Programs and data for "The generalized Fermat equation xᵖ + yᵖ = z⁵ and relative class numbers"

This package contains the programs, with their stored outputs, for the finite computations listed in Appendix B of the paper as (C1)–(C9) (the proof of Proposition 3.10 uses (C3), Appendix A uses (C6)–(C8), (C2) recomputes the dimensions of Lemma 3.5, and the others give the counts of Section 10 and support the statements in the text where they are cited): the primes 7 ≤ p < 10⁵, p ≢ 1 (mod 5), with 5 ∤ h⁻(ℚ(ζ₅ₚ)); Hilbert cusp forms over ℚ(√5) of level q₅ᵉ, e ≤ 2 (the paper also cites e = 3 as a check; the programs go to e = 5); Frobenius polynomials of genus-2 curves over 𝔽₁₉ and 𝔽₇₉; the 5-adic valuations of h⁻(M′) and h⁻(L) and the kernel of localization at 5 for p < 60; the counts of Table 3 and of Sections 1 and 10, with the values of the heuristic in Remark 10.1; the Frobenius traces of the Frey curve at the primes above 11, the Hilbert newforms over ℚ(√5) at the levels dividing q₂²q₅³ (from the LMFDB and from the Brandt module of the icosian ring) and the newform 2.2.5.1-2000.1-h, by the programs of the earlier paper "The generalized Fermat equation x¹¹ + y¹¹ = z⁵", rerun; and the worked examples for p = 7 and p = 19 of Example 8.8. Section, statement, table and appendix numbers are those of the paper. All arithmetic is exact (integers, rational numbers, finite fields, number fields given by explicit polynomials, and quaternion orders), apart from one cross-check of exact class numbers noted under (C1), a second evaluation in floating point of exact products in `localization/s05_hminus_ML.gp`, and the probabilities of the heuristic of Remark 10.1, which are computed in floating point; no computation assumes the generalized Riemann hypothesis: whether 5 divides a relative class number is decided from generalized Bernoulli numbers through the analytic class number formula, without computing a class group. Each computation was repeated by a separately written program; the repeats are stored next to the first programs, and the README (section "Second programs") names them, item by item. The README gives, for each computation, the commands that run it, their running times and their output files; `rerun_sample.sh` reruns a sample in about four minutes and compares it with the stored outputs. Requirements: PARI/GP 2.17; Python 3.12 with NumPy, SymPy and python-flint 0.8.