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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (5)

Label Occurrences
Klein quartic canonical 5
Hurwitz curve 1
Hurwitz curves 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 (9)

Full triples — surface form annotated when it differs from this entity's canonical label.

Felix Klein → notableWork → Klein quartic ⓘ
Klein quartic → relatedTo → Klein quartic self-linksurface differs ⓘ
this entity surface form: modular curve X(7)
Hurwitz bound on automorphism groups of curves → relatedConcept → Klein quartic ⓘ
this entity surface form: Hurwitz curve
Hurwitz bound on automorphism groups of curves → attainedBy → Klein quartic ⓘ
this entity surface form: Hurwitz curves
Hurwitz bound on automorphism groups of curves → example → Klein quartic ⓘ
this entity surface form: The Klein quartic has 168 = 84(3 − 1) automorphisms
PSL(2,7) → fullAutomorphismGroupOf → Klein quartic ⓘ
PSL(2,7) → actsFaithfullyOn → Klein quartic ⓘ
(2,3,7) triangle group → isRelatedTo → Klein quartic ⓘ
(2,3,7) triangle group → uniformizes → Klein quartic ⓘ