If Conjecture 1 is true, then this provides a fast way of computing the CuspidalClassGroup,
see:  http://www.math.fsu.edu/~hoeij/files/X1N/cusp_divisors_program   for a Maple
implementation.

Sage can compute the CuspidalClassGroup without using Conjecture 1.  If the orders of the
groups match for X1(N) then the conjecture is verified for N. As an additional test, we
compare the group structure as well.

The file verify_conjecture1.sage checks N in the range 13..100.

It obtains the conjecture-based-CuspidalClassGroup by downloading the file
cusp_divisors.  It compares this group with the actual CuspidalClassGroup.

(Note: instead of downloading the file cusp_divisors, you can also download the program 
https://github.com/koffie/mdsage/blob/master/modular_unit_divisors.py   which is a Sage
version of the Maple program cusp_divisors_program.  In the notation of Section 2 of
the paper, the program computes the divisors of the modular units F_k while the file
cusp_divisors lists the divisors of the f_k. But these generate the same Z-module, so
either can be used to find the conjecture-based-CuspidalClassGroup).

The reason the N's don't appear sorted in the output is because it runs 4 N's at the
same time with the @parallel(ncpus=4) command.

We also checked 101..120 with a similar computation, and verified 3..12 as well.
So Conjecture 1 is true for N <= 120, a strong indication that it may be true in general.

For prime numbers p, there is a formula for the order of the CuspidalClassGroup.
In the file verify_conjecture1.maple we use that formula to test the conjecture
for all primes < 1000.