Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-115445829930x-13236321467612844\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-115445829930xz^2-13236321467612844z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-149617795589955x-617551570126011003714\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $22786506865002494728411623664386048$ | = | $2^{37} \cdot 3^{8} \cdot 7^{8} \cdot 11^{10} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1138013177660908404625}{152393548560334848} \) | = | $2^{-37} \cdot 3^{-8} \cdot 5^{3} \cdot 7 \cdot 11^{2} \cdot 13^{-2} \cdot 199^{3} \cdot 1109^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $5.3201776667109730430233248188$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0246581733421220529014700082$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0376440725257465$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.74851636673561$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.0082629428512245168923183769340$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.29746594264408260812346156962 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.297465943 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.008263 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 0.297465943\end{aligned}$$
Modular invariants
Modular form 462462.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4726149120 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{37}$ | nonsplit multiplicative | 1 | 1 | 37 | 37 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $11$ | $1$ | $II^{*}$ | additive | -1 | 2 | 10 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2G | 8.2.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.b.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $32$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
| $37$ | good | $2$ | \( 231231 = 3 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462462.bc consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 462462.jo1, its twist by $77$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.