alkahest.pari

A Galois orbit of normalised Hecke eigenforms of the new subspace.

field is None for rational eigenvalues, otherwise the NumberField Q[y]/(polynomial) containing them. coefficients is [a_1, ..., a_nmax] with a_n an int / Fraction (rational orbit) or a NumberFieldElement.

degree: int

polynomial: list[int]

field: Any

coefficients: list[Any]

What elliptic_curve_invariants() returns.

ainvs (a1, a2, a3, a4, a6) as Fraction s. discriminant, conductor int s (discriminant is of the given model). ap {p: a_p} for primes p <= ap_bound (a_p for bad p is the PARI convention 0/±1). torsion {"order", "structure", "generators"}; generators are exact (x, y) Fraction pairs on the given model. root_number ±1. analytic_rank / analytic_rank_status PARI's numerical analytic rank; status "proven" when Alkahest's own series evaluation shows the leading L-value is non-zero for rank 0 or 1, otherwise "numerical". l_value L(E, 1) (rank 0) or L'(E, 1) (rank 1; L(E,1) = 0 exactly by the functional equation) as an ArbBall computed from a_n with a rigorous truncation bound, or None when the analytic rank is >= 2 (value not enclosed). mw_rank_bounds / mw_rank_status (lower, upper) from ellrank (2-descent); status "proven" when they agree, else "bounds_only" (the true rank may be anywhere in the range; only conditional statements such as BSD give more). mw_points Independent points of infinite order found by ellrank. They need not generate the Mordell–Weil group (no saturation).

ainvs: tuple[Fraction, ...]

discriminant: int

conductor: int

ap: dict[int, int]

torsion: dict[str, Any]

root_number: int

analytic_rank: int | None

analytic_rank_status: str

l_value: Any

mw_rank_bounds: tuple[int, int] | None

mw_rank_status: str

mw_points: list[tuple[Fraction, Fraction]]

verification: dict[str, Any]

Cusp forms S_k(Gamma_0(N)) with trivial character.

dimension dim S_k(Gamma_0(N)) (full space, old forms included). basis dim lists [a_0, ..., a_nmax] of exact Fraction / int q-expansion coefficients (PARI's basis; not an eigenbasis). new_dimension / orbits New-subspace dimension and its Hecke eigenform orbits.

level: int

weight: int

nmax: int

dimension: int

basis: list[list[Any]]

new_dimension: int

orbits: list[EigenformOrbit]

verification: dict[str, Any]

What number_field_invariants() returns.

field

The NumberField the data is expressed in.

discriminant

d_K, an int.

signature

(r1, r2).

integral_basis

A Z-basis of the ring of integers as NumberFieldElement s.

class_number, class_group

h and the cyclic invariants [d1, d2, ...] (d1 | d2 | ...) or None when class_group=False.

torsion_units

(w, zeta): the order of the roots of unity in K and a generator.

fundamental_units

r1 + r2 - 1 multiplicatively independent units (NumberFieldElement).

regulator

PARI's decimal string for the regulator (not an exact object).

primes

{p: [PrimeIdeal, ...]} for the primes requested.

galois

{"order", "parity", "name", "transitive_index"} or None.

certification

"certified" (bnfcertify succeeded: h, units, regulator are proven), "grh_conditional" (not run, too expensive, or failed), or "not_computed" when class_group=False.

verification

{check name: True} for every check Alkahest ran, and a "not_checked" list naming what rests on PARI alone.

field: Any

discriminant: int

signature: tuple[int, int]

integral_basis: list[Any]

class_number: int | None

class_group: list[int] | None

torsion_units: tuple[int, Any] | None

fundamental_units: list[Any] | None

regulator: str | None

primes: dict[int, list[PrimeIdeal]]

galois: dict[str, Any] | None

certification: str

verification: dict[str, Any]

A prime ideal P = (p, alpha) above a rational prime p.

e is the ramification index, f the residue degree, norm = p**f and generators = (p, alpha) a two-element presentation with alpha a NumberFieldElement of the field passed in.

p: int

e: int

f: int

norm: int

generators: tuple[Any, Any]

Base error of pari; .code is the stable E-PARI-NNN.

No PARI binding (cypari2 / cypari) is installed (E-PARI-001).

True when a PARI binding (cypari2 or cypari) can be loaded.

Name of the binding in use ("cypari2" / "cypari") or None.

Conductor, a_p, torsion, rank and L-value of an elliptic curve over Q.

