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.

AI illustration

How this image was made

AI-generated illustration of Klein quartic

This AI-generated illustration was produced by black-forest-labs/FLUX.2-dev (1024x1024) from a prompt written by openai/gpt-oss-120b from the entity's label + description.

Prompt

Generate an image of the 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.)

All labels observed (5)

How this entity was disambiguated

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 ⓘ
modular curve X(7) ⓘ
linked to: Klein quartic
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

Referenced by (10)

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

Felix Klein → notableWork → Klein quartic ⓘ
Klein quartic → relatedTo → modular curve X(7) ⓘ
linked to: Klein quartic
Hurwitz bound on automorphism groups of curves → relatedConcept → Hurwitz curve ⓘ
linked to: Klein quartic
Hurwitz bound on automorphism groups of curves → attainedBy → Hurwitz curves ⓘ
linked to: Klein quartic
Hurwitz bound on automorphism groups of curves → example → The Klein quartic has 168 = 84(3 − 1) automorphisms ⓘ
linked to: Klein quartic
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 ⓘ
Hurwitz surface → hasExample → Klein quartic ⓘ
subject linked to: Hurwitz surfaces