numthy · NumberFieldOrder

15 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Reproduce Fisher's order-five elliptic-function normalization uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""

from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup


def main() -> None:
    coefficients = RationalFunctionField(Q, "lambda")
    parameter = coefficients.generator()
    curve = EllipticCurve(
        coefficients,
        [1 - parameter, -parameter, -parameter, 0, 0],
    )
    point = curve.point(0, 0)
    point.verify_exact_order(5)

    miller = curve.miller_function(point, 5)
    divisor = WeilDivisor(
        curve.underlying_curve,
        [
            (curve.origin().underlying_curve_point.as_closed_point(), 5),
            (point.underlying_curve_point.as_closed_point(), -5),
        ],
    )
    function = divisor.riemann_roch_space().basis[0]
    assert (function / miller.function).as_constant() is not None

    evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
    evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
    assert evaluation_at_two.value is not None
    assert evaluation_at_four.value is not None
    classes = NthPowerClassGroup(coefficients, 5)
    normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
        evaluation_at_four.value
    )
    assert normalization.is_identity()
    print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/26_elliptic_descent_functions.py.

numthy.NumberFieldOrder

A validated exact order in an absolute number field.

PythonRust

NumberFieldOrder(field: NumberField, basis: Sequence[int | Fraction | NumberFieldElement | Sequence[int | Fraction]])

noether::numthy::NumberFieldOrder · Direct Rust declaration

pub struct NumberFieldOrder { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/numthy/test_number_field_facade.py:10 · validation/unit/numthy/test_number_field_facade.py:115 · validation/unit/numthy/test_number_field_facade.py:125 · validation/unit/numthy/test_number_field_facade.py:98

numthy.NumberFieldOrder.__call__

Call self as a function.

PythonRust

(self, value: int | Fraction | NumberFieldElement | Sequence[int | Fraction]) -> NumberFieldElement

noether::numthy::NumberFieldOrder · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/infra/test_discovery.py:57 · validation/unit/infra/test_public_api_index.py:155 · validation/unit/infra/test_public_api_index.py:169

numthy.NumberFieldOrder.__contains__

Return bool(key in self).

Detailed contract

Whether an exact field element lies in this integral lattice.

PythonRust

(self, value: int | Fraction | NumberFieldElement | Sequence[int | Fraction]) -> bool

noether::numthy::NumberFieldOrder::contains · Direct Rust declaration

pub fn contains(&self, element: &NumberFieldElement) -> bool
fn contains(&self, value: &Self::Element) -> bool

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/infra/test_discovery.py:200

numthy.NumberFieldOrder.basis

Canonical integral basis.

Detailed contract

Canonical integral basis as exact field elements.

PythonRust

(self) -> list[NumberFieldElement]

noether::numthy::NumberFieldOrder::basis · Direct Rust declaration

pub fn basis(&self) -> Vec<NumberFieldElement>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/26_elliptic_descent_functions.py:28 · examples/30_modules_and_resolutions.py:27 · examples/30_modules_and_resolutions.py:28 · examples/30_modules_and_resolutions.py:29 · examples/33_simplicial_cohomology_operations.py:26 · examples/33_simplicial_cohomology_operations.py:27

numthy.NumberFieldOrder.discriminant

Exact order discriminant.

Detailed contract

Exact discriminant of this order.

PythonRust

(self) -> int

noether::numthy::NumberFieldOrder::discriminant · Direct Rust declaration

pub fn discriminant(&self) -> Integer

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/31_local_fields_and_precision.py:37 · validation/unit/numthy/test_acceptance_workload.py:53 · validation/unit/numthy/test_number_field_facade.py:101 · validation/unit/numthy/test_number_field_facade.py:126 · validation/unit/numthy/test_number_field_facade.py:143 · validation/unit/numthy/test_number_field_facade.py:144

numthy.NumberFieldOrder.field

Ambient exact number field.

Detailed contract

The ambient number field.

PythonRust

(self) -> NumberField

noether::numthy::NumberFieldOrder::field · Direct Rust declaration

pub fn field(&self) -> &NumberField

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/08_finite_field_extensions.py:10 · examples/08_finite_field_extensions.py:11 · examples/08_finite_field_extensions.py:12 · examples/08_finite_field_extensions.py:13 · examples/08_finite_field_extensions.py:14 · examples/08_finite_field_extensions.py:3

numthy.NumberFieldOrder.fractional_ideal

Construct the nonzero fractional ideal generated by exact field elements.

Detailed contract

Construct the nonzero fractional ideal generated by generators.

PythonRust

(self, generators: Sequence[int | Fraction | NumberFieldElement | Sequence[int | Fraction]]) -> FractionalIdeal

noether::numthy::NumberFieldOrder::fractional_ideal · Direct Rust declaration

pub fn fractional_ideal( &self, generators: &[NumberFieldElement], ) -> Result<FractionalIdeal, FractionalIdealError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/12_number_field_ideals.py:17 · validation/unit/numthy/test_acceptance_workload.py:78 · validation/unit/numthy/test_acceptance_workload.py:79 · validation/unit/numthy/test_number_field_facade.py:173 · validation/unit/numthy/test_number_field_facade.py:186 · validation/unit/numthy/test_number_field_facade.py:187

numthy.NumberFieldOrder.from_json

Restore and revalidate a canonical order representation.

