Fungrim entry: 1b2d8a

\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\}

TeX:

\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\}

Definitions:

Fungrim symbol	Notation	Short description
`ModularJ`	$j\!\left(\tau\right)$	Modular j-invariant
`HH`	$\mathbb{H}$	Upper complex half-plane
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ModularGroupAction`	$\gamma \circ \tau$	Action of modular group
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)

Source code for this entry:

Entry(ID("1b2d8a"),
    Formula(Equal(Zeros(ModularJ(tau), tau, Element(tau, HH)), SetBuilder(ModularGroupAction(gamma, Exp(Div(Mul(Mul(2, ConstPi), ConstI), 3))), gamma, Element(gamma, PSL2Z)))))

Topics using this entry

Modular j-invariant