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numthy · EllipticCurve

16 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.EllipticCurve

A nonsingular generalized Weierstrass elliptic curve.

EllipticCurve(base: E, coefficients: Sequence[EllipticCoefficient])

noether::numthy::EllipticCurve · Direct Rust declaration

pub struct EllipticCurve<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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:13 · examples/26_elliptic_descent_functions.py:7 · validation/unit/infra/test_public_api_index.py:180 · validation/unit/numthy/test_elliptic_curve_facade.py:19 · validation/unit/numthy/test_elliptic_curve_facade.py:26 · validation/unit/numthy/test_elliptic_curve_facade.py:34

numthy.EllipticCurve.base

Return the base recorded by this EllipticCurve.

Detailed contract

Exact base field context.

(self) -> E

noether::numthy::EllipticCurve::base · Direct Rust declaration

pub fn base(&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/EllipticCurve@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:59 · validation/unit/commalg/test_acceptance_workload.py:60 · validation/unit/commalg/test_base_change.py:112 · validation/unit/commalg/test_base_change.py:113 · validation/unit/commalg/test_polynomial_ideal_facade.py:398 · validation/unit/commalg/test_polynomial_ideal_facade.py:399

numthy.EllipticCurve.coefficients

Return the coefficients recorded by this EllipticCurve.

Detailed contract

Coefficients (a1,a2,a3,a4,a6).

(self) -> list[EllipticCoefficient]

noether::numthy::EllipticCurve::coefficients · Direct Rust declaration

pub fn coefficients(&self) -> &[R::Element; 5]
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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:11 · examples/26_elliptic_descent_functions.py:12 · examples/26_elliptic_descent_functions.py:14 · examples/26_elliptic_descent_functions.py:35 · examples/31_local_fields_and_precision.py:11 · examples/31_local_fields_and_precision.py:12

numthy.EllipticCurve.contains

Compute contains for this EllipticCurve.

Detailed contract

Whether a checked point belongs to this exact elliptic context.

(self, point: EllipticCurvePoint[E]) -> bool

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

pub fn contains(&self, point: &EllipticCurvePoint<R>) -> 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/EllipticCurve@1

Executable usage: examples/38_exact_linear_systems.py:73 · validation/unit/commalg/test_local_algebra.py:63 · validation/unit/commalg/test_prime_localization.py:50 · validation/unit/commalg/test_prime_localization.py:55 · validation/unit/groups/test_acceptance_workload.py:56 · validation/unit/groups/test_permutation_group_facade.py:81

numthy.EllipticCurve.discriminant

Return the discriminant recorded by this EllipticCurve.

Detailed contract

Exact nonzero Weierstrass discriminant.

(self) -> EllipticCoefficient

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

pub fn discriminant(&self) -> &R::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/EllipticCurve@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.EllipticCurve.from_json

Deserialize and revalidate a EllipticCurve from typed JSON.

Detailed contract

Restore the coefficient context and recheck nonsingularity.

(cls, value: str) -> EllipticCurve[EllipticBase]

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

pub fn from_json(value: &str) -> Result<Self, NoetherError>where R: DeserializeOwned, R::Element: DeserializeOwned,
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/EllipticCurve@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.EllipticCurve.line_function

Compute line function for this EllipticCurve.

Detailed contract

The chord, tangent, or vertical function through two affine points.

(self, left: EllipticCurvePoint[E], right: EllipticCurvePoint[E]) -> CurveFunction[E]

noether::numthy::EllipticCurve::line_function · Direct Rust declaration

pub fn line_function( &self, left: &EllipticCurvePoint<R>, right: &EllipticCurvePoint<R>, ) -> Result<CurveFunction<R>, EllipticCurveError>
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/EllipticCurve@1

numthy.EllipticCurve.miller_function

Compute Miller's function with divisor n(P) - (nP) - (n-1)(O).

Detailed contract

Compute Miller’s function with divisor n(P) - (nP) - (n-1)(O) for positive n.

