Solve an equation or system of equations

Multiply two matrices and return the result as nested lists

Linear algebra on one matrix: determinant, inverse, eigenvalues, rank, reduced row echelon form, transpose. Prefer this over evaluate_sage.

Boolean polynomials over GF(2): evaluate, list variables, degree, and zero/one tests. Prefer this over evaluate_sage for boolean algebra.

Polynomial ring operations: construct rings and compute Groebner bases, ideals, quotients

Differentiate an expression with respect to a variable

Integrate an expression (indefinite or definite with bounds)

Compute the limit of an expression

Compute a Taylor/Laurent series expansion

Solve an ordinary differential equation of any order, returning the general solution with arbitrary constants. Prefer this over evaluate_sage.

Closed form of a symbolic sum or product over an index variable, including infinite series. Prefer this over evaluate_sage for summations.

Vector calculus operations: gradient, divergence, curl, laplacian

Run arbitrary SageMath code in a persistent session; variables persist across calls.

LAST RESORT. A dedicated tool exists for most tasks and should be preferred: it validates arguments and returns a typed result instead of a repr string. Reach for one of these first:

• calculus: differentiate_expression, integrate_expression, limit_expression, series_expansion, symbolic_sum, solve_ode

• algebra: solve_equation, simplify_expression, expand_expression, factor_expression, find_root

• linear algebra: matrix_operation (determinant, inverse, eigenvalues, rank, rref, transpose), matrix_multiply

• discrete: number_theory_operation, combinatorics_operation (binomial, partitions, catalan, fibonacci, bell), graph_operation, group_operation

• specialised: elliptic_curve_operation, coding_theory_operation, polynomial_ring_operation, boolean_algebra_operation, geometry_operation, vector_calculus_operation

• data: statistics_summary, distribution_operation

• plots: plot_expression, plot3d_expression, plot_multi_expression

Use evaluate_sage only for what those do not cover, for example:

Transforms: var('t s'); laplace(sin(t), t, s); inverse_laplace(1/(s^2+1), s, t) Modular arithmetic: Mod(17, 5); power_mod(3, 100, 97) Recurrences: var('n'); f = function('f'); desolve_rec(f(n+2)-f(n+1)-f(n), f, [0, 1]) Continued fractions: continued_fraction(pi).convergents()[:10] Number fields: K. = NumberField(x^3 - 2); K.class_number() Multi-step work that builds on values defined earlier in the same session.

Evaluate a SageMath expression and return numeric/string forms

Simplify a mathematical expression

Expand a mathematical expression

Factor a mathematical expression or integer

Find a numeric root of an expression in a given interval

Execute SageMath code and stream intermediate print() output line by line. Final result is returned as usual.

Number theory: primality testing, integer factorisation, the next prime above n, gcd and lcm. Prefer this over evaluate_sage for any of these.

Combinatorics: binomial coefficients, permutations, combinations, integer partitions, factorial, Catalan, Fibonacci and Bell numbers. Prefer this over evaluate_sage for any of these.

Graph theory: create named graphs and compute properties (chromatic_number, is_connected, diameter, etc.)

Group theory: construct groups and query properties (order, is_abelian, center, etc.)

Elliptic curves over Q: rank, torsion order, discriminant, j-invariant, conductor and generators, from Weierstrass coefficients. Prefer this over evaluate_sage for curve invariants.

Error-correcting codes: length, dimension, minimum distance, rate and generator matrix for Hamming and generalized Reed-Solomon codes. Prefer this over evaluate_sage for code parameters.

Plot a 3D surface of a two-variable expression as base64 PNG

Plot multiple expressions overlaid on a single 2D graph

Plot an expression and return a base64-encoded PNG image

Computational geometry on point sets: euclidean distance, polygon area, polytope volume, convex hull vertices and convexity tests. Prefer this over evaluate_sage for these.

Reset the SageMath session state for the current MCP session

Interrupt a running Sage computation while keeping variables defined so far

Cancel any running Sage computation and restart the worker

Start a named Sage workspace with its own independent variables

List the named Sage workspaces belonging to this client

Stop a named Sage workspace and release its worker

Descriptive statistics for a list of numbers: mean, median, population and sample variance and standard deviation, min and max. Prefer this over evaluate_sage for summary statistics.

Probability distribution operations: PDF, CDF, quantile, mean, variance, sampling