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.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hauptmodul
ⓘ
classical modular function ⓘ complex analytic function ⓘ invariant of elliptic curves ⓘ modular function ⓘ |
| appearsIn | Hilbert class polynomial NERFINISHED ⓘ |
| 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)
NERFINISHED
ⓘ
modular group NERFINISHED ⓘ |
| isAlgebraicFunctionOf | lambda modular function on appropriate covers ⓘ |
| isHauptmodulFor | SL(2,Z) NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Eisenstein series E6 NERFINISHED ⓘ Monster group via monstrous moonshine ⓘ Weierstrass ℘-function NERFINISHED ⓘ 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 NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.