In mathematics, Felix Klein's j-invariant or j function is a modular function of weight zero for the special linear group SL ⁡ ( 2 , Z ) {\displaystyle...

of the normalized j-invariant exapand as, 1728   j ( τ ) = 1 / q + 744 + 196884 q + 21493760 q 2 + ⋯ {\displaystyle 1728{\text{ }}j(\tau...

The j-invariant of an elliptic curve given by the Weierstrass equation y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} is given by the formula: j (...

J-function may refer to: The Klein j-invariant or j function in mathematics Leverett J-function in petroleum engineering This disambiguation page lists...

Reshetikhin–Turaev invariant Tau-invariant I-Invariant Klein J-invariant Quantum isotopy invariant Ermakov–Lewis invariant Hermitian invariant Goussarov–Habiro theory...

affine line, the coarse moduli space of elliptic curves, given by the j-invariant of an elliptic curve. It is a classical observation that every elliptic...

invariants are invariants introduced by Vladimir Arnold in 1994 for studying the topology and geometry of plane curves. The three main invariants— J +...

comes from the phrase "singular values of the j {\displaystyle j} -invariant" used for values of the j-invariant for which a complex elliptic curve has complex...

of the j-invariant. In what follows, j(z) denotes the j-invariant of the complex number z. Briefly, j ( 1 + − d 2 ) {\displaystyle \textstyle j\left({\frac...

equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant. The curve is sometimes called X0(n), though...

only if their j {\displaystyle j} -invariants agree, the affine space A j 1 {\displaystyle \mathbb {A} _{j}^{1}} parameterizing j-invariants of elliptic...

sense of algebraic geometry. Reversing this is accomplished by the j-invariant j(E), which can be used to determine τ and hence a torus. The set of all...

degree term of the expansion of Klein's j-invariant, and the zeroth degree term of the Laurent series of the J-invariant. Furthermore, 744 = 3 × 248 where 248...

smooth curve appears is described by the j-invariant in the table. Over the complex numbers, the curve with j-invariant 0 is the unique elliptic curve with...

coordinate: that is, both the Laplace–Beltrami operator and the curvature are invariant under isometries. Thus, for example, let S be a Riemann surface with metric...

that the first few terms in the Fourier expansion of the normalized J-invariant (sequence A014708 in the OEIS) could be expressed in terms of linear...

j, the second imaginary unit of a quaternion j, an index variable in a matrix The j-invariant, a modular function J, Joule, the SI unit of energy J,...

as cubic and quartic twists. The curve and its twists have the same j-invariant. Applications of twists include cryptography, the solution of Diophantine...

Fundamental region of the modular j-invariant in the upper half-plane (shaded gray), with modular discriminant | τ | ≥ 1 {\displaystyle |\tau |\geq 1}...

elliptic curves. Let j = g 2 3 g 2 3 − 27 g 3 2 {\displaystyle j={\frac {g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}}} be the j-invariant with g 2 {\displaystyle...

tiling of the Poincaré disk is given in a natural way by the J-invariant, which is invariant under the modular group, and attains every complex number once...

product formula for the j-invariant: j ( p ) − j ( q ) = ( 1 p − 1 q ) ∏ n , m = 1 ∞ ( 1 − p n q m ) c n m . {\displaystyle j(p)-j(q)=\left({1 \over p}-{1...

there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane H 2 {\displaystyle \mathbb {H}...

Stanford University, April 1988. Milne, A., Sethares, W.A. and Plamondon, J., Invariant Fingerings Across a Tuning Continuum, Computer Music Journal, Winter...

lambda function Closely related are the modular forms, which include J-invariant Dedekind eta function Airy function Bessel functions: Defined by a differential...

745. 744 is a semiperfect number. It is also an abundant number. The j-invariant, an important function in the study of modular forms and Monstrous moonshine...

the one given above. The term "singular elliptic curve" (or "singular j-invariant") was at one times used to refer to complex elliptic curves whose ring...

In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another...

class invariant (or type invariant) is an invariant used for constraining objects of a class. Methods of the class should preserve the invariant. The class...

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view...