Fermat curve
E146193
algebraic curve
curve over the rational numbers
geometrically irreducible curve
nonsingular curve
plane curve
projective curve
smooth 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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Fermat curve canonical | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
curve over the rational numbers ⓘ geometrically irreducible curve ⓘ nonsingular curve ⓘ plane curve ⓘ projective curve ⓘ smooth curve ⓘ |
| ambientSpace |
affine plane
ⓘ
projective plane ⓘ |
| BelyiType | three-point branched cover of ℙ¹ ⓘ |
| classification |
elliptic curve for n = 3 in suitable form
ⓘ
non-hyperelliptic for n ≥ 4 ⓘ |
| definedByEquation |
X^n + Y^n = Z^n in projective coordinates
ⓘ
x^n + y^n = 1 ⓘ |
| degree | n ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| genus | (n − 1)(n − 2)/2 for n ≥ 3 ⓘ |
| hasAutomorphismGroup | large finite group depending on n ⓘ |
| hasComplexPoints | compact Riemann surface for n ≥ 3 ⓘ |
| hasCoverings | covers of the projective line branched at three points ⓘ |
| hasJacobian | abelian variety decomposing into factors with complex multiplication ⓘ |
| hasProperty |
may be singular in characteristic p dividing n
ⓘ
smooth over fields of characteristic not dividing n ⓘ |
| hasSymmetryGroup | group of permutations of coordinates and n-th roots of unity ⓘ |
| hasWeierstrassPoints | points with special gap sequences depending on n ⓘ |
| namedAfter | Pierre de Fermat ⓘ |
| overField |
complex numbers ℂ
ⓘ
rational numbers ℚ ⓘ |
| parameter | positive integer n ≥ 3 ⓘ |
| rationalPointsProperty | for n ≥ 4 has only trivial rational points (by Fermat’s Last Theorem) ⓘ |
| relatedObject |
Fermat surface
ⓘ
superelliptic curve ⓘ |
| relatedTo |
Kummer extensions
ⓘ
cyclotomic fields ⓘ |
| relatedToConjecture |
Fermat's Last Theorem
ⓘ
surface form:
Fermat’s Last Theorem
|
| specialCase | unit circle for n = 2 ⓘ |
| specialCaseOf | superelliptic curve y^m = f(x) ⓘ |
| studiedFor |
Diophantine properties
ⓘ
Galois representations ⓘ Jacobian variety structure ⓘ modular forms connections ⓘ |
| trivialRationalPoints | (±1,0) and (0,±1) in affine form ⓘ |
| usedIn |
examples in arithmetic geometry
ⓘ
examples of curves with many automorphisms ⓘ testing conjectures on rational points ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.