Is infinity a Weierstrass point on X0(N)?

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The Weierstrass point bound enumerated here is an upper bound on the difference of the genus g of X0(N) and the dimension of the space spanned by the Hecke operators T1,...,Tg. If the bound is 0, then infinity is not a Weierstrass point; if the bound is nonzero, then, with high probability, infinity is a Weierstrass point.

The following table contains all the data for levels <= 3314: 

   weierstrass_point_bound.txt
   weierstrass_point_bound.m
Enter a semicolon-separated list of levels N to obtain a list of lists of "Weierstrass point bounds" for the corresponding modular curves X0(N).

WARNING:When X0(N) has genus 0 or 1, I consider infinity to not be a Weierstrass point.

These computations imply that for every square-free integer N<=3223, infinity is NOT a Weierstrass point.

Conjecture: If N is square free, then infinity is not a Weierstrass point.

If you are interested in general results about whether or not infinity is a Weierstrass point, see Atkin's 1967 Annals of Mathematics paper. For analogous computations on X0(p)+, see this page.

I have computed this bound at least for all integers N up to 3314. Click the "List known levels" button below to see exactly what I've computed.




Output format:           HUMAN                MAGMA