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Fungrim entry: 298bb1

(E2k ⁣(τ)=0  and  τ{i,e2πi/3})        (τQ)\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:kZ2  and  τFk \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathcal{F}
\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}
Fungrim symbol Notation Short description
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
AlgebraicNumbersQ\overline{\mathbb{Q}} Algebraic numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
Source code for this entry:
    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))),

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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