SageMath MCP Server

Elliptic Curve Operation

elliptic_curve_operation

Idempotent

Compute elliptic curve invariants over Q from Weierstrass coefficients: rank, torsion order, discriminant, j-invariant, conductor, and generators.

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

A3.8/5.0

Behavior2/5

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

Annotations already state idempotentHint=true and destructiveHint=false, and the description adds no behavioral detail beyond the operation list. The readOnlyHint=false annotation is not addressed: the description implies a pure computation, while the annotation suggests possible side effects (e.g., session/workspace changes). No permissions, rate limits, or mutation behavior are disclosed.

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 short sentences convey the full purpose and routing guidance. The first sentence front-loads the operation list and curve domain; the second provides the sibling-tool preference. No filler or redundancy.

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?

Given the output schema exists and annotations cover idempotence/destructiveness, the description is largely sufficient: it names the curve domain, lists all operations, and points to the preferred tool. The only minor gap is that it does not clarify the session/workspace side-effect behavior hinted by readOnlyHint=false.

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 schema fully documents all three parameters. The description's 'from Weierstrass coefficients' mirrors the coefficients schema and its operation list repeats the operation schema, adding marginal semantic value. This meets the baseline for high schema coverage but does not go beyond it.

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

Purpose4/5

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

The description clearly identifies the resource (elliptic curves over Q) and enumerates the available invariants (rank, torsion order, discriminant, j-invariant, conductor, generators), and it explicitly distinguishes itself from evaluate_sage. However, it lacks a direct verb like 'computes' or 'returns', relying on a noun phrase with a colon, so it stops just short of the strongest clarity standard.

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

Usage Guidelines5/5

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

The description explicitly says 'Prefer this over evaluate_sage for curve invariants.' This tells the agent when to use this tool and names the alternative, leaving no ambiguity about the routing decision for elliptic curve invariant calculations.

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