Fungrim entry: d5f569

j\!\left(\frac{a \tau + b}{c \tau + d}\right) = j(\tau)

Assumptions:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

j\!\left(\frac{a \tau + b}{c \tau + d}\right) = j(\tau)

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`ModularJ`	$j(\tau)$	Modular j-invariant
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("d5f569"),
    Formula(Equal(ModularJ(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), ModularJ(tau))),
    Variables(a, b, c, d, tau),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

Topics using this entry

Modular j-invariant

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC