Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-386425663x-2923653373915\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-386425663xz^2-2923653373915z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-500807659275x-136398459698480826\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23155, 725970)$ | $6.9433549215471805235204728905$ | $\infty$ |
| $(-45021/4, 45017/8)$ | $0$ | $2$ |
Integral points
\( \left(23155, 725970\right) \), \( \left(23155, -749126\right) \)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $766775749701936771084288$ | = | $2^{12} \cdot 3 \cdot 7^{6} \cdot 11^{12} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{30618029936661765625}{3678951124992} \) | = | $2^{-12} \cdot 3^{-1} \cdot 5^{6} \cdot 11^{-6} \cdot 13^{-2} \cdot 17^{6} \cdot 433^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6077274296691261127741776389$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4358247187422841881905294782$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1301238642654245$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.437688974691689$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.9433549215471805235204728905$ |
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| Real period: | $\Omega$ | ≈ | $0.034051495921323099650781551542$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.348717845985399431493389844 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.348717846 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.034051 \cdot 6.943355 \cdot 192}{2^2} \\ & \approx 11.348717846\end{aligned}$$
Modular invariants
Modular form 462462.2.a.gq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 119439360 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $13$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
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 $96$, genus $1$, and generators
$\left(\begin{array}{rr} 8579 & 0 \\ 0 & 12011 \end{array}\right),\left(\begin{array}{rr} 12001 & 12 \\ 12000 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11962 & 12003 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 7643 & 8568 \\ 8106 & 3359 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4586 & 6867 \\ 6265 & 3424 \end{array}\right),\left(\begin{array}{rr} 925 & 3444 \\ 7266 & 8653 \end{array}\right),\left(\begin{array}{rr} 1002 & 8449 \\ 1001 & 5006 \end{array}\right)$.
The torsion field $K:=\Q(E[12012])$ is a degree-$33476463820800$ 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 | $4$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 77077 = 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $72$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | 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, 3 and 6.
Its isogeny class 462462gq
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 858b4, 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.