Klein quartic
E50328
Hurwitz surface
algebraic curve
compact Riemann surface
complex algebraic curve
projective algebraic curve
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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| modular curve X(7) | 1 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
modular curve X(7)