SageMath MCP Server

elliptic_curve_operation

Idempotent

Compute elliptic curve invariants over Q—rank, torsion order, discriminant, j-invariant, conductor, and generators—from Weierstrass coefficients. Use this instead of evaluate_sage for curve invariants.

Instructions

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

Input Schema

TableJSON Schema

Name	Required	Description	Default
`session`	No	Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'.	default
`operation`	Yes	One of: rank, torsion_order, discriminant, j_invariant, conductor, gens
`coefficients`	Yes	Curve coefficients [a1,a2,a3,a4,a6] or short Weierstrass [a,b] for y^2 = x^3 + a*x + b

Output Schema

TableJSON Schema

Name	Required	Description	Default
No arguments

Schema Changelog

Changes observed during successful MCP inspections.

Changed1 schema field changedv0.7.0Sep 12, 2026

changedInput schema / properties / session / description

Previous value: -"Named workspace to use. Workspaces have independent variables; omit for 'default'."New value: +"Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'."

Changed3 schema fields changedv0.5.0Aug 14, 2026

addedInput schema / properties / coefficients / items / anyOf

Added value: +[
+  {
+    "type": "integer"
+  },
+  {
+    "type": "string"
+  }
+]

removedInput schema / properties / coefficients / items / type
```
Removed value: -"integer"
```

addedInput schema / properties / session

Added value: +{
+  "default": "default",
+  "description": "Named workspace to use. Workspaces have independent variables; omit for 'default'.",
+  "type": "string"
+}

First observedv0.3.1Jul 31, 2026

TDQS

A4/5.0

Behavior3/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations declare openWorldHint=false, idempotentHint=true, destructiveHint=false but readOnlyHint=false, which is non-obvious for a pure computation tool and is left unexplained by the description. The description adds nothing about workspace state creation, session scoping, or whether computation can modify a workspace, so it does not resolve that ambiguity.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

Two sentences, zero filler. The output enumeration and the disambiguation against evaluate_sage are both front-loaded, and nothing is repeated for padding.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

With a full output schema, the description need not describe return values, and it correctly covers what is computed and the alternative tool. It is slightly incomplete on workspace/session side effects and the mixed readOnly/idempotent signal, but nothing essential to invoking the tool is missing.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema coverage is 100%, so the baseline is 3. The phrase 'from Weierstrass coefficients' loosely ties to the coefficients parameter, but the operation list and coefficient format are already fully documented in the schema, so the description adds no new parameter meaning.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

Names a specific class of operation (Elliptic curves over Q) and enumerates the exact results returned: rank, torsion order, discriminant, j-invariant, conductor, generators. It also names the sibling it supersedes, so an agent can distinguish it from evaluate_sage without opening either schema.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines4/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

Explicitly routes the agent: 'Prefer this over evaluate_sage for curve invariants.' That covers the primary alternative-selection decision. It does not state exclusions or when a generic evaluator would still be preferable (e.g. non-invariant computations), so it falls short of a full when/when-not pair.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.