numthy · EllipticCurvePoint

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.EllipticCurvePoint

An affine point or the chosen origin on an elliptic curve.

PythonRust

class EllipticCurvePoint

noether::numthy::EllipticCurvePoint · Direct Rust declaration

pub struct EllipticCurvePoint<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/EllipticCurvePoint@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:41 · validation/unit/numthy/test_elliptic_curve_facade.py:43 · validation/unit/numthy/test_elliptic_curve_facade.py:45 · validation/unit/numthy/test_elliptic_curve_facade.py:9 · validation/unit/numthy/test_elliptic_curve_facade.py:97

numthy.EllipticCurvePoint.__add__

Add another compatible value to this value.

Detailed contract

Returns the argument unchanged.

PythonRust

(self, other: EllipticCurvePoint[E]) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint::add · Direct Rust declaration

fn add(self, rhs: &EllipticCurvePoint<R>) -> Self::Output

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/EllipticCurvePoint@1

Executable usage: validation/unit/infra/test_discovery.py:211 · validation/unit/infra/test_public_api_index.py:293

numthy.EllipticCurvePoint.__mul__

Multiply this value by another compatible value.

PythonRust

(self, scalar: int) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint · 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/EllipticCurvePoint@1

Executable usage: validation/unit/infra/test_public_api_index.py:214

numthy.EllipticCurvePoint.__neg__

Return the additive inverse of this value.

PythonRust

(self) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint::neg · Direct Rust declaration

fn neg(self) -> Self::Output

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/EllipticCurvePoint@1

numthy.EllipticCurvePoint.__rmul__

Multiply this value by a compatible value on its left.

PythonRust

(self, scalar: int) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint · 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/EllipticCurvePoint@1

numthy.EllipticCurvePoint.__sub__

Subtract another compatible value from this value.

PythonRust

(self, other: EllipticCurvePoint[E]) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint::sub · Direct Rust declaration

fn sub(self, rhs: &EllipticCurvePoint<R>) -> Self::Output

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/EllipticCurvePoint@1

Executable usage: validation/unit/infra/test_discovery.py:212

numthy.EllipticCurvePoint.affine_coordinates

Return the affine coordinates recorded by this EllipticCurvePoint.

Detailed contract

Affine coordinates, or None at infinity.

PythonRust

(self) -> tuple[EllipticCoefficient, EllipticCoefficient] | None

noether::numthy::EllipticCurvePoint::affine_coordinates · Direct Rust declaration

pub fn affine_coordinates(&self) -> Option<(&R::Element, &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/EllipticCurvePoint@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:64 · validation/unit/numthy/test_elliptic_curve_facade.py:65 · validation/unit/numthy/test_elliptic_curve_facade.py:66 · validation/bench/magma_release.py:1533 · validation/corpus/numthy/miura-elliptic-addition.py:21 · validation/corpus/numthy/rational-cubic-two-cycle.py:20

numthy.EllipticCurvePoint.curve

Return the curve recorded by this EllipticCurvePoint.

Detailed contract

Owning elliptic-curve context.

PythonRust

(self) -> EllipticCurve[E]

noether::numthy::EllipticCurvePoint::curve · Direct Rust declaration

pub fn curve(&self) -> &EllipticCurve<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/EllipticCurvePoint@1

Executable usage: examples/26_elliptic_descent_functions.py:13 · examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:22 · examples/26_elliptic_descent_functions.py:24 · examples/37_complete_intersection_theory.py:27

numthy.EllipticCurvePoint.from_json

Deserialize and revalidate a EllipticCurvePoint from typed JSON.

Detailed contract

Restore and recheck an affine point or the chosen origin.

PythonRust

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

noether::numthy::EllipticCurvePoint::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/EllipticCurvePoint@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.EllipticCurvePoint.has_exact_order

Test whether this EllipticCurvePoint has exact order.

Detailed contract

Decide whether candidate is this point’s exact positive order.

PythonRust

(self, candidate: int) -> bool

noether::numthy::EllipticCurvePoint::has_exact_order · Direct Rust declaration

pub fn has_exact_order( &self, candidate: &Integer, ) -> Result<bool, 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/EllipticCurvePoint@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:22 · validation/bench/magma_release.py:1523 · validation/bench/magma_release.py:1527

numthy.EllipticCurvePoint.is_infinity

Test whether this EllipticCurvePoint is infinity.

Detailed contract

Whether this is the chosen point at infinity.

PythonRust

(self) -> bool

noether::numthy::EllipticCurvePoint::is_infinity · Direct Rust declaration

pub fn is_infinity(&self) -> 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/EllipticCurvePoint@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:21 · validation/unit/numthy/test_elliptic_curve_facade.py:28 · validation/unit/numthy/test_elliptic_curve_facade.py:36 · validation/unit/numthy/test_elliptic_curve_facade.py:69

numthy.EllipticCurvePoint.scalar_mul

Compute scalar mul for this EllipticCurvePoint.

Detailed contract

Scalar multiplication by an arbitrary-precision signed integer.

PythonRust

(self, scalar: int) -> EllipticCurvePoint[E]

noether::numthy::EllipticCurvePoint::scalar_mul · Direct Rust declaration

pub fn scalar_mul(&self, scalar: &Integer) -> Result<Self, 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/EllipticCurvePoint@1

Executable usage: validation/unit/commalg/test_intersection_theory.py:30 · validation/unit/commalg/test_intersection_theory.py:37 · validation/unit/commalg/test_intersection_theory.py:51 · validation/unit/commalg/test_power_series_facade.py:152 · validation/unit/numthy/test_elliptic_curve_facade.py:36 · validation/bench/macaulay2_derived.py:693

numthy.EllipticCurvePoint.to_json

Serialize this EllipticCurvePoint to its canonical typed JSON representation.

Detailed contract

Canonical typed JSON; point membership is recomputed.

PythonRust

(self) -> str

noether::numthy::EllipticCurvePoint::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/EllipticCurvePoint@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.EllipticCurvePoint.underlying_curve_point

Return the underlying curve point recorded by this EllipticCurvePoint.

Detailed contract

Checked projective point on the underlying commutative-algebra curve.

PythonRust

(self) -> CurvePoint[E]

noether::numthy::EllipticCurvePoint::underlying_curve_point · Direct Rust declaration

pub fn underlying_curve_point(&self) -> &CurvePoint<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/EllipticCurvePoint@1

Executable usage: examples/26_elliptic_descent_functions.py:24 · examples/26_elliptic_descent_functions.py:25 · examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32 · validation/unit/numthy/test_elliptic_curve_facade.py:76 · validation/unit/numthy/test_elliptic_curve_facade.py:77

numthy.EllipticCurvePoint.verify

Replay and verify the mathematical evidence carried by this EllipticCurvePoint.

Detailed contract

Recheck incidence, affine normalization, and the chosen origin.

PythonRust

(self) -> None

noether::numthy::EllipticCurvePoint::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/EllipticCurvePoint@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.EllipticCurvePoint.verify_exact_order

Compute verify exact order for this EllipticCurvePoint.

Detailed contract

Fail unless candidate is the exact positive order.

PythonRust

(self, candidate: int) -> None

noether::numthy::EllipticCurvePoint::verify_exact_order · Direct Rust declaration

pub fn verify_exact_order( &self, candidate: &Integer, ) -> Result<(), 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/EllipticCurvePoint@1

Executable usage: examples/26_elliptic_descent_functions.py:18 · validation/unit/numthy/test_elliptic_curve_facade.py:23 · validation/unit/numthy/test_elliptic_curve_facade.py:59