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Tate's Algorithm

Classifying an elliptic curve at a selected place through local model changes, divisibility and residue-root tests.

Version
v1 · 2026-10-17 · History
Domain-specific #
14330
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Elliptic Curves, Local Arithmetic → Mathematics
Aliases
Tate algorithm

Core Idea

Tate's algorithm classifies an elliptic curve at a chosen local place. Starting with a Weierstrass model, it prepares coordinates and follows ordered divisibility and residue-root tests. It also checks local minimality. Cremona's rational-prime account gives a Kodaira type, conductor exponent, local index, and a corrected model when needed.[^ref-16d97a8a6dc0]

Scope of Application

Its roles are a nonsingular curve model, a chosen prime or function-field place, applicable local arithmetic, ordered tests and minimality control, and an interpreted finding. Without these, a displayed type is only a label.[ref-16d97a8a6dc0][ref-ffceb454b4ac][^ref-d7c31b7c0003]

Two documented settings show the same roles in use:

  • Number field: Sage describes a curve over Q(i) at the prime ideal generated by 2+i. Its routine supplies the procedure path; it reports type I0∗, Tamagawa number 4, and a locally minimal equation. The documentation prints neither the branch trace nor a conductor number for this example.[^ref-ffceb454b4ac]
  • Function field: Lai and coauthors enumerate bad places of a base-changed curve and use Magma's LocalInformation to aggregate local minimal-discriminant valuations, reporting degree 24. The curve and places are inputs; the named routine supplies the path and minimality handling; valuation fields feed the aggregate. The paper's field and twist signs disagree, so 24 remains an attributed result. Individual symbols and traces are unreported.[ref-9ee7a3a31936][ref-d7c31b7c0003]

Cremona's printed pseudocode uses integer coefficients and a rational prime; it should not be applied unchanged to every residue field.[^ref-16d97a8a6dc0]

Clarity

A discriminant valuation does not determine every local index. Over Q, split type I_n has index n; nonsplit I_n has index 2 for even n and 1 for odd n. Sage's curve 14a1 at 2 has nonsplit I6 and Tamagawa number 2, despite discriminant valuation 6.[ref-16d97a8a6dc0][ref-ffceb454b4ac]

Scaling can change invariant valuations, so a nonminimal equation cannot simply supply the final local result.[^ref-16d97a8a6dc0]

Manages Complexity

The procedure tests good, multiplicative, and additive cases in sequence. Some need quadratic or cubic root tests; a nonminimal branch rescales and restarts.[^ref-16d97a8a6dc0] The final symbol does not retain every coordinate change or splitting decision.

Abstract Reasoning

For a reported answer, ask: Which model and place? Which root tests? Was the model rescaled? These distinguish the method from a bare Kodaira symbol or separate minimization procedure.[^ref-16d97a8a6dc0]

It is a strict kind of Algorithm on its declared effective domain: ordered operations terminate with local outputs and a symbolic resource account. This claims neither polynomial bit complexity nor correctness for arbitrary non-effective fields. Curve, place, and reduction interpretation make it domain-specific.

Knowledge Transfer

The questions transfer to another valid curve and place, subject to the software contract. Sage's algorithm='pari' option is limited to rational curves; current PARI documentation separately permits elllocalred over number fields.[ref-ffceb454b4ac][ref-22dea966f7e3]

Outside elliptic-curve arithmetic, ordered testing is only an analogy. These sources do not certify historical priority, an alleged original-paper misprint, or a general complete-discrete-valuation-ring theorem.

Relationships to Other Abstractions

Local relationship map for Tate's AlgorithmParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Tate's AlgorithmDOMAINPrime abstraction: Algorithm — is a kind ofAlgorithmPRIME

Current abstraction Tate's Algorithm Domain-specific

Parents (1) — more general patterns this builds on

  • Tate's Algorithm is a kind of Algorithm Prime

    Tate's algorithm is an effective terminating local-arithmetic decision procedure with a finite resource account on its declared valid input domain.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Tate's Algorithm sits in a sparse region of the domain-specific corpus (89th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

References

[^ref-16d97a8a6dc0]: Cremona, John E. (1997). Algorithms for Modular Elliptic Curves. Second edition, author-hosted Chapter III. Relevant locators: §3.1 printed pp. 62–63 / PDF zero-based indices 0–1, Weierstrass invariants and weighted coordinate changes; §3.2 printed pp. 65–68 / indices 3–6, local outputs, distinct minimization algorithm and Tate pseudocode. Full authored exposition and implementation account; not a direct inspection of Tate's 1975 original paper. The explicit pseudocode discussed here is over the integers and rational primes.

[^ref-ffceb454b4ac]: SageMath Development Team. (2026). “Local data for elliptic curves over number fields”. Official reference manual, retained capture accessed 2026-10-16. Relevant locators: introduction and Gaussian-field example; EllipticCurveLocalData constructor's algorithm option and 14a1 example. The documented results were inspected, not independently rerun; the wrapper's PARI option is restricted to the rational base field.

[^ref-9ee7a3a31936]: King-Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki-Seng Tan and Fabien Trihan, “On the mu-invariants of abelian varieties over function fields of positive characteristic”, original research preprint, arXiv version 4, 20 October 2020. Relevant locators: §7.3 setup printed p. 39 / PDF zero-based index 38; explicit LocalInformation sum and reported degree 24 printed p. 40 / index 39. The narrative/code sign discrepancy is preserved; no independently reproduced numeric result or unprinted per-place output is claimed. The separately retained July 2020 university manuscript is another edition of this work, with an extra wrapper page.

[^ref-d7c31b7c0003]: Computational Algebra Group, University of Sydney, “Local Computations”, Magma Handbook, V2.29 footer dated 3 August 2026, official web text accessed 2026-10-16. Relevant locator: LocalInformation(E, Pl), named Tate-algorithm statement, input preparation and tuple/minimal-model output contract. Official full web text was inspected; no successful local byte capture or independent execution is claimed.

[^ref-22dea966f7e3]: PARI/GP Development Team. (2026). “Catalogue of GP/PARI Functions — Elliptic curves”. Current official web manual accessed 2026-10-16. Relevant locator: elllocalred(E, {p}), accepted base fields and prime specification, four-component result and minimality coordinate change. Current interface contract only; not a historical implementation claim or a computation rerun here.