Fungrim entry: 298bb1

\left(E_{2 k}\!\left(\tau\right) = 0 \;\mathbin{\operatorname{and}}\; \tau \notin \left\{i, {e}^{2 \pi i / 3}\right\}\right) \;\implies\; \left(\tau \notin \overline{\mathbb{Q}}\right)

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathcal{F}

References:

https://doi.org/10.14992/00008713

TeX:

\left(E_{2 k}\!\left(\tau\right) = 0 \;\mathbin{\operatorname{and}}\; \tau \notin \left\{i, {e}^{2 \pi i / 3}\right\}\right) \;\implies\; \left(\tau \notin \overline{\mathbb{Q}}\right)

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathcal{F}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group

Source code for this entry:

Entry(ID("298bb1"),
    Formula(Implies(And(Equal(EisensteinE(Mul(2, k), tau), 0), NotElement(tau, Set(ConstI, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))), NotElement(tau, AlgebraicNumbers))),
    Variables(tau, k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, ModularGroupFundamentalDomain))),
    References("https://doi.org/10.14992/00008713"))

Topics using this entry

Eisenstein series