modular j-invariant
E656674
Hauptmodul
classical modular function
complex analytic function
invariant of elliptic curves
modular function
The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
All labels observed (2)
| Label | Occurrences |
|---|---|
| modular j-invariant canonical | 3 |
| j-invariant is a rigid analytic function of q | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338393 — 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: modular j-invariant Context triple: [Monster group, relatedTo, modular j-invariant]
-
A.
Jacobi theta functions
Jacobi theta functions are special functions in complex analysis and number theory that encode modular and elliptic properties, playing a central role in the theory of elliptic functions, modular forms, and various applications in mathematical physics.
-
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.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
E.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
- 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: modular j-invariant Target entity description: The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
-
A.
Jacobi theta functions
Jacobi theta functions are special functions in complex analysis and number theory that encode modular and elliptic properties, playing a central role in the theory of elliptic functions, modular forms, and various applications in mathematical physics.
-
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.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
E.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hauptmodul
ⓘ
classical modular function ⓘ complex analytic function ⓘ invariant of elliptic curves ⓘ modular function ⓘ |
| appearsIn | Hilbert class polynomial ⓘ |
| associatedWith |
elliptic curves
ⓘ
lattices in the complex plane ⓘ |
| characterizes | isomorphism classes of complex elliptic curves ⓘ |
| classifies | complex elliptic curves up to isomorphism ⓘ |
| codomain | complex numbers ⓘ |
| definedOn | upper half-plane ⓘ |
| domain | complex upper half-plane ⓘ |
| hasFormula |
j(τ) = 1728 E4(τ)^3 / (E4(τ)^3 - E6(τ)^2)
ⓘ
j(τ) = 1728 g_2(τ)^3 / (g_2(τ)^3 - 27 g_3(τ)^2) ⓘ |
| hasFourierExpansion | j(τ) = q^{-1} + 744 + 196884 q + 21493760 q^2 + … ⓘ |
| hasGrowth | |j(τ)| → ∞ as Im(τ) → ∞ ⓘ |
| hasPoleAt | i∞ ⓘ |
| hasPoleOrder | 1 at i∞ ⓘ |
| hasProperty | two complex elliptic curves are isomorphic iff they have the same j-invariant ⓘ |
| holomorphicOn | upper half-plane ⓘ |
| inducesBijectionBetween | SL(2,Z)\H and C ⓘ |
| invariantUnder |
SL(2,Z)
ⓘ
linked to:
SL(2,ℤ)
modular group ⓘ
linked to:
modular group PSL(2,Z)
|
| isAlgebraicFunctionOf | lambda modular function on appropriate covers ⓘ |
| isHauptmodulFor |
SL(2,Z)
ⓘ
linked to:
SL(2,ℤ)
|
| meromorphicOn | extended upper half-plane ⓘ |
| normalization |
constant term 744 in its q-expansion
ⓘ
q^{-1} term has coefficient 1 in its q-expansion ⓘ |
| relatedTo |
Eisenstein series E4
ⓘ
linked to:
Eisenstein series
Eisenstein series E6 ⓘ
linked to:
Eisenstein series
Monster group via monstrous moonshine ⓘ Weierstrass ℘-function ⓘ
linked to:
Weierstrass elliptic functions
modular discriminant Δ ⓘ |
| satisfies | j(τ) = j(γτ) for all γ in SL(2,Z) ⓘ |
| takesAlgebraicValuesAt | CM points ⓘ |
| usedIn |
arithmetic geometry
ⓘ
class field theory ⓘ complex analysis ⓘ complex multiplication theory of elliptic curves ⓘ monstrous moonshine ⓘ number theory ⓘ theory of modular forms ⓘ |
| usedToDefine | singular moduli ⓘ |
| usedToDistinguish | non-isomorphic elliptic curves over C ⓘ |
| usedToParametrize | moduli space of elliptic curves over C ⓘ |
| usesVariable | τ in the upper half-plane ⓘ |
| valuesGenerate | class fields of imaginary quadratic fields ⓘ |
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: modular j-invariant Description of subject: The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: modular j-invariant