The curve is y^2 + a1 xy + a3 y = x^3 + a2 x^2 + a4 x + a6.

ainvs

[a1, a2, a3, a4, a6] or [a4, a6] (integers or Fractions; the curve must be non-singular). Non-integral coefficients are refused with E-PARI-002.

ap_bound

Return a_p for all primes p <= ap_bound.

rank

Run ellrank (2-descent) and ellanalyticrank.

l_value

Evaluate the leading L-value as a ball.

discriminant is recomputed from the b-invariants; the conductor's primes must divide it; a_p is recounted by brute-force point counting for good primes p <= 1000 and checked against Hasse; each torsion generator is checked to lie on the curve and its order is recomputed in Alkahest's own Pic^0 (FunctionField / DivisorClass) on the isomorphic short model — it must equal the order PARI claims (this certifies the listed points, not that no further torsion exists); the analytic rank must match the parity of the root number; L(E,1) (or L'(E,1)) is recomputed from the a_n with a rigorous truncation bound and compared with PARI's lfun; each Mordell–Weil point lies on the curve and is shown to have infinite order by Pic^0.

analytic_rank_status / mw_rank_status as described on EllipticCurveData; modularity (the theorem that L(E,s) has the functional equation used above) is assumed.

>>> from alkahest import pari
>>> e = pari.elliptic_curve_invariants([0, -1, 1, -10, -20])   # 11a1  # doctest: +SKIP
>>> e.conductor, e.torsion["order"], e.analytic_rank           # doctest: +SKIP
(11, 5, 0)

q-expansions and Hecke eigenvalues of S_k(Gamma_0(N)) via PARI's mf* suite.

N, k

Level N >= 1 and integer weight k >= 1 (trivial character).

nmax

Number of coefficients to return (a_0..a_nmax / a_1..a_nmax).

Returns exact coefficients: int / Fraction for the basis and for rational eigenforms, NumberFieldElement in the orbit's coefficient field otherwise.

Every eigenform is re-checked in exact arithmetic: a_1 = 1, full multiplicativity and the prime-power Hecke recursion a_{p^{r+1}} = a_p a_{p^r} - [p ∤ N] p^{k-1} a_{p^{r-1}} for all indices <= nmax; each a_n is an algebraic integer (integral minimal polynomial); the orbit degrees sum to dim S_k^new. For (N, k) = (1, 12)` the form is compared with `q prod (1 - q^n)^24 (Ramanujan's tau). A failure raises E-PARI-003. The list of orbits being complete is PARI's claim (not re-derived).

>>> from alkahest import pari
>>> pari.modular_forms(1, 12).basis[0][:4]      # tau(n)  # doctest: +SKIP
[0, 1, -24, 252]

Ring of integers, class group, units, prime splitting and Galois group of a number field.

field_or_poly

An experimental.NumberField, or an ascending integer coefficient list of a monic irreducible polynomial.

primes

Rational primes whose prime-ideal factorisation to return.

class_group

Run bnfinit (class group, units, regulator). Disable for large discriminants.

galois

Compute the Galois group with polgalois (degree <= 7 in a stock PARI; refused with E-PARI-002 above that unless the galdata package is installed, in which case PARI answers up to degree 11).

certify

True always runs bnfcertify (can be very slow), False never, "auto" (default) runs it when degree <= 6 and |d_K| < 10**15. The outcome decides certification.

All of the following are recomputed in Alkahest's exact arithmetic and a failure raises E-PARI-003: the integral basis spans an order of discriminant d_K (Gram determinant of Tr(w_i w_j)), is closed under multiplication, consists of algebraic integers, disc(f) = index^2 d_K, d_K has the right sign and residue mod 4; every unit has norm ±1 and the torsion generator has the stated order; each prime ideal's two-element presentation spans a lattice of index exactly p**f, sum e f = n, and p | d_K exactly when some e > 1; the Galois parity matches whether disc(f) is a square; for imaginary quadratic fields h is recounted from reduced binary quadratic forms.

Maximality of the order, fundamentality of the units and the value of the class group are not re-derived (they are listed under verification["not_checked"]); they hold unconditionally only when certification == "certified".

PariUnavailable

No PARI binding (E-PARI-001).

PariError

E-PARI-002 for non-monic or reducible input, E-PARI-003 for a failed check, E-PARI-004 when PARI itself fails.

>>> from alkahest import pari
>>> d = pari.number_field_invariants([5, 0, 1])      # Q(sqrt(-5))   # doctest: +SKIP
>>> d.discriminant, d.class_number                                   # doctest: +SKIP
(-20, 2)

The PARI library version string, or None when unavailable.

Ramanujan's tau(n) from the product q prod (1 - q^m)^24, exactly.

Pure Python (no PARI); the independent oracle used to check Delta.