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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| SL(2,Z) | 4 |
| SL(2,ℤ) canonical | 3 |
| SL₂(ℤ) | 1 |
| modular group SL(2,Z) | 1 |
| modular group SL(2,ℤ) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338618 — 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.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: SL(2,ℤ) Context triple: [PSL(2,ℤ), isQuotientOf, SL(2,ℤ)]
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
-
C.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
D.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: SL(2,ℤ) Target entity description: 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.
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
-
C.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
D.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- F. None of above. chosen
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)
ⓘ
linked to:
SL(2,R)
|
| 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) ⓘ |
| relation |
(ST)^3 = S^2
ⓘ
S^4 = I ⓘ |
| roleInEllipticCurves | classifies complex elliptic curves up to isomorphism via j-invariant ⓘ |
| symbol |
SL(2,Z)
ⓘ
linked to:
SL(2,ℤ)
SL₂(ℤ) ⓘ
linked to:
SL(2,ℤ)
|
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.
Instruction
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.
Input
Subject: SL(2,ℤ) Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
modular group PSL(2,Z)
subject linked to:
modular group PSL(2,Z)
linked to: SL(2,ℤ)
linked to: SL(2,ℤ)
linked to: SL(2,ℤ)
linked to: SL(2,ℤ)
subject linked to:
T:z ↦ z+1
linked to: SL(2,ℤ)