Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-809432x+178453001\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-809432xz^2+178453001z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1049023899x+8341638581430\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(787, 5051)$ | $3.9771035231840993285171549383$ | $\infty$ |
| $(-995, 497)$ | $0$ | $2$ |
| $(237, -119)$ | $0$ | $2$ |
Integral points
\( \left(-995, 497\right) \), \( \left(237, -119\right) \), \( \left(787, 5051\right) \), \( \left(787, -5839\right) \), \( \left(30855, 5402257\right) \), \( \left(30855, -5433113\right) \)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $20131430989086392976$ | = | $2^{4} \cdot 3^{6} \cdot 7^{8} \cdot 11^{6} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{281397674377}{96589584} \) | = | $2^{-4} \cdot 3^{-6} \cdot 7^{-2} \cdot 13^{-2} \cdot 6553^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4059724534586252188961911564$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.23406974253178329431254299570$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9475546814672546$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.019057227431869$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9771035231840993285171549383$ |
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| Real period: | $\Omega$ | ≈ | $0.19881068230766307763918937503$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.651050640839064145451532388 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.651050641 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.198811 \cdot 3.977104 \cdot 256}{4^2} \\ & \approx 12.651050641\end{aligned}$$
Modular invariants
Modular form 462462.2.a.hm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11796480 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12012 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 6051 & 3278 \\ 6886 & 8735 \end{array}\right),\left(\begin{array}{rr} 8009 & 6556 \\ 7282 & 1101 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12009 & 4 \\ 12008 & 5 \end{array}\right),\left(\begin{array}{rr} 7643 & 0 \\ 0 & 12011 \end{array}\right),\left(\begin{array}{rr} 6007 & 3278 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10449 & 8734 \\ 10142 & 3277 \end{array}\right)$.
The torsion field $K:=\Q(E[12012])$ is a degree-$66952927641600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12012\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462462.hm
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 546.b2, its twist by $77$.
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.