sagebrush

Fast, certified engines for research mathematics, in Rust, for Python. Permissively licensed (MIT OR Apache-2.0): use them anywhere, in anything, from agents and notebooks to products. No C libraries; one abi3 wheel per platform covers Python 3.9 and later.

This is an early release of Sagebrush, which also runs in the browser at sagebrush.space.

>>> from sagebrush import nf, mf, ap, poly, linalg

>>> nf.bnf([-11, 0, 0, 1])              # Q[x]/(x^3 - 11): class group, regulator
{'cyc': [2], 'degree': 3, 'disc': -3267, 'h': 2, 'r1': 1, 'r2': 1, 'regulator': '5.5872066260609077619', 'w': 2}

>>> nf.factor_integer(2**128 + 1)       # ECM
[(59649589127497217, 1), (5704689200685129054721, 1)]

>>> nf.primes_above([-11, 0, 0, 1], 5)  # 5 = P_1 P_2 with residue degrees 1, 2
[{'e': 1, 'f': 1, 'p': 5, 'pi': [-1, 1, 0], 'pi_den': 1}, {'e': 1, 'f': 2, 'p': 5, 'pi': [1, 1, 1], 'pi_den': 1}]

>>> nf.lll([[1, 1, 1], [-1, 0, 2], [3, 5, 6]])
[[0, 1, 0], [1, 0, 1], [-1, 0, 2]]

>>> poly.factor([5, -1, -1, 1])         # x^3 - x^2 - x + 5 over Z
(1, [([5, -1, -1, 1], 1)])

>>> linalg.det([[2, 7, 1], [8, 2, 8], [1, 8, 2]]), linalg.charpoly([[0, 1], [-1, 0]])
(-114, [1, 0, 1])

>>> linalg.solve([[2, 1], [1, 3]], [[1], [2]])   # exact, over QQ
[[Fraction(1, 5)], [Fraction(3, 5)]]

>>> d = mf.newforms(389, 2, bound=6)    # newform orbits of S_2(Gamma0(389))
>>> d["status"], [(o["letter"], o["dim"]) for o in d["newforms"]]
('proven', [('a', 1), ('b', 2), ('c', 3), ('d', 6), ('e', 20)])

Polynomials are coefficient lists, constant term first.

Sage-compatible: sagebrush.sage

For Sage users, sagebrush.sage is a namespace with Sage's names, behaviour and printing, for what Sagebrush implements: number fields, polynomials, matrices, modular forms and symbols, elliptic curves, Dirichlet characters, integers, plotting.

>>> from sagebrush.sage import *
>>> factor(2026)
2 * 1013
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**3 + 17*x + 1, 'a')
>>> K.class_group()
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^3 + 17*x + 1
>>> K.regulator()
2.83341678218613
>>> K.factor(11)
(Fractional ideal (11, a + 3))^2 * (Fractional ideal (11, a + 5))
>>> Integer(2026).is_prime(), Integer(2026) / 4
(False, 1013/2)
>>> EllipticCurve('389a1').rank()
2

This is plain Python, without Sage's preparser: write x**3 (x^3 is xor in Python, and raises an error here as in Sage), Integer(2)/3 for a rational (2/3 is a float), and K = NumberField(f, 'a'); a = K.gen() for K.<a> = NumberField(f). Integers from Integer(n) or ZZ(n) have Sage's methods. The output is checked line by line against Sage itself: Sage's own preparser turns Sagebrush's Sage test files into plain Python, and their output under sagebrush.sage must equal Sage's.

Engines

• sagebrush.nf: number fields: maximal orders, prime ideals, class groups, units and regulators (assuming GRH, with a field-specific bound for the generators), quadratic class groups to |D| ~ 10^60; integer factoring (ECM); Hermite and Smith forms, LLL; complex roots to any precision.
• sagebrush.mf: modular forms of weight k >= 2 with character: dimensions, exact Hecke characteristic polynomials, newform orbits, trace forms.
• sagebrush.modsym: weight-2 modular symbols for Gamma0(N).
• sagebrush.ap: traces of Frobenius a_p of elliptic curves over Q.
• sagebrush.poly: factoring in Z[x] and F_p[x]; products, gcds and exact division in Z[x].
• sagebrush.linalg: exact matrices over ZZ and QQ: det, rank, rref, solve, inverse, charpoly, kernel (multimodular and p-adic, every answer certified).

Results are checked against independent systems: PARI for class groups, Sage and Magma for the rest, and LMFDB for modular forms.

License

Copyright (c) 2026 SageMath, Inc. Licensed under either of the Apache License, Version 2.0 or the MIT license, at your option.