EllipticCurves 3.7.0

EllipticCurves

A managed C# library for elliptic curves over the rationals and finite fields, targeting .NET Standard 2.0. Native computations run offline without PARI, Sage, Magma or other mathematical software. LMFDB integration is optional.

This package contains the library. The Windows desktop Elliptic Curves Explorer and the Console command-line tool are separate applications in the repository.

using EllipticCurves;

var e = new EllipticCurveQ(0, 0, 1, -1, 0);  // 37.a1
var p = new EllipticCurvePoint(0, 0);

var twice = e.Double(p);
var conductor = e.Conductor;                 // 37
var bounds = e.GetRankBounds();
var rank = bounds.ExactRank;                 // null unless proved
var local = e.GetLocalData(37);
var height = e.CanonicalHeight(p);           // certified rational enclosure
var periods = e.GetPeriods();                // on the global minimal model
var count = e.CountPoints(5);                // 8, including infinity
var finite = e.ReduceModuloPrime(5);

Capabilities

• Exact rational arithmetic, invariants, point arithmetic, rational torsion, minimal models, isomorphisms and quadratic twists.
• Conductors with prime/exponent factorizations from the same calculation, configurable parallel factorization, root numbers, local reduction data and Tamagawa numbers.
• General 2-descent and descent by 2-isogeny, with proved algebraic rank bounds.
• Certified rank lower bounds from supplied rational points, without full descent.
• Numerical central L-function derivatives and analytic rank estimates, with separate rigorous certificates for analytic ranks 0 and 1.
• Certified canonical/local heights, subgroup regulators, periods, and Faltings heights.
• Subgroup saturation at explicitly requested primes and exact rational division.
• Fourier coefficients, Frobenius traces, rational CM recognition, Velu isogenies with pointwise rational kernels, and dual 2-isogenies.
• Prime-field curves, arithmetic, direct point counting and enumeration, including characteristics 2 and 3.
• Exact finite extensions F_(p^k) with proved irreducible defining polynomials, curve arithmetic, counting by x-coordinate and bounded enumeration of points.
• Numerical elliptic logarithms of rational points on both real components.
• Optional cached LMFDB data, including generators, heights, periods and isogeny-class metadata, with offline snapshot parsing.

Scope and limits

Point searches are bounded. Saturation certifies only the requested primes and does not establish a full Mordell-Weil basis. Work limits can leave rank bounds incomplete. Numerical estimates and stored LMFDB values are distinguished from proofs; the library does not compute the Tate-Shafarevich group order.

Finite-field counting uses direct search, not SEA. Extension-field point counting visits q x-coordinates, with a default limit of 1000000. Point enumeration still checks q^2 coordinate pairs with the same default limit. General isogeny-class discovery, higher descents and curves over general number fields are outside this release.

See the repository README for examples and the algorithm notes for ranks, heights, periods and saturation, coefficients, division, isogenies and prime fields, and Faltings heights and finite extensions.

Licensed under MIT.