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Comment:

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Deletions are marked like this.  Additions are marked like this. 
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I = C.cuspidal_subgroup().invariants()  I = J.cuspidal_subgroup().invariants() 
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def cuspidal_subgroup_J0(N):  def cuspidal_subgroup_J1(N): 
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I = C.cuspidal_subgroup().invariants()  I = J.cuspidal_subgroup().invariants() 
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BUT WAIT  isn't there an ''a priori'' formula for this structure/order? Yes  Ligozat, but not really  that gives only rational cuspidal subgroup, and might be just as hard. Anyway, I'm computing a few of these here, as a test of the modular symbols code, etc., since this is easy: http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J0N/ http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J1N/ 
Specific Tables
Contents
Component Groups of J0(N)(R) and J1(N)(R)
URL: http://wstein.org/Tables/real_tamagawa/ and http://wstein.org/Tables/compgrp/. The second page has much more extensive data and a conjecture.
 New Code:
This function computes the J_0(N) real component groups.
def f(N): M = ModularSymbols(N).cuspidal_subspace() d = M.dimension()//2 S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list())  1 return 2^(S.nullity()d)
For J_1(N) it is:
def f(N): M = ModularSymbols(Gamma1(N)).cuspidal_subspace() d = M.dimension()//2 S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list())  1 return 2^(S.nullity()d)
Future extension: one could replace Gamma1(N) by GammaH(N,...). One could also do the new subspace.
And note Frank's conjecture:
Conjecture (Frank Calegari):
Let m = #odd prime factors of N + {1, if N = 0 mod 8 {0, otherwise. Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m  1.
the above conjecture is wrong, but the following matches our data (up to level N<=2723):
Conjecture (BoothbyStein): Let m = #odd prime factors of N  {1, if N != 0 mod 8 {0, otherwise. Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m  1, unless N=1,2,4, in which case the component is
Soroosh  the prime level case is known. See Calegari <insert link> which *just* cites Agashe and Merel (http://wstein.org/home/wstein/days/17/Merel_Laccouplement_de_Weil_entre_le_sousgroup.pdf  page 12).
Cuspidal Subgroup
Computing the structure of the cuspidal subgroup of J0(N) and J1(N) (say).
URL: http://wstein.org/Tables/cuspgroup/ (the displayed formula is backwards at the top)
 New Sage code:
def cuspidal_subgroup_J0(N): J = J0(N) I = J.cuspidal_subgroup().invariants() # maybe pickle J return I
def cuspidal_subgroup_J1(N): J = J1(N) I = J.cuspidal_subgroup().invariants() # maybe pickle J return I
BUT WAIT  isn't there an a priori formula for this structure/order? Yes  Ligozat, but not really  that gives only rational cuspidal subgroup, and might be just as hard.
Anyway, I'm computing a few of these here, as a test of the modular symbols code, etc., since this is easy:
http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J0N/
http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J1N/
Discriminants of Hecke Algebra
Computation of discriminants of various Hecke algebras.
Amazingly, it seems that there is "discriminants of Hecke algebras" implementation in Sage! Here is a straightforward algorithm:
The input is the level N.
Chose a random vector v in the space M of cuspidal modular symbols of level N.
Compute the sturm bound B.
Compute the products T_1(v), ..., T_B(v), and find a basis b_i for the ZZmodule they span.
Find Hecke operators S_1, ..., S_n such that S_i(v) = b_i. (This is linear algebra  inverting a matrix and a matrix multiply.)
Compute the determinant det ( Trace(S_i * S_j) ). That is the discriminant. This also gives a basis for the Hecke algebra, which is very useful for lots of things.
Note: See http://trac.sagemath.org/sage_trac/ticket/6768 for very slow code for computing a basis for the Hecke algebra.
Here is a more complicated algorithm, but it might suck because of hidden denseness!
The input is the level N.
If N is divisible by a prime p^3 and X_0(N/p^3) has positive genus, then the discriminant is 0, as one can see by taking images of forms of level N/p^3.
I think the above is an if and only if condition for when the discriminant is 0. See I think ColemanVoloch.
 The actual algorithm now.
Find a random Hecke operator t such that the charpoly of t has nonzero discriminant.
Choose a random vector v in the space of cuspidal modular symbols.
Let B be the Sturm bound.
Compute the images T_n(v) for n up to the Sturm Bound.