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 turns a local elliptic-curve problem into an ordered decision procedure: choose a curve model and a place, make suitable coordinate changes, and test divisibility and residue-field roots until a reduction case is identified.^[1] In Cremona's explicit rational-prime formulation, the outputs include a Kodaira symbol, conductor exponent and local index, together with detection and correction of a nonminimal model.^[1] The recurring relation is model at a place → normalized tests → local reduction finding, rather than the classification of one distinguished curve.

The place is part of the question. Sage's number-field interface describes reduction type, a local minimal equation, Tamagawa number and conductor exponent at a finite place.^[2] A function-field implementation can expose related local data at a place of a curve's base field; Magma explicitly identifies its LocalInformation(E, Pl) routine with Tate's algorithm.^[3] The common operation is local even when its results are later combined into a global invariant.

Structural Signature¶

The procedure has five functional roles:

Elliptic-curve model. A Weierstrass equation supplies the coefficients; its discriminant must be nonzero for the curve to be elliptic.^[1]
Selected place and arithmetic domain. A prime, prime ideal or function-field place supplies the valuation and residue setting in which the local question is asked.^[2]^[3]
Ordered normalization and tests. Coordinate changes prepare divisibility conditions and auxiliary quadratic or cubic root tests distinguish subsequent branches, so each later test depends on the conditions established earlier.^[1]
Minimality control. An already minimal model need not be rescaled; at the nonminimal branch, the rational-prime algorithm divides the coefficient of weight i by the corresponding power of the prime and restarts.^[1]
Local reduction finding. The result records the reduction case and associated local information, but an application may consume only one part of it: the function-field research example below uses local minimal-discriminant valuations in an aggregate without printing every local symbol.^[4]^[3]

Remove the model or place and there is no determinate local input. Remove the procedure or its established implementation path and a displayed Kodaira symbol alone does not establish how it was computed. Remove minimality control and excess valuation belonging to a nonminimal equation can be mistaken for intrinsic local data.

What It Is Not¶

It is not a universal shortcut from the valuations of j and the discriminant to every local output. Cremona's multiplicative branch tests splitting: for type In over the rational-prime setting, the local index is n in the split case, but is 2 for even n and 1 for odd n in the nonsplit case.^[1] Sage supplies a concrete warning: curve 14a1 at 2 has nonsplit multiplicative type I6 and minimal-discriminant valuation 6, yet Tamagawa number 2.^[2]

It is not identical to any procedure that finds a minimal equation. Cremona separately presents the Laska–Kraus–Connell minimization algorithm before Tate's fuller local-data algorithm.^[1] Nor is it the Enriques–Kodaira classification of compact complex surfaces: a local elliptic-fibre reduction symbol and a global surface classification answer different questions.

Scope of Application¶

The verified habitats include elliptic curves over number fields at finite places and elliptic curves over function fields at selected places.^[2]^[3] Cremona's explicit pseudocode is written for integer Weierstrass coefficients and a rational prime; its coordinate calculations distinguish residue characteristics 2 and 3.^[1] A presentation over integers should not silently be treated as an executable recipe for every possible residue field.

Input requirements also depend on the interface. Magma's function-field routine permits a nonintegral input model and returns an integral model minimal at the selected place.^[3] Thus an integral mathematical starting model and an implementation that first prepares one are compatible descriptions, rather than competing definitions.

Clarity¶

Separate an invariant's value from its valuation. Under Cremona's coordinate transformation with scale u, the relations are u^4 c4′ = c4, u^6 c6′ = c6 and u^12 Δ′ = Δ, while j′ = j.^[1] Consequently, transformations with u = 1 leave those values unchanged; a valuation-unit scale preserves their valuations without generally preserving their numerical values. Nonunit rescaling changes valuations according to the displayed weights.

Also separate a geometric root condition from arithmetic splitting. In the prepared cubic branch for I0∗, Cremona tests that the cubic discriminant is nonzero modulo the prime, then determines the index as one plus the number of roots in the residue field.^[1] Distinct roots over an algebraic closure do not mean all three roots belong to that residue field.

