Klein quartic
E50328
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Klein quartic canonical | 5 |
| Hurwitz curve | 1 |
| Hurwitz curves | 1 |
| The Klein quartic has 168 = 84(3 − 1) automorphisms | 1 |
| modular curve X(7) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T397946 — 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.
Target entity: Klein quartic Context triple: [Felix Klein, notableWork, Klein quartic]
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A.
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.
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B.
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.
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C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
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D.
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.
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E.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Klein quartic Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
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.
-
E.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Hurwitz surface
ⓘ
algebraic curve ⓘ compact Riemann surface ⓘ complex algebraic curve ⓘ projective algebraic curve ⓘ |
| ambientSpace | complex projective plane P^2(C) ⓘ |
| associatedCayleyGraph | Cayley graph of PSL(2,7) ⓘ |
| automorphismGroup |
PSL(2,7)
ⓘ
projective special linear group of 2x2 matrices over field with 7 elements ⓘ |
| automorphismGroupOrder | 168 ⓘ |
| definedOver |
algebraic numbers
ⓘ
complex numbers ⓘ rational numbers ⓘ |
| degree | 4 ⓘ |
| edgesInMinimalTriangulation | 84 ⓘ |
| equation | x^3 y + y^3 z + z^3 x = 0 ⓘ |
| EulerCharacteristic | -4 ⓘ |
| facesInMinimalTriangulation | 56 ⓘ |
| FuchsianGroupSignature | (2,3,7) ⓘ |
| genus | 3 ⓘ |
| hasBelyiMap | yes ⓘ |
| hasHyperbolicArea | 8π ⓘ |
| hasRealizationAs | quotient of upper half-plane by (2,3,7) triangle group ⓘ |
| hasRealModel | highly symmetric genus-3 surface embedded in R^3 ⓘ |
| hasSpecialRoleIn |
algebraic geometry
ⓘ
complex geometry ⓘ group theory ⓘ theory of Riemann surfaces ⓘ |
| hasSymmetryGroupOrder | 168 ⓘ |
| hasTessellation | regular {3,7} triangulation ⓘ |
| hasWeierstrassPoints | 24 ⓘ |
| HurwitzBoundForGenus3 | 168 ⓘ |
| HurwitzBoundValue | 84(g-1) ⓘ |
| isBelyiCurve | yes ⓘ |
| isIsomorphicTo | modular curve X(7) over C ⓘ |
| isQuotientOf | hyperbolic plane by a Fuchsian group ⓘ |
| isRegularMap | yes ⓘ |
| maximalAutomorphismsForGenus | yes ⓘ |
| moduliSpacePoint | Teichmüller curve ⓘ |
| namedAfter | Felix Klein ⓘ |
| relatedTo |
Fano plane
ⓘ
finite projective plane of order 2 ⓘ Klein quartic self-linksurface differs ⓘ
surface form:
modular curve X(7)
|
| relatedToGroupTheory | simple group of order 168 ⓘ |
| relatedToTriangleGroup | (2,3,7) triangle group ⓘ |
| satisfiesHurwitzBound | yes ⓘ |
| uniformizedBy | hyperbolic plane ⓘ |
| verticesInMinimalTriangulation | 24 ⓘ |
| yearIntroduced | 1879 ⓘ |
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.
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.
Subject: Klein quartic Description of subject: 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.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.