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Modular j-invariant

Table of contents: Definitions - Illustrations - Modular transformations - Special values - Connection formulas - Analytic properties - Hilbert class polynomials

Definitions

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Symbol: ModularJ — j ⁣(τ)j\!\left(\tau\right) — Modular j-invariant

Illustrations

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

Modular transformations

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j ⁣(τ+1)=j ⁣(τ)j\!\left(\tau + 1\right) = j\!\left(\tau\right)
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j ⁣(−1τ)=j ⁣(τ)j\!\left(-\frac{1}{\tau}\right) = j\!\left(\tau\right)
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j ⁣(aτ+bcτ+d)=j ⁣(τ)j\!\left(\frac{a \tau + b}{c \tau + d}\right) = j\!\left(\tau\right)

Special values

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j ⁣(eπi/3)=0j\!\left({e}^{\pi i / 3}\right) = 0
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j ⁣(i)=1728j\!\left(i\right) = 1728
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j ⁣(2i)=663=287496j\!\left(2 i\right) = {66}^{3} = 287496
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j ⁣(2i)=203=8000j\!\left(\sqrt{2} i\right) = {20}^{3} = 8000
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j ⁣(3i)=64(2+3)2(21+203)3j\!\left(3 i\right) = 64 {\left(2 + \sqrt{3}\right)}^{2} {\left(21 + 20 \sqrt{3}\right)}^{3}
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j ⁣(4i)=27(724+5132)3j\!\left(4 i\right) = 27 {\left(724 + 513 \sqrt{2}\right)}^{3}
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j ⁣(12(1+7i))=−153j\!\left(\frac{1}{2} \left(1 + \sqrt{7} i\right)\right) = -{15}^{3}
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j ⁣(12(1+11i))=−323j\!\left(\frac{1}{2} \left(1 + \sqrt{11} i\right)\right) = -{32}^{3}
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j ⁣(12(1+19i))=−963j\!\left(\frac{1}{2} \left(1 + \sqrt{19} i\right)\right) = -{96}^{3}
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j ⁣(12(1+43i))=−9603j\!\left(\frac{1}{2} \left(1 + \sqrt{43} i\right)\right) = -{960}^{3}
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j ⁣(12(1+67i))=−52803j\!\left(\frac{1}{2} \left(1 + \sqrt{67} i\right)\right) = -{5280}^{3}
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j ⁣(12(1+163i))=−6403203j\!\left(\frac{1}{2} \left(1 + \sqrt{163} i\right)\right) = -{640320}^{3}

Connection formulas

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j ⁣(τ)=32((θ2 ⁣(0,τ))8+(θ3 ⁣(0,τ))8+(θ4 ⁣(0,τ))8)3(θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ))8j\!\left(\tau\right) = \frac{32 {\left({\left(\theta_2\!\left(0, \tau\right)\right)}^{8} + {\left(\theta_3\!\left(0, \tau\right)\right)}^{8} + {\left(\theta_4\!\left(0, \tau\right)\right)}^{8}\right)}^{3}}{{\left(\theta_2\!\left(0, \tau\right) \theta_3\!\left(0, \tau\right) \theta_4\!\left(0, \tau\right)\right)}^{8}}
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j ⁣(τ)=((η ⁣(τ)η ⁣(2τ))8+28(η ⁣(2τ)η ⁣(τ))16)3j\!\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}

Analytic properties

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HolomorphicDomain⁡ ⁣(j ⁣(τ),τ,H)=H\operatorname{HolomorphicDomain}\!\left(j\!\left(\tau\right), \tau, \mathbb{H}\right) = \mathbb{H}
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zeros⁡ τ∈Fj ⁣(τ)={e2πi/3}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} j\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}
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zeros⁡ τ∈Hj ⁣(τ)={γ∘e2πi/3:γ∈PSL⁡2(Z)}\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\}
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{j ⁣(τ):τ∈F}=C\left\{ j\!\left(\tau\right) : \tau \in \mathcal{F} \right\} = \mathbb{C}
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∣solutions⁡ τ∈F[j ⁣(τ)=z]∣=1\left|\mathop{\operatorname{solutions}\,}\limits_{\tau \in \mathcal{F}} \left[j\!\left(\tau\right) = z\right]\right| = 1

Hilbert class polynomials

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Symbol: PrimitiveReducedPositiveIntegralBinaryQuadraticForms — QD∗\mathcal{Q}^{*}_{D} — Primitive reduced positive integral binary quadratic forms
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QD∗={(a,b,c):a∈Z≥1 and⁡ b∈Z and⁡ c∈Z and⁡ b2−4ac=D and⁡ ∣b∣≤a≤c and⁡ ((∣b∣=a or⁡ a=c)  ⟹  (b≥0)) and⁡ gcd⁡ ⁣(a,b,c)=1}\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\}
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Symbol: HilbertClassPolynomial — HD ⁣(x)H_{D}\!\left(x\right) — Hilbert class polynomial
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HD ⁣(x)=∏(a,b,c)∈QD∗(x−j ⁣(−b+D2a))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)
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Table of HD ⁣(x)H_{D}\!\left(x\right) for −D≤68-D \le 68

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

2019-06-18 07:49:59.356594 UTC