Validate Euler factor links to abelian varieties over Fq on L-function pages

[roed-math]

L-function pages of motivic weight 1 link each good-prime Euler factor to the corresponding
isogeny class of abelian varieties over F_p (feature introduced in LMFDB#6573). This PR hardens that
linking: a new Lfactor_to_gq check confirms the factor is genuinely the reciprocal Weil
polynomial of an abelian variety over F_p (even degree, integral, lead p^g, self-dual) before a
label is displayed, so malformed or non-Weil factors now give an empty cell instead of a
ValueError-driven 500 or a wrong/garbage label, while out-of-range (g,q) still shows a plain
unlinked label. The av/Fq coverage lookup now happens lazily at first use instead of at module
import, and the row's prime is passed in so the label can be checked against it. Verified
against av_fq_isog data (11.a links 1.2.c at p=2 with 5 = #E(F_2) points; genus 2 curve 169.a
links 2.2.d_f, its reduced Jacobian) and covered by new tests in test_lfunctions.py; the full
lfunctions suite passes.
Addresses LMFDB#4535.
Addresses LMFDB#4922.

🤖 Generated with Claude Code

[roed-math]

Addressed both of the external review's P2 findings: commit b0a99a2.

Weil condition: the validator now requires the reversed factor to pass Sage's is_weil_polynomial() — exact arithmetic (squared-roots trace polynomial + Sturm counting via PARI polsturm, no floats). The reviewer's [1,10,2] example is rejected in both flat and factored form (regression tests added); the supersingular boundary case (x+2)² over F₄ is correctly accepted. Measured cost 26–53 µs/call, ~1 ms/page worst case.

Beyond the review: while verifying, the agent found that Weil ⟹ exists-in-DB only holds for prime q (Honda–Tate): x²+25 is Weil but 1.25.a doesn't exist. Labels for proper prime-power q are now confirmed against av_fq_isog before linking — a cold path, since current pages always pass prime p, so zero added queries on real pages.

Empty/malformed payloads: polynomial_unroll now raises ValueError on empty/non-list input instead of IndexError, an exponent-0 factorization no longer hits an uncaught NotImplementedError (a second 500 path found en route), and the caller catches these → empty cell, not a 500.

27/27 lfunctions tests pass; av/Fq links verified identical before/after on six real pages (11.a→1.2.c etc.).

[roed-math]

Addressed the review in b3962a1. Behavior for valid local factors is unchanged throughout: valid in-range factors still link to av/Fq, valid factors outside database coverage still show a plain label, and bad primes still get no link.

Required change 1: the row renderer. row() inside lfuncEPhtml indexed poly[0] and picked a formatter before its try, so poly=[] raised IndexError and malformed factorizations like [[]] and [[[1,2,2]]] raised ValueError (which the except IndexError did not catch either) straight out of the page. The fallback also used the outer loop variable j instead of the row's prime, emitted duplicated closing tags, and produced the wrong number of cells.

Row preparation now lives in a module-level euler_factor_row_data(poly, p, display_galois, complex_coefficients, isogeny_label) returning (factors, gal_groups, isog_class). All indexing, the None in poly test, formatter selection and formatter invocation happen inside its exception boundary, which catches only TypeError/ValueError/ArithmeticError/IndexError (the same set Lfactor_to_gq already used) rather than a bare except Exception. row() builds the HTML once from that triple, so there is no second inconsistent row. Malformed data give one valid row with the row's own p, the factor not available, and an empty isogeny cell.

One extra fix in the same area, worth a look since it wasn't in the review: the show more / show less toggle rows built last_entry accounting for the Galois column but not the isogeny column, so those rows were one cell short of the header whenever the isogeny column was displayed. The new test asserts the toggle rows span the table, which is what caught it.

Required change 2: exponents. polynomial_unroll expanded factorizations with [fac] * exponent, and Python turns both zero and negative repetitions into an empty list, so a malformed factor could disappear while a valid one remained. Every entry is now validated (two elements, nonempty coefficient list, strictly positive integer exponent, bool rejected explicitly since it is an int subclass and True in ZZ), and the factors are accumulated by exponentiation rather than list expansion. Both cases from the review now return None:

Lfactor_to_gq([[[1, 2, 2], 1], [[1, 10, 2], 0]], 2) is None
Lfactor_to_gq([[[1, 2, 2], 1], [[1, 10, 2], -1]], 2) is None

Test hardening. test_euler_factor_isogeny_class_out_of_range no longer assumes dimension 3 data stop at q=25; it patches AbvarExists to return False and asserts the plain label with no anchor, then checks the unpatched call does produce the link. The real-page assertions for a known link are kept, minus the one that would break as av_fq_isog grows.

New tests: test_Lfactor_exponent_validation (zero, negative, nonintegral, True, plus the valid repeated-factor cases), test_euler_factor_row_data (the extracted helper, including that gal_groups is never empty since the caller indexes gal_groups[0]), and test_euler_product_table_malformed_factor, which goes through lfuncEPhtml itself for both 2-11-1.1-c1-0-0 (no Galois column) and 4-13e2-1.1-c1e2-0-0 (Galois column), asserting the fallback row uses the prime 3 and that the row and toggle rows match the header's column count. Those three tests fail on d415e64 and pass here.

Verification. sage -python -m pytest lmfdb/lfunctions/test_lfunctions.py gives 30 passed. pyflakes over lmfdb/, pylint and ruff (the tox -e lint targets) are clean. The elliptic curve and genus 2 isogeny class pages also call Lfactor_to_label/AbvarExists, but with unfactored coefficient lists, so they never reach the changed factorization path; I confirmed /EllipticCurve/Q/11/a/ and /Genus2Curve/Q/169/a/ still render with their av/Fq links.

[roed314]

GPT signed off.

[roed-math]

Superseded by LMFDB#7165, opened upstream from this same branch. Closing here; review continues upstream.