Modular j-invariant

Definitions

Symbol: ModularJ $j\!\left(\tau\right)$ Modular j-invariant

Illustrations

Image: X-ray of $j\!\left(\tau\right)$ on $\tau \in \left[-1, 1\right] + \left[0, 2\right] i$ with $\mathcal{F}$ highlighted

Modular transformations

$j\!\left(\tau + 1\right) = j\!\left(\tau\right)$
$j\!\left(-\frac{1}{\tau}\right) = j\!\left(\tau\right)$
$j\!\left(\frac{a \tau + b}{c \tau + d}\right) = j\!\left(\tau\right)$

Special values

$j\!\left({e}^{\pi i / 3}\right) = 0$
$j\!\left(i\right) = 1728$
$j\!\left(2 i\right) = {66}^{3} = 287496$
$j\!\left(\sqrt{2} i\right) = {20}^{3} = 8000$
$j\!\left(3 i\right) = 64 {\left(2 + \sqrt{3}\right)}^{2} {\left(21 + 20 \sqrt{3}\right)}^{3}$
$j\!\left(4 i\right) = 27 {\left(724 + 513 \sqrt{2}\right)}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{7} i\right)\right) = -{15}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{11} i\right)\right) = -{32}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{19} i\right)\right) = -{96}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{43} i\right)\right) = -{960}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{67} i\right)\right) = -{5280}^{3}$
$j\!\left(\frac{1}{2} \left(1 + \sqrt{163} i\right)\right) = -{640320}^{3}$

Connection formulas

$j\!\left(\tau\right) = \frac{32 {\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3}}{{\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}}$
$j\!\left(\tau\right) = {\left({\left(\frac{\eta\!\left(\tau\right)}{\eta\!\left(2 \tau\right)}\right)}^{8} + {2}^{8} {\left(\frac{\eta\!\left(2 \tau\right)}{\eta\!\left(\tau\right)}\right)}^{16}\right)}^{3}$
$j\!\left(\tau\right) = \frac{E_{4}^{3}\!\left(\tau\right)}{\eta^{24}\!\left(\tau\right)}$

Derivatives

$j'(\tau) = -2 \pi i \frac{E_{14}\!\left(\tau\right)}{\eta^{24}\!\left(\tau\right)}$
$j'(\tau) = -2 \pi i \frac{E_{6}\!\left(\tau\right)}{E_{4}\!\left(\tau\right)} j\!\left(\tau\right)$

Analytic properties

$\operatorname{HolomorphicDomain}\!\left(j\!\left(\tau\right), \tau, \mathbb{H}\right) = \mathbb{H}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} j\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} j\!\left(\tau\right) = \left\{ \gamma \circ {e}^{2 \pi i / 3} : \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}$
$\left\{ j\!\left(\tau\right) : \tau \in \mathcal{F} \right\} = \mathbb{C}$
$\# \mathop{\operatorname{solutions}\,}\limits_{\tau \in \mathcal{F}} \left[j\!\left(\tau\right) = z\right] = 1$

Hilbert class polynomials

Symbol: PrimitiveReducedPositiveIntegralBinaryQuadraticForms $\mathcal{Q}^{*}_{D}$ Primitive reduced positive integral binary quadratic forms
$\mathcal{Q}^{*}_{D} = \left\{ \left(a, b, c\right) : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {b}^{2} - 4 a c = D \,\mathbin{\operatorname{and}}\, \left|b\right| \le a \le c \,\mathbin{\operatorname{and}}\, \left(\left(\left|b\right| = a \,\mathbin{\operatorname{or}}\, a = c\right) \implies \left(b \ge 0\right)\right) \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, b, c\right) = 1 \right\}$
Symbol: HilbertClassPolynomial $H_{D}\!\left(x\right)$ Hilbert class polynomial
$H_{D}\!\left(x\right) = \prod_{\left(a, b, c\right) \in \mathcal{Q}^{*}_{D}} \left(x - j\!\left(\frac{-b + \sqrt{D}}{2 a}\right)\right)$
Table of $H_{D}\!\left(x\right)$ for $-D \le 68$

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

2019-09-20 18:07:53.062439 UTC