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Fungrim entry: 10f3b2

E6 ⁣(τ)=12(b12+c123a8(b4+c4))   where a=θ2 ⁣(0,τ),b=θ3 ⁣(0,τ),c=θ4 ⁣(0,τ)E_{6}\!\left(\tau\right) = \frac{1}{2} \left({b}^{12} + {c}^{12} - 3 {a}^{8} \left({b}^{4} + {c}^{4}\right)\right)\; \text{ where } a = \theta_{2}\!\left(0 , \tau\right),\,b = \theta_{3}\!\left(0 , \tau\right),\,c = \theta_{4}\!\left(0 , \tau\right)
Assumptions:τH\tau \in \mathbb{H}
TeX:
E_{6}\!\left(\tau\right) = \frac{1}{2} \left({b}^{12} + {c}^{12} - 3 {a}^{8} \left({b}^{4} + {c}^{4}\right)\right)\; \text{ where } a = \theta_{2}\!\left(0 , \tau\right),\,b = \theta_{3}\!\left(0 , \tau\right),\,c = \theta_{4}\!\left(0 , \tau\right)

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("10f3b2"),
    Formula(Equal(EisensteinE(6, tau), Where(Mul(Div(1, 2), Sub(Add(Pow(b, 12), Pow(c, 12)), Mul(Mul(3, Pow(a, 8)), Add(Pow(b, 4), Pow(c, 4))))), Equal(a, JacobiTheta(2, 0, tau)), Equal(b, JacobiTheta(3, 0, tau)), Equal(c, JacobiTheta(4, 0, tau))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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2019-08-19 14:38:23.809000 UTC