Fix bug with pari interface in classical_modular_polynomial

[drago-96]

There is a bug in the new classical_modular_polynomial function created in #36190.

A simple example to reproduce the bug is the following

p = 2^216 * 3^137 - 1
F.<i> = GF(p^2, modulus=[1,0,1])
E = EllipticCurve(F, [0, 6, 0, 1, 0])
classical_modular_polynomial(2, E.j_invariant())

This will fail with a TypeError.

The bug is due to the interface with the pari function polmodular. In particular, contrary to the documentation, the polmodular function will only evaluate $\Phi_\ell(j, Y)$ for $j$-invariants that are defined over $\mathbb F_p$, and not over any generic finite field.
If however j.parent() is a generic finite field, but j itself is defined over the prime field, pari will evaluate that and the current implementation fails to convert back to a sage polynomial.

The proposed fix is to cast the result of the pari call to ZZ['Y'], since the result of polmodular currently returns a polynomial in the correct $\mathbb Z/n\mathbb Z[Y]$.

#sd123

[fchapoton]

you could use the syntax GF((p,2),...) to avoid factoring.

[drago-96]

Thanks, I didn't know this constructor!

[github-actions]

Documentation preview for this PR (built with commit b372660; changes) is ready! 🎉

[pjbruin]

Shouldn't the result be converted into a polynomial over Z/pZ where p is the characteristic of the finite field?

[grhkm21]

Shouldn't the result be converted into a polynomial over Z/pZ where p is the characteristic of the finite field?

Why? I think the PR is fine, it returns a polynomial over parent(j), which is what you want normally anyways (to compute its roots later etc.).

[pjbruin]

You are right. I was puzzled by the conversion to Z[Y], and did not notice that changing Z to Z/pZ would not make a difference, due to the final conversion to R.

The fact that polmodular returns a polynomial over a different field than the one in which j lies does seem to be a possible PARI bug and should probably also be reported there.

[pjbruin]

Likely a PARI bug, the patch looks like a good workaround to me

[pjbruin]

Now fixed in PARI: https://pari.math.u-bordeaux.fr/cgi-bin/bugreport.cgi?bug=2533

[drago-96]

Thanks for the reporting and fixing in PARI!