CoCalc Shared Filessupport / 2017-02-13-103620-cuspidal.sagews
Author: William A. Stein
Description: Examples for support purposes.

From: Yun Cheng

I'm a Ph.D student from UChicago, studying with Prof. Emerton. Currently I'm trying to understand the cuspidal subgroup, especially the order of $(0)-(\infty)$, for $J_0(pq)$, where $p$ and $q$ are distinct primes.

I found in the page of Prof. Stein http://wstein.org/Tables/cuspgroup/index.html where it's mentioned that prof. Ogg has done calculation for the cuspidal subgroup for this case, but I tried to search online and couldn't find any reference for that.

def ord0oo(N, sign=0):
"""
Compute the order of $(0)-(\infty)$ in $J_0(N)$.

INPUT:
- N - positive integer
- sign - 0 or 1; if 0 gives correct answer.
- if 1, then code may be 10x faster, but power of 2 that divides the order may be WRONG.
OUTPUT:
- an integer, the order of the class of 0-oo.
"""
M = ModularSymbols(N, sign=sign)
z = M([0,oo])
S = M.cuspidal_subspace()
phi = S.integral_period_mapping()
return phi(z).denominator()

for N in [11..100]:
print N, ord0oo(N)

11 5 12 1 13 1 14 6 15 4 16 1 17 4 18 1 19 3 20 6 21 4 22 5 23 11 24 4 25 1 26 21 27 3 28 6 29 7 30 24 31 5 32 4 33 10 34 12 35 24 36 6 37 3 38 45 39 28 40 12 41 10 42 48 43 7 44 15 45 8 46 22 47 23 48 8 49 2 50 15 51 48 52 42 53 13 54 9 55 20 56 24 57 30 58 35 59 29 60 24 61 5 62 120 63 48 64 4 65 42 66 120 67 11 68 72 69 44 70 144 71 35 72 12 73 6 74 171 75 40 76 45 77 60 78 168 79 13 80 24 81 9 82 70 83 41 84 48 85 24 86 231 87 140 88 60 89 22 90 24 91 56 92 66 93 80 94 92 95 180 96 16 97 8 98 42 99 40 100 30
%time ord0oo(389)

97 CPU time: 1.31 s, Wall time: 1.36 s