Execute arbitrary SageMath code within a persistent session. Variables and definitions persist across calls. Use this for any computation not covered by the specialized helpers. Examples by domain:

Combinatorics: binomial(10, 3); Permutations(4).cardinality(); Combinations([1,2,3,4], 2).list() Graph theory: G = graphs.PetersenGraph(); G.chromatic_number() Number theory: prime_range(100); euler_phi(60); continued_fraction(pi, nterms=10) Geometry: polytopes.cube().volume(); EllipticCurve([0,0,1,-1,0]).rank() Probability: RealDistribution('gaussian', 1).cum_distribution_function(1.96) Group theory: SymmetricGroup(5).order(); AlternatingGroup(4).is_abelian() Polynomial rings: R.<a,b> = PolynomialRing(QQ); (a+b)^3 Coding theory: codes.HammingCode(GF(2), 3).minimum_distance() Symbolic sums: var('n'); sum(1/n^2, n, 1, oo) Transforms: laplace(sin(t), t, s); inverse_laplace(1/(s^2+1), s, t) Modular arithmetic: Mod(17, 5); power_mod(3, 100, 97) Vector calculus: var('x y z'); f = x^2+y^2+z^2; diff(f,x), diff(f,y), diff(f,z) Numeric root finding: find_root(x - cos(x), 0, 1) Recurrences: desolve_rsolve(f(n+2)-f(n+1)-f(n), f, [0, 1])

Reset the SageMath session state for the current MCP session

Cancel any running Sage computation and restart the worker

Evaluate a SageMath expression and return numeric/string forms

Solve an equation or system of equations

Differentiate an expression with respect to a variable

Integrate an expression (indefinite or definite with bounds)

Compute descriptive statistics for a dataset

Multiply two matrices and return the result as nested lists

Simplify a mathematical expression

Expand a mathematical expression

Factor a mathematical expression or integer

Compute the limit of an expression

Compute a Taylor/Laurent series expansion

Perform a matrix operation (det, inverse, eigenvalues, ...)

Solve an ordinary differential equation

Number theory operations: is_prime, factor_integer, next_prime, gcd, lcm

Compute a symbolic sum or product over an index variable

Combinatorics: binomial, permutations, combinations, partitions, and more

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

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

Find a numeric root of an expression in a given interval

Plot multiple expressions overlaid on a single 2D graph

Vector calculus operations: gradient, divergence, curl, laplacian

Plot an expression and return a base64-encoded PNG image

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 curve operations: rank, torsion, discriminant, j_invariant, rational points

Coding theory: construct error-correcting codes and compute properties

Boolean algebra: operations on boolean polynomials

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

Geometry: distances, areas, volumes, convex hulls for points and polytopes

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