Fungrim home page

Modular j-invariant

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

Definitions

70eb98
Symbol: ModularJ — j(τ)j(\tau) — Modular j-invariant

Illustrations

8c2862
Image: X-ray of j(τ)j(\tau) 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

a997f2
j ⁣(τ+1)=j(τ)j\!\left(\tau + 1\right) = j(\tau)
42a909
j ⁣(−1τ)=j(τ)j\!\left(-\frac{1}{\tau}\right) = j(\tau)
d5f569
j ⁣(aτ+bcτ+d)=j(τ)j\!\left(\frac{a \tau + b}{c \tau + d}\right) = j(\tau)

Special values

9aa62c
j ⁣(eπi/3)=0j\!\left({e}^{\pi i / 3}\right) = 0
ad228f
j(i)=1728j(i) = 1728
229c97
j ⁣(2i)=663=287496j\!\left(2 i\right) = {66}^{3} = 287496
1356e4
j ⁣(2i)=203=8000j\!\left(\sqrt{2} i\right) = {20}^{3} = 8000
8be46c
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}
3189b9
j ⁣(4i)=27(724+5132)3j\!\left(4 i\right) = 27 {\left(724 + 513 \sqrt{2}\right)}^{3}
29c095
j ⁣(12(1+7i))=−153j\!\left(\frac{1}{2} \left(1 + \sqrt{7} i\right)\right) = -{15}^{3}
a498dd
j ⁣(12(1+11i))=−323j\!\left(\frac{1}{2} \left(1 + \sqrt{11} i\right)\right) = -{32}^{3}
3ee358
j ⁣(12(1+19i))=−963j\!\left(\frac{1}{2} \left(1 + \sqrt{19} i\right)\right) = -{96}^{3}
5b108e
j ⁣(12(1+43i))=−9603j\!\left(\frac{1}{2} \left(1 + \sqrt{43} i\right)\right) = -{960}^{3}
951017
j ⁣(12(1+67i))=−52803j\!\left(\frac{1}{2} \left(1 + \sqrt{67} i\right)\right) = -{5280}^{3}
1cb24e
j ⁣(12(1+163i))=−6403203j\!\left(\frac{1}{2} \left(1 + \sqrt{163} i\right)\right) = -{640320}^{3}

Connection formulas

cedcfc
j(τ)=32(θ28 ⁣(0,τ)+θ38 ⁣(0,τ)+θ48 ⁣(0,τ))3(θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ))8j(\tau) = \frac{32 {\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3}}{{\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}}
664b4c
j(τ)=((η(τ)η ⁣(2τ))8+28(η ⁣(2τ)η(τ))16)3j(\tau) = {\left({\left(\frac{\eta(\tau)}{\eta\!\left(2 \tau\right)}\right)}^{8} + {2}^{8} {\left(\frac{\eta\!\left(2 \tau\right)}{\eta(\tau)}\right)}^{16}\right)}^{3}
dc8251
j(τ)=E43 ⁣(τ)η24 ⁣(τ)j(\tau) = \frac{E_{4}^{3}\!\left(\tau\right)}{\eta^{24}\!\left(\tau\right)}

Derivatives

f0f53b
j′(τ)=−2πiE14 ⁣(τ)η24 ⁣(τ)j'(\tau) = -2 \pi i \frac{E_{14}\!\left(\tau\right)}{\eta^{24}\!\left(\tau\right)}
348b26
j′(τ)=−2πiE6 ⁣(τ)E4 ⁣(τ)j(τ)j'(\tau) = -2 \pi i \frac{E_{6}\!\left(\tau\right)}{E_{4}\!\left(\tau\right)} j(\tau)

Analytic properties

27f9d2
j(τ) is holomorphic on τ∈Hj(\tau) \text{ is holomorphic on } \tau \in \mathbb{H}
ea3e3c
zeros⁡ τ∈Fj(τ)={e2πi/3}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} j(\tau) = \left\{{e}^{2 \pi i / 3}\right\}
1b2d8a
zeros⁡ τ∈Hj(τ)={γ∘e2πi/3:γ∈PSL⁡2(Z)}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} j(\tau) = \left\{ \gamma \circ {e}^{2 \pi i / 3} : \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}
dcc8b1
{j(τ):τ∈F}=C\left\{ j(\tau) : \tau \in \mathcal{F} \right\} = \mathbb{C}
441301
#solutions⁡ τ∈F[j(τ)=z]=1\# \mathop{\operatorname{solutions}\,}\limits_{\tau \in \mathcal{F}} \left[j(\tau) = z\right] = 1

Hilbert class polynomials

36eb82
Symbol: PrimitiveReducedPositiveIntegralBinaryQuadraticForms — QD∗\mathcal{Q}^{*}_{D} — Primitive reduced positive integral binary quadratic forms
0b4d4b
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\}
fd72e0
Symbol: HilbertClassPolynomial — HD ⁣(x)H_{D}\!\left(x\right) — Hilbert class polynomial
dd5681
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)
20b6d2
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.

2021-03-15 19:12:00.328586 UTC