Weierstrass form
E831077
Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Weierstrass equations | 2 |
| Weierstrass form canonical | 2 |
| Weierstrass elliptic curve | 1 |
| Weierstrass equation of elliptic curves | 1 |
| general Weierstrass form | 1 |
| short Weierstrass form | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9931907 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Weierstrass form Context triple: [brainpool curves, curveType, Weierstrass form]
-
A.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
B.
Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
-
C.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Fermat surface
A Fermat surface is an algebraic surface in projective space defined by a homogeneous equation where each variable appears with the same exponent, generalizing the notion of Fermat curves to higher dimensions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Weierstrass form Target entity description: Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
-
A.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
B.
Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
-
C.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Fermat surface
A Fermat surface is an algebraic surface in projective space defined by a homogeneous equation where each variable appears with the same exponent, generalizing the notion of Fermat curves to higher dimensions.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve representation
ⓘ
elliptic curve model ⓘ mathematical concept ⓘ |
| allows | definition of group law on elliptic curve ⓘ |
| belongsTo | classical analysis tradition ⓘ |
| contrastedWith |
Edwards form
ⓘ
Hessian form ⓘ Montgomery form ⓘ |
| definedOver |
complex numbers
ⓘ
field ⓘ finite field ⓘ number field ⓘ |
| enables | efficient arithmetic formulas on elliptic curves ⓘ |
| ensures | curve is nonsingular ⓘ |
| equivalentUpToIsomorphismTo | any elliptic curve over a field of characteristic not 2 or 3 ⓘ |
| hasAffineChart | equation in variables x and y ⓘ |
| hasDomain | projective plane ⓘ |
| hasGeneralEquation | y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 ⓘ |
| hasParameter |
a_1
ⓘ
a_2 ⓘ a_3 ⓘ a_4 ⓘ a_6 ⓘ |
| hasProperty | birationally equivalent to other elliptic curve models ⓘ |
| hasShortEquation | y^2 = x^3 + ax + b ⓘ |
| hasVariant |
general Weierstrass form
ⓘ
linked to:
Weierstrass form
long Weierstrass form ⓘ short Weierstrass form ⓘ
linked to:
Weierstrass form
|
| historicalPeriod | 19th century mathematics ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| relatedTo |
elliptic curve discriminant
ⓘ
j-invariant ⓘ |
| requiresCondition | discriminant nonzero ⓘ |
| specialCaseOf | plane cubic curve ⓘ |
| usedFor |
ECDH
ⓘ
linked to:
ECC
ECDSA ⓘ
linked to:
Elliptic Curve Digital Signature Algorithm
classification of elliptic curves up to isomorphism ⓘ computing invariants of elliptic curves ⓘ elliptic curve cryptographic protocols ⓘ point addition on elliptic curves ⓘ scalar multiplication on elliptic curves ⓘ |
| usedIn |
algebraic geometry
ⓘ
computational number theory ⓘ elliptic curve cryptography ⓘ
linked to:
Elliptic Curve Cryptography
elliptic curve theory ⓘ number theory ⓘ public-key cryptography ⓘ |
| usedInStandard |
ANSI X9.62 elliptic curve standards
ⓘ
FIPS 186 elliptic curve specifications ⓘ
linked to:
FIPS 186
SEC 2 recommended elliptic curves ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weierstrass form Description of subject: Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Weierstrass form
linked to: Weierstrass form
linked to: Weierstrass form
linked to: Weierstrass form
linked to: Weierstrass form
Tate’s non-archimedean uniformization of elliptic curves
→
relatesTo
→
Weierstrass equation of elliptic curves
ⓘ
linked to: Weierstrass form