Manages Complexity¶

The procedure replaces an undifferentiated bad-reduction problem with successive exits. Cremona's account moves from good and multiplicative tests through additive cases, using a more involved refinement loop for Im∗ and a separate path for the remaining starred cases or nonminimality.^[1] This organization makes the prerequisites of an answer inspectable: which coordinates were prepared, which divisibility test failed, and which residue-root question remained?

The compression has a cost. A short label summarizes a substantial local calculation, and the label does not by itself record splitting, the coordinate changes or the field in which it was obtained. The useful simplification is the structured finding, with its local context retained, rather than erasing the calculation's conditions.

Abstract Reasoning¶

The split-versus-nonsplit distinction supports a direct diagnostic. If an index was inferred solely from the discriminant valuation in a multiplicative case, ask which root test establishes splitting before accepting the value. The 14a1 example shows why matching the valuation to the subscript of I6 is insufficient for inferring its Tamagawa number.^[2]

Minimality creates a second diagnostic. In Cremona's procedure, a nonminimal exit triggers weighted scaling and a restart.^[1] Comparing valuations before and after that step requires tracking the model change; the original coefficient list is not a permanent reference for the final local answer. Neither diagnostic supplies an unreported branch trace for a software example.

Knowledge Transfer¶

Literal transfer stays within the local arithmetic of elliptic curves: substitute another valid model and place while retaining the applicable valuation, coordinate and residue-root rules. Sage's number-field case and the function-field research case demonstrate unlike uses of that structure.^[2]^[4]^[3] They do not make the software interfaces interchangeable.

Implementation names need separate checking. Sage's documented algorithm='pari' choice applies only over the rationals, while its general number-field implementation is a distinct path.^[2] Current PARI documentation separately allows elllocalred over a number field with a suitable prime ideal.^[5] A restriction on one wrapper's selection option is not a restriction on every version of the underlying library's API.

Outside elliptic-curve arithmetic, ordered tests and normalization are useful analogies. Without a Weierstrass model, local place and reduction interpretation, however, they are generic decision procedures rather than Tate's algorithm.

Examples¶

Number-field reduction at a Gaussian prime¶

Sage's official example constructs the field Q(i), a curve with short Weierstrass coefficients (2+i)^2 and (2+i)^7, and the prime ideal generated by 2+i.^[2] Its local-data call reports bad but not multiplicative reduction, Kodaira type I0∗ and Tamagawa number 4; its minimal-model call displays y^2 = x^3 + (4i+3)x + (-29i-278).^[2] These are maintainer-documented results, not a computation independently rerun here. This particular example does not print a conductor exponent or its internal branch trace.

Mapped back: the coefficient list supplies the model; the Gaussian prime supplies the local place; the documented local-data interface supplies the established procedure path; the returned minimal equation supplies the minimality result; and I0∗ with Tamagawa 4 supplies the displayed local finding. The map does not require inventing which coordinate transformations were executed.

Local data inside a function-field calculation¶

Lai and coauthors' arXiv v4 §7.3 studies a Legendre elliptic curve over F7(x), base-changed to a quadratic function field.^[4] Their displayed code enumerates BadPlaces(EK) and forms a degree-weighted sum of the second field of LocalInformation(EK,v), reporting discriminant degree 24.^[4] Magma's official contract identifies the second tuple field as the valuation of the local minimal discriminant and describes the routine as performing Tate's algorithm.^[3]

The paper's narrative and displayed field/twist code contain sign discrepancies.^[4] The number 24 is therefore retained as the authors' reported result, not certified here by silently selecting a corrected equation. No individual place's Kodaira symbol, conductor exponent or internal branch trace is printed in this calculation.^[4] The characteristic-three example in the following subsection concerns a different curve.^[4]

Mapped back: the base-changed elliptic curve is the model; each enumerated bad place is a local place; the explicit named routine establishes the procedure path; the routine's contract supplies local minimality control; and the selected valuation fields feed the reported aggregate finding. Contract fields not displayed by the paper are not promoted into observed outputs.

Structural Tensions¶

T1 — compact invariant summaries versus splitting information. A valuation summary can make a local case easy to recognize, while the arithmetic index may still depend on residue-root information. Compressing the answer to a few numbers can discard precisely the distinction needed for a downstream inference. Diagnostic: Which part of this answer follows from valuations, and which still depends on splitting?

