NOTE: For this assignment, make sure you know how to make
for loops in Python or you will be screwed. If not read the sage tutorial, especially http://doc.sagemath.org/html/en/tutorial/programming.html#loops-functions-control-statements-and-comparisons
Let be a weight 2 modular form for some congruence subgroup . Check by hand that for all we have . Here acts by linear fractional transformations.
Let be the weight cusp form .
(a) Supersingular primes: Find all primes such that .
(b) A big one?: There is one more known big prime with . Find it using the Internet.
(c) A congruence: Find a prime such that for all .
Noam Elkies provided that if is an elliptic curve, then there are infinitely many primes such that .
Write down an elliptic curve of conductor 43. How many primes are supersingular for ? (Hint: use
Plot a histogram showing the gaps between supersingular primes. (Hint: use
Compute the class groups of all of the number fields generated by the coefficients of the newforms of weight on for each . Make a conjecture.