(self, point: EllipticCurvePoint[E], multiplier: int) -> MillerFunction[E]

noether::numthy::EllipticCurve::miller_function · Direct Rust declaration

pub fn miller_function( &self, point: &EllipticCurvePoint<R>, multiplier: &Integer, ) -> Result<MillerFunction<R>, EllipticCurveError>
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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:20 · validation/unit/infra/test_public_api_index.py:180 · validation/unit/numthy/test_elliptic_curve_facade.py:68

numthy.EllipticCurve.origin

Compute origin for this EllipticCurve.

Detailed contract

Chosen origin, the point [0:1:0].

(self) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurve::origin · Direct Rust declaration

pub fn origin(&self) -> Result<EllipticCurvePoint<R>, EllipticCurveError>
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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:24 · validation/unit/numthy/test_elliptic_curve_facade.py:76 · validation/unit/topology/test_eilenberg_maclane.py:121 · validation/unit/topology/test_eilenberg_maclane.py:161 · validation/unit/topology/test_eilenberg_maclane.py:207 · validation/unit/topology/test_eilenberg_maclane.py:208

numthy.EllipticCurve.point

Compute point for this EllipticCurve.

Detailed contract

Construct and validate one affine point (x,y).

(self, x: EllipticCoefficient, y: EllipticCoefficient) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurve::point · Direct Rust declaration

pub fn point( &self, x: R::Element, y: R::Element, ) -> Result<EllipticCurvePoint<R>, EllipticCurveError>
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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:18 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:25 · examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32

numthy.EllipticCurve.to_json

Serialize this EllipticCurve to its canonical typed JSON representation.

Detailed contract

Canonical typed JSON; nonsingularity is recomputed on restoration.

(self) -> str

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

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize, R::Element: Serialize,
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/EllipticCurve@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.EllipticCurve.underlying_curve

Return the underlying curve recorded by this EllipticCurve.

Detailed contract

Checked underlying smooth projective curve.

(self) -> Curve[E]

noether::numthy::EllipticCurve::underlying_curve · Direct Rust declaration

pub fn underlying_curve(&self) -> &Curve<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/EllipticCurve@1

Executable usage: examples/26_elliptic_descent_functions.py:22 · validation/unit/numthy/test_elliptic_curve_facade.py:37 · validation/unit/numthy/test_elliptic_curve_facade.py:74

numthy.EllipticCurve.verify

Replay and verify the mathematical evidence carried by this EllipticCurve.

Detailed contract

Reconstruct the curve, chosen origin, and nonzero discriminant.

(self) -> None

noether::numthy::EllipticCurve::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/EllipticCurve@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

numthy.EllipticCurve.vertical_line_function

Compute vertical line function for this EllipticCurve.

Detailed contract

The vertical-line function x-x(P) through an affine point.

(self, point: EllipticCurvePoint[E]) -> CurveFunction[E]

noether::numthy::EllipticCurve::vertical_line_function · Direct Rust declaration

pub fn vertical_line_function( &self, point: &EllipticCurvePoint<R>, ) -> Result<CurveFunction<R>, EllipticCurveError>
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/EllipticCurve@1

numthy.EllipticCurve.x_function

Compute x function for this EllipticCurve.

Detailed contract

Distinguished x coordinate in the underlying curve function field.

(self) -> CurveFunction[E]

noether::numthy::EllipticCurve::x_function · Direct Rust declaration

pub fn x_function(&self) -> Result<CurveFunction<R>, EllipticCurveError>
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/EllipticCurve@1

numthy.EllipticCurve.y_function

Compute y function for this EllipticCurve.

Detailed contract

Distinguished y coordinate in the underlying curve function field.

(self) -> CurveFunction[E]

noether::numthy::EllipticCurve::y_function · Direct Rust declaration

pub fn y_function(&self) -> Result<CurveFunction<R>, EllipticCurveError>
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/EllipticCurve@1