T2 — short decision paths versus transformation bookkeeping. Early exits are economical; exceptional branches require carefully prepared coordinates and repeated local tests. Tracking only values confuses unit scaling with u = 1, while tracking only valuations can lose the coefficients needed for a later root test. Diagnostic: Have the prerequisites survived the last coordinate change, and which quantities were preserved as values rather than only as valuations?

T3 — an autonomous arithmetic method versus reduction to a generic procedure. Model, place, tests and finding expose a portable decision structure. Removing elliptic-curve arithmetic makes that structure easier to generalize but removes the named method's discriminating content. Keeping every implementation as an isolated tool hides the common local operation. Diagnostic: What remains Tate-specific after replacing the software, and what disappears if the Weierstrass and reduction conditions are removed?

Structural–Framed Character¶

Tate's algorithm is structural within a mathematical frame. Once the model, place and applicable arithmetic are fixed, the branches concern divisibility, coordinate equivalence and roots rather than a judgment of aesthetic quality. Human practice enters in choosing the representation, implementation and reporting conventions; a person need not choose the outcome of each valid branch.

Its evaluative weight is limited: computational economy and readable implementations are aims, not the definition of correct reduction. Its institutional origin belongs to mathematical research and software implementation, but no particular package is essential. Its vocabulary travels only partly: model, test and branch are portable, while Weierstrass coefficients, local minimality, Tamagawa index and Kodaira type retain their arithmetic obligations.

Import versus recognition favors importing those obligations before recognizing an instance. A table of familiar symbols or an application named after Tate is insufficient without the curve/place/procedure relation. Its character: a reusable formal method whose source-dependent representations and implementation choices sit inside a sharply bounded mathematical domain.

Structural Core vs. Domain Accent¶

The skeletal core is preparation followed by ordered tests, with a restart when the representation is unsuitable and an interpretable finding on exit. That shape can guide comparisons with other algorithms. The live Algorithm prime is a tested strict parent for the admitted effective local procedure: specified valid inputs, finite ordered operations, termination on that domain, local outputs and a symbolic resource account are necessary to it. The parent relation does not claim polynomial bit complexity or a theorem for arbitrary non-effective local fields.

The domain-bound content supplies the identity: an elliptic curve in Weierstrass form, a discrete local place, weighted coordinate changes, residue-root questions and a reduction/minimality interpretation. Substituting a different curve or software package can preserve the method. Substituting arbitrary records and category names does not.

This is why the named method remains domain-specific. Its cross-domain reach becomes an analogy to normalization and decision procedures once the local arithmetic is removed; those broader patterns do not carry the same admissible inputs or conclusions.

This entry is a kind of Algorithm.

Algorithm is the tested strict parent: the calculation has specified inputs, effective ordered operations, terminating exits and a symbolic resource account on its declared valid domain. Classification illuminates assignment to local reduction types, but the method also handles model preparation and associated arithmetic data. Transformation clarifies the coordinate changes, provided the distinction between preserved values and preserved valuations is maintained.

The Algorithm relation uses the complete tested procedural signature on the declared valid domain. Classification and Transformation remain comparisons rather than added DAG edges; a shared word does not establish a second necessary relation or erase the method's arithmetic conditions.

Relationships to Other Abstractions¶

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

Tate's Algorithm → Algorithm → Function (Mapping)

Neighborhood in Abstraction Space¶

Tate's Algorithm sits in a sparse region of the domain-specific corpus (89^th 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

Not to Be Confused With¶

A minimal equation is a result; Tate's algorithm is one procedure capable of finding local data while handling minimality. A Kodaira symbol is part of a classification vocabulary, rather than evidence of the particular computation that produced it.

A local index is not merely the number in the subscript of a multiplicative symbol: the nonsplit example separates them. A function-field discriminant degree is an aggregate of local contributions, not one local reduction type.

A software routine is an implementation with a stated domain and output contract. Its name does not certify that every cited worked example used it, or that every field supported by another package is supported through the same interface.

References¶

[1] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l

[2] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[3] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[4] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[5] 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. registry ↩