Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+2418x+13182\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+2418xz^2+13182z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3133053x+568020222\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(299/4, 5993/8)$ | $2.3760698472742082670006517567$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-968239578306$ | = | $-1 \cdot 2 \cdot 3 \cdot 7^{2} \cdot 11^{7} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{17999471}{11154} \) | = | $2^{-1} \cdot 3^{-1} \cdot 7 \cdot 11^{-1} \cdot 13^{-2} \cdot 137^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98901233288641237923466838275$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53425366168865844364719553014$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8123298963819588$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.6820137611957104$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3760698472742082670006517567$ |
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| Real period: | $\Omega$ | ≈ | $0.54443371843187837320444852226$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1744501688214502102817162583 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.174450169 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.544434 \cdot 2.376070 \cdot 4}{1^2} \\ & \approx 5.174450169\end{aligned}$$
Modular invariants
Modular form 462462.2.a.bf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 875520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 145 & 2 \\ 145 & 3 \end{array}\right),\left(\begin{array}{rr} 133 & 2 \\ 133 & 3 \end{array}\right),\left(\begin{array}{rr} 199 & 2 \\ 199 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 263 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 89 & 2 \\ 89 & 3 \end{array}\right),\left(\begin{array}{rr} 263 & 2 \\ 262 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$486604800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $14$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $72$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462462bf consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 42042cf1, its twist by $-11$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.