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 | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1281488 — 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: Fermat curve Context triple: [Pierre de Fermat, notableWork, Fermat curve]
-
A.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
C.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
D.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
E.
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.
- 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: Fermat curve Target entity description: 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.
-
A.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
C.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
D.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
E.
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.
- F. None of above. chosen
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
ⓘ
linked to:
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 ⓘ |
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: Fermat curve Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.