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Author: Kevin Lui
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%auto %default_mode magma

newform in abvar.py

Newform(JOne(11));
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + O(q^12)
for S in Decomposition(JZero(33)) do Newform(S); end for;
q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 4*q^7 - 3*q^8 + q^9 - 2*q^10 + q^11 + O(q^12) q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + O(q^12) q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + O(q^12)

conductor in abvar.py

Factorization(Conductor(JZero(23)));
[ <23, 2> ]
Factorization(Conductor(JOne(25)));
[ <5, 24> ]
Factorization(Conductor(JZero(11^2)));
[ <11, 10> ]
Factorization(Conductor(JZero(33)(2)));
[ <11, 1> ]

number_of_rational_points in abvar.py

NumberOfRationalPoints(JZero(23));
11 11
NumberOfRationalPoints(JZero(29));
7 7
NumberOfRationalPoints(JZero(37));
Infinity Infinity
JZero(12); // NumberOfRationalPoints(JZero(12)); bug in Magma
Modular abelian variety JZero(12) of dimension 0 and level 2^2*3 over Q
NumberOfRationalPoints(JOne(17)); // sage does better
1 584
NumberOfRationalPoints(JOne(16));
1 20

frobenius_polynomial in abvar.py

FrobeniusPolynomial(JZero(39)(2), 5);
$.1^4 + 2*$.1^2 + 25
FrobeniusPolynomial(JZero(23), 997);
$.1^4 + 20*$.1^3 + 1374*$.1^2 + 19940*$.1 + 994009
FrobeniusPolynomial(JZero(33),7);
$.1^6 + 9*$.1^4 - 16*$.1^3 + 63*$.1^2 + 343
FrobeniusPolynomial(JZero(19), 3);
$.1^2 + 2*$.1 + 3
FrobeniusPolynomial(JZero(3), 11);
1
FrobeniusPolynomial(JOne(27)(2), 11);
$.1^24 - 3*$.1^23 - 15*$.1^22 + 126*$.1^21 - 201*$.1^20 - 1488*$.1^19 + 7145*$.1^18 - 1530*$.1^17 - 61974*$.1^16 + 202716*$.1^15 - 19692*$.1^14 - 1304451*$.1^13 + 4526883*$.1^12 - 14348961*$.1^11 - 2382732*$.1^10 + 269814996*$.1^9 - 907361334*$.1^8 - 246408030*$.1^7 + 12657803345*$.1^6 - 28996910448*$.1^5 - 43086135081*$.1^4 + 297101409066*$.1^3 - 389061369015*$.1^2 - 855935011833*$.1 + 3138428376721

__call__ in lseries.py

Evaluate(LSeries(JZero(23)), 1, 5000);
0.248431866590599681207250339320
Evaluate(LSeries(JZero(389)(1)), 1); LRatio(JZero(389)(1), 1);
0.000000000000000000000000000000 0
Evaluate(LSeries(JOne(11) * JZero(11)), 1);
0.0644356903227915209449222862420
Evaluate(LSeries(JH(17, 2)), 1);
0.386769938387780043302396512438

vanishes_at_1 in lseries.py

IsZeroAt(LSeries(JZero(389)(1)), 1);
true
IsZeroAt(LSeries(JOne(23)(1)), 1);
false

rational_part in abvar.py

LRatio(LSeries(JZero(43)(1)),1);
0
LRatio(LSeries(JZero(43)(2)),1);
2/7

order in torsion_subgroup.py

TorsionLowerBound(JZero(11)); TorsionMultiple(JZero(11));
5 5
TorsionLowerBound(JZero(23)); TorsionMultiple(JZero(23));
11 11
TorsionLowerBound(JZero(37)(2)); TorsionMultiple(JZero(37)(2));
3 3
// TorsionLowerBound(JOne(13)); not in Magma TorsionMultiple(JOne(13));
19
// TorsionLowerBound(JOne(23)); not in Magma TorsionMultiple(JOne(23));
408991

possible_orders in torsion_subgroup.py

// TorsionLowerBound(JOne(13)); not in Magma TorsionMultiple(JOne(13));
19
TorsionMultiple(JOne(16));
20

multiple_of_order in torsion_subgroup.py

TorsionMultiple(JOne(11));
5
TorsionMultiple(JZero(17));
4
TorsionMultiple(JOne(23));
408991

multiple_of_order_using_frobp in torsion_subgroup.py

TorsionMultiple(JZero(11) * JZero(33));
1000
TorsionMultiple(JOne(23));
408991
TorsionMultiple(JOne(19) * JZero(21));
35064