SL(2,ℤ)

E656692
discrete group finitely generated group group infinite group lattice in Lie group linear group matrix group non-abelian group

SL(2,ℤ) is the group of 2×2 integer matrices with determinant 1, fundamental in number theory, geometry, and the theory of modular forms.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form Occurrences
modular group SL(2,Z) 1

Statements (49)

Predicate Object
instanceOf discrete group ⓘ
finitely generated group ⓘ
group ⓘ
infinite group ⓘ
lattice in Lie group ⓘ
linear group ⓘ
matrix group ⓘ
non-abelian group ⓘ
abelianizationIs cyclic group of order 12 ⓘ
actionTypeOnUpperHalfPlane fractional linear transformations ⓘ
actsOn set of lattices in ℂ ⓘ
upper half-plane ℍ ⓘ
centerIs {±I} ⓘ
congruenceSubgroups Γ(N) ⓘ
containsSubgroupIsomorphicTo free group on two generators ⓘ
covolumeInSL2R finite ⓘ
definedAs group of 2×2 integer matrices with determinant 1 ⓘ
determinantCondition determinant = 1 ⓘ
fundamentalDomainForActionOn upper half-plane ℍ ⓘ
fundamentalIn hyperbolic geometry ⓘ
number theory ⓘ
theory of modular forms ⓘ
generatedBy S = [[0,-1],[1,0]] ⓘ
T = [[1,1],[0,1]] ⓘ
hasPropertyT false ⓘ
identityElement 2×2 identity matrix ⓘ
isCountable true ⓘ
isLatticeIn SL(2,ℝ) NERFINISHED ⓘ
isLinear true ⓘ
isNonAmenable true ⓘ
isomorphicTo free product C₄ *_{C₂} C₆ ⓘ
isPerfect false ⓘ
isResiduallyFinite true ⓘ
isUniversalCoverOf PSL(2,ℤ) up to center ⓘ
matrixSize 2×2 ⓘ
modularGroup true ⓘ
operation matrix multiplication ⓘ
over integers ℤ ⓘ
presentation ⟨S,T | S^4 = I, S^2 = (ST)^3⟩ ⓘ
principalCongruenceSubgroupDefinition kernel of reduction mod N homomorphism SL(2,ℤ) → SL(2,ℤ/Nℤ) ⓘ
PSL2ZIsomorphicTo free product C₂ * C₃ ⓘ
PSL2ZPresentation ⟨S̄,T̄ | S̄^2 = (S̄T̄)^3 = 1⟩ ⓘ
quotientByCenterIs PSL(2,ℤ) ⓘ
relatedObject modular curve X(1) NERFINISHED ⓘ
relation (ST)^3 = S^2 ⓘ
S^4 = I ⓘ
roleInEllipticCurves classifies complex elliptic curves up to isomorphism via j-invariant ⓘ
symbol SL(2,Z) NERFINISHED ⓘ
SL₂(ℤ) NERFINISHED ⓘ

Referenced by (3)

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

modular group PSL(2,Z) → commensurableWith → SL(2,ℤ) ⓘ
subject surface form: PSL(2,ℤ)
modular group PSL(2,Z) → isQuotientOf → SL(2,ℤ) ⓘ
subject surface form: PSL(2,ℤ)
Ramanujan theta function → relatedTo → SL(2,ℤ) ⓘ
this entity surface form: modular group SL(2,Z)