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

>>> 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)])

>>> 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.

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].

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.