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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Mordell equation | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
elliptic curve ⓘ plane cubic curve ⓘ |
| appearsIn | Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| hasNameOrigin | Louis J. Mordell NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| studiedIn |
Diophantine geometry
NERFINISHED
ⓘ
arithmetic geometry NERFINISHED ⓘ number theory ⓘ |
| usedToStudy |
Mordell’s theorem
NERFINISHED
ⓘ
rational points on elliptic curves ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Mordell equation