Mordell curve
E640411
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Mordell curve canonical | 1 |
| Mordell equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7078802 — 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: Mordell curve Context triple: [Louis Mordell, knownFor, Mordell curve]
-
A.
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.
-
B.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
C.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
D.
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.
-
E.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
- 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: Mordell curve Target entity description: 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.
-
A.
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.
-
B.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
C.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
D.
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.
-
E.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
elliptic curve ⓘ plane cubic curve ⓘ |
| appearsIn |
Birch and Swinnerton-Dyer conjecture
ⓘ
linked to:
Birch and Swinnerton-Dyer Conjecture
|
| definedOver | field ⓘ |
| hasAffineModel | y^2 = x^3 + k ⓘ |
| hasAutomorphism | (x,y) ↦ (ζx, y) for ζ^3 = 1 ⓘ |
| hasComplexMultiplication |
by ring of integers of ℚ(√-3)
ⓘ
when j = 0 ⓘ |
| hasComplexPoints | solutions (x,y) ∈ ℂ² of y^2 = x^3 + k ⓘ |
| hasConductorDependingOn | k ⓘ |
| hasCoordinateRing | field[x,y]/(y^2 - x^3 - k) ⓘ |
| hasDefiningEquation | y^2 = x^3 + k ⓘ |
| hasDegree | 3 ⓘ |
| hasDiscriminant | Δ = -27k^2 ⓘ |
| hasGenus | 1 ⓘ |
| hasGoodReductionOutside | primes dividing 6k (over ℚ) ⓘ |
| hasGroupLaw | elliptic curve group law ⓘ |
| hasIdentityElement | point at infinity ⓘ |
| hasIntegralPoints | finite set for fixed nonzero k over ℤ ⓘ |
| hasJInvariant | j = 0 ⓘ |
| hasLFunction |
Hasse–Weil L-function
ⓘ
linked to:
Hasse–Weil zeta function
|
| hasNameOrigin |
Louis J. Mordell
ⓘ
linked to:
Louis Mordell
|
| hasParameter | k ⓘ |
| hasProjectiveClosure | Y^2Z = X^3 + kZ^3 ⓘ |
| hasRank | nonnegative integer ⓘ |
| hasRationalPoints | solutions (x,y) ∈ ℚ² of y^2 = x^3 + k ⓘ |
| hasRationalPointsForming | finitely generated abelian group ⓘ |
| hasRealPoints | solutions (x,y) ∈ ℝ² of y^2 = x^3 + k ⓘ |
| hasShortWeierstrassCoefficients | a = 0, b = k ⓘ |
| hasSymmetry | (x,y) ↦ (x,-y) ⓘ |
| hasTorsionPoints | finite set over ℚ ⓘ |
| isCentralTo |
study of cubic twists of elliptic curves
ⓘ
study of curves with j-invariant 0 ⓘ |
| isDefinedByPolynomial | x^3 + k - y^2 ⓘ |
| isExampleOf |
Diophantine equation
ⓘ
cubic Diophantine equation ⓘ |
| isIsomorphicOverAlgebraicClosureTo | curve y^2 = x^3 + 1 (for k ≠ 0) ⓘ |
| isNonsingularFor | k ≠ 0 ⓘ |
| isSpecialCaseOf | elliptic curve in short Weierstrass form ⓘ |
| isWeierstrassFormOf | elliptic curve ⓘ |
| relatedTo | Mordell–Weil group ⓘ |
| studiedIn |
Diophantine geometry
ⓘ
arithmetic geometry ⓘ
linked to:
Diophantine geometry
number theory ⓘ |
| usedToStudy |
Mordell’s theorem
ⓘ
linked to:
Mordell–Weil theorem
rational points on 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: Mordell curve Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Mordell curve