Triple

T7338618
Position Surface form Disambiguated ID Type / Status
Subject PSL(2,ℤ) E169191 entity
Predicate isQuotientOf P39249 FINISHED
Object SL(2,ℤ)
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.
E656692 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: SL(2,ℤ) | Statement: [PSL(2,ℤ), isQuotientOf, SL(2,ℤ)]
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.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: SL(2,ℤ)
Triple: [PSL(2,ℤ), isQuotientOf, SL(2,ℤ)]
Generated 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.
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

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69c68a57710481909f0c1f3c6ebdb6f2 completed March 27, 2026, 1:47 p.m.
NER Named-entity recognition batch_69c6f0d702108190a00a3681ff6e67d4 completed March 27, 2026, 9:04 p.m.
NED1 Entity disambiguation (via context triple) batch_69c7ef266fd0819096cf3ece3fff6b90 completed March 28, 2026, 3:09 p.m.
NEDg Description generation batch_69c7efa4f5148190842f30988cbea94c completed March 28, 2026, 3:11 p.m.
NED2 Entity disambiguation (via description) batch_69c7f0092bac819080ded1863f99290a completed March 28, 2026, 3:13 p.m.
Created at: March 27, 2026, 3:04 p.m.