Detailed contract

Restore and validate an exact order from canonical JSON.

PythonRust

(cls, value: str) -> NumberFieldOrder

noether::numthy::NumberFieldOrder::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

numthy.NumberFieldOrder.index

Index [super_order : self], requiring containment.

Detailed contract

Index [super_order : self] when self is contained in super_order.

PythonRust

(self, super_order: NumberFieldOrder) -> int

noether::numthy::NumberFieldOrder::index · Direct Rust declaration

pub fn index( &self, super_order: &NumberFieldOrder, ) -> Result<Integer, NumberFieldOrderError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/38_exact_linear_systems.py:79 · validation/unit/combinatorics/test_kashiwara_crystal.py:34 · validation/unit/combinatorics/test_kashiwara_crystal.py:35 · validation/unit/combinatorics/test_kashiwara_crystal.py:36 · validation/unit/commalg/test_base_change.py:90 · validation/unit/commalg/test_base_change.py:93

numthy.NumberFieldOrder.inverse_unit

Invert a unit of this order.

PythonRust

(self, value: NumberFieldElement) -> NumberFieldElement

noether::numthy::NumberFieldOrder::inverse_unit · Direct Rust declaration

fn inverse_unit( &self, value: &Self::Element, ) -> Result<Self::Element, UnitError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/numthy/test_number_field_facade.py:104 · validation/unit/numthy/test_number_field_facade.py:112 · validation/unit/numthy/test_padic_local_field_facade.py:199

numthy.NumberFieldOrder.is_maximal

Whether this is the maximal order of its ambient field.

Detailed contract

Whether this order is the maximal order of its ambient number field.

PythonRust

(self) -> bool

noether::numthy::NumberFieldOrder::is_maximal · Direct Rust declaration

pub fn is_maximal(&self) -> Result<bool, NumberFieldOrderError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	released
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/31_local_fields_and_precision.py:25 · validation/unit/numthy/test_number_field_facade.py:145 · validation/unit/numthy/test_number_field_facade.py:161 · validation/unit/numthy/test_number_field_facade.py:162 · validation/unit/numthy/test_padic_local_field_facade.py:177 · validation/unit/numthy/test_padic_local_field_facade.py:188

numthy.NumberFieldOrder.one

Multiplicative identity.

PythonRust

(self) -> NumberFieldElement

noether::numthy::NumberFieldOrder::one · Direct Rust declaration

fn one(&self) -> Self::Element

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/02_ball_arithmetic.py:13 · examples/11_qadic_witt_vectors.py:11 · examples/24_homology_and_persistence.py:3 · examples/31_local_fields_and_precision.py:27 · validation/unit/combinatorics/test_poset.py:3 · validation/unit/commalg/test_acceptance_workload.py:37

numthy.NumberFieldOrder.prime_ideals_above

Checked prime ideals above a positive rational prime.

Detailed contract

Decompose (p) into checked prime ideals of this maximal order. Kummer–Dedekind factorization is the fast path when the distinguished equation order is p-maximal. Native residue-algebra splitting handles index primes and rational presentations. Every result is verified as a field quotient, and the product is checked to reconstruct (p).

PythonRust

(self, prime: int) -> list[PrimeIdeal[NumberFieldOrder]]

noether::numthy::NumberFieldOrder::prime_ideals_above · Direct Rust declaration

pub fn prime_ideals_above( &self, prime: &Integer, ) -> Result<Vec<PrimeIdeal<NumberFieldOrder>>, PrimeDecompositionError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	released
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/12_number_field_ideals.py:14 · validation/unit/numthy/test_acceptance_workload.py:56 · validation/unit/numthy/test_acceptance_workload.py:66 · validation/unit/numthy/test_acceptance_workload.py:67 · validation/unit/numthy/test_number_field_facade.py:210 · validation/unit/numthy/test_number_field_facade.py:232

numthy.NumberFieldOrder.to_json

Canonical JSON representation.

Detailed contract

Serialize this exact order to canonical JSON.

PythonRust

(self) -> str

noether::numthy::NumberFieldOrder::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

numthy.NumberFieldOrder.zero

Additive identity.

PythonRust

(self) -> NumberFieldElement

noether::numthy::NumberFieldOrder::zero · Direct Rust declaration

fn zero(&self) -> Self::Element

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/NumberFieldOrder@1

Executable usage: examples/08_finite_field_extensions.py:11 · examples/10_padic_hensel_lifting.py:15 · examples/40_exterior_algebra.py:26 · examples/distributed/f2_linear_algebra.py:28 · examples/distributed/f2_linear_algebra.py:29 · examples/distributed/f2_linear_algebra.py:30