
2305Expansion time: 15.141
2306Reducing differentials modulo cohomology relations.
2307317 -8180959439659097002455664090975829418728238319179 + O(317^20) 0
2308Computing (y^Frobenius)^(-1)
2309Expansion time: 15.5
2310Reducing differentials modulo cohomology relations.
2311331 106324002766280001349902268973298443532764328404373 + O(331^20) 0
2312Computing (y^Frobenius)^(-1)
2313Expansion time: 15.559
2314Reducing differentials modulo cohomology relations.
2315337 1487641797148146783174749484529981177684649280*337^-2 + O(337^16) -2
2316Computing (y^Frobenius)^(-1)
2317Expansion time: 15.01
2318Reducing differentials modulo cohomology relations.
2319347 288881056768437771513194445718327926423466873535164 + O(347^20) 0
2320Computing (y^Frobenius)^(-1)
2321Expansion time: 14.851
2322Reducing differentials modulo cohomology relations.
2323349 -324627174783822167409446159532805108501548546483875 + O(349^20) 0
2324Computing (y^Frobenius)^(-1)
2325Expansion time: 15.359
2326Reducing differentials modulo cohomology relations.
2327353 -150041228084538689303273049361188586637161356864518 + O(353^20) 0
2328Computing (y^Frobenius)^(-1)
2329Expansion time: 15.1
2330Reducing differentials modulo cohomology relations.
2331359 508244171233401628697951496404505641462642110066758 + O(359^20) 0
2332Computing (y^Frobenius)^(-1)
2333Expansion time: 15.769
2334Reducing differentials modulo cohomology relations.
2335367 -416527758588057821093991552213449352177065535454474 + O(367^20) 0
2336Computing (y^Frobenius)^(-1)
2337Expansion time: 19.52
2338Reducing differentials modulo cohomology relations.
2339373 -1185597183209223899081873937523134791093301772229805 + O(373^20) 0
2340Computing (y^Frobenius)^(-1)
2341Expansion time: 19.569
2342Reducing differentials modulo cohomology relations.
2343379 642664726719383534578067014671603665079200147802845 + O(379^20) 0
2344Computing (y^Frobenius)^(-1)
2345Expansion time: 19.611
2346Reducing differentials modulo cohomology relations.
2347383 -1980679639877875794142899451152825399982395507331373 + O(383^20) 0
2348Computing (y^Frobenius)^(-1)
2349Expansion time: 19.711
2350Reducing differentials modulo cohomology relations.
2351389 912910025622472388115214247783732215543517477434958 + O(389^20) 0
2352Computing (y^Frobenius)^(-1)
2353Expansion time: 19.819
2354Reducing differentials modulo cohomology relations.
2355397 -2289985440897127570150400503318597019362572185376739 + O(397^20) 0
2356Computing (y^Frobenius)^(-1)
2357Expansion time: 19.92
2358Reducing differentials modulo cohomology relations.
2359401 2662460691462671991566057040285291569696404461033180 + O(401^20) 0
2360Computing (y^Frobenius)^(-1)
2361Expansion time: 22.52
2362Reducing differentials modulo cohomology relations.
2363409 -6035761433262382557092768771901984647658378844632414 + O(409^20) 0
2364Computing (y^Frobenius)^(-1)
2365Expansion time: 24.969
2366Reducing differentials modulo cohomology relations.
2367419 6966665477845050730634394930098503632468159092330232 + O(419^20) 0
2368Computing (y^Frobenius)^(-1)
2369Expansion time: 24.98
2370Reducing differentials modulo cohomology relations.
2371421 -8722498142436839434340378205352120473632018692238306 + O(421^20) 0
2372Computing (y^Frobenius)^(-1)
2373Expansion time: 25.289
2374Reducing differentials modulo cohomology relations.
2375431 -23154404799097883257764550108478971805586495350456898 + O(431^20) 0
2376Computing (y^Frobenius)^(-1)
2377Expansion time: 25.36
2378Reducing differentials modulo cohomology relations.
2379433 100101112887074728257664136622452394504319579883*433^-2 + O(433^16) -2
2380Computing (y^Frobenius)^(-1)
2381Expansion time: 25.449
2382Reducing differentials modulo cohomology relations.
2383439 27100444100108260647618106076004738147820891403030283 + O(439^20) 0
2384Computing (y^Frobenius)^(-1)
2385Expansion time: 25.471
2386Reducing differentials modulo cohomology relations.
2387443 6219765964400288347185296345919434522982680131332921 + O(443^20) 0
2388Computing (y^Frobenius)^(-1)
2389Expansion time: 25.609
2390Reducing differentials modulo cohomology relations.
2391449 2367703250647633532500453909151780788048681517906996 + O(449^20) 0
2392Computing (y^Frobenius)^(-1)
2393Expansion time: 25.731
2394Reducing differentials modulo cohomology relations.
2395457 -48835807557429023353520353913148898222094238891889268 + O(457^20) 0
2396Computing (y^Frobenius)^(-1)
2397Expansion time: 25.8
2398Reducing differentials modulo cohomology relations.
2399461 -26453166166427471420028949259756585259124118028066374 + O(461^20) 0
2400Computing (y^Frobenius)^(-1)
2401Expansion time: 25.879
2402Reducing differentials modulo cohomology relations.
2403463 57576177554821815956381222050012596877468570136176904 + O(463^20) 0
2404Computing (y^Frobenius)^(-1)
2405Expansion time: 31.471
2406Reducing differentials modulo cohomology relations.
2407467 113311978683408496093899189133874063724678183358071625 + O(467^20) 0
2408Computing (y^Frobenius)^(-1)
2409Expansion time: 31.721
2410Reducing differentials modulo cohomology relations.
2411487 5978459439427873076881614773786500598115284065894479 + O(487^20) 0
2412Computing (y^Frobenius)^(-1)
2413Expansion time: 31.929
2414Reducing differentials modulo cohomology relations.
2415499 -337366512483466327430173577004337987723710229937060602 + O(499^20) 0
2416>