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numthy · NthPowerClassGroup

8 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.NthPowerClassGroup

An effective multiplicative quotient context K*/K*^n.

NthPowerClassGroup(base: P, exponent: int)

noether::numthy::NthPowerClassGroup · Direct Rust declaration

pub struct NthPowerClassGroup<R: Field> { /* 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/NthPowerClassGroup@1

Executable usage: examples/26_elliptic_descent_functions.py:35 · examples/26_elliptic_descent_functions.py:7 · validation/unit/numthy/test_elliptic_curve_facade.py:104 · validation/unit/numthy/test_elliptic_curve_facade.py:111 · validation/unit/numthy/test_elliptic_curve_facade.py:13 · validation/unit/numthy/test_elliptic_curve_facade.py:89

numthy.NthPowerClassGroup.class_of

Compute class of for this NthPowerClassGroup.

(self, value: EllipticCoefficient) -> NthPowerClass[P]

noether::numthy::NthPowerClassGroup · 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/NthPowerClassGroup@1

Executable usage: examples/26_elliptic_descent_functions.py:36 · validation/unit/numthy/test_elliptic_curve_facade.py:105 · validation/unit/numthy/test_elliptic_curve_facade.py:108 · validation/unit/numthy/test_elliptic_curve_facade.py:113 · validation/unit/numthy/test_elliptic_curve_facade.py:90

numthy.NthPowerClassGroup.exponent

Return the exponent recorded by this NthPowerClassGroup.

Detailed contract

Quotient exponent n.

(self) -> int

noether::numthy::NthPowerClassGroup::exponent · Direct Rust declaration

pub fn exponent(&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/NthPowerClassGroup@1

Executable usage: examples/31_local_fields_and_precision.py:38 · validation/unit/commalg/test_base_change.py:93 · validation/unit/commalg/test_groebner_verification_budget.py:32 · validation/unit/commalg/test_groebner_verification_budget.py:33 · validation/unit/commalg/test_groebner_verification_budget.py:34 · validation/unit/commalg/test_groebner_verification_budget.py:35

numthy.NthPowerClassGroup.field

Return the field recorded by this NthPowerClassGroup.

Detailed contract

Exact source field.

(self) -> P

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

pub fn field(&self) -> &R
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/NthPowerClassGroup@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.NthPowerClassGroup.from_json

Deserialize and revalidate a NthPowerClassGroup from typed JSON.

Detailed contract

Restore Q/Q^n and recheck the exponent. Restore the exact Q(t) presentation and exponent.

(cls, value: str) -> NthPowerClassGroup[PowerClassBase]

noether::numthy::NthPowerClassGroup::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/NthPowerClassGroup@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.NthPowerClassGroup.identity

Compute identity for this NthPowerClassGroup.

Detailed contract

Trivial class.

(self) -> NthPowerClass[P]

noether::numthy::NthPowerClassGroup::identity · Direct Rust declaration

pub fn identity(&self) -> NthPowerClass<RationalField>
pub fn identity(&self) -> NthPowerClass<RationalFunctionField<RationalField>>
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/NthPowerClassGroup@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:12 · examples/01_bezout_certificates.py:14 · examples/01_bezout_certificates.py:9 · examples/02_ball_arithmetic.py:3 · examples/distributed/f2_linear_algebra.py:20

numthy.NthPowerClassGroup.to_json

Serialize this NthPowerClassGroup to its canonical typed JSON representation.

Detailed contract

Canonical typed JSON for this effective quotient context.

(self) -> str

noether::numthy::NthPowerClassGroup::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/NthPowerClassGroup@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.NthPowerClassGroup.verify

Replay and verify the mathematical evidence carried by this NthPowerClassGroup.

Detailed contract

Recheck the admitted field context and exponent.

(self) -> None

noether::numthy::NthPowerClassGroup::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>
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/NthPowerClassGroup@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12