Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+358120x-764244600\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+358120xz^2-764244600z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+29007693x-557047290294\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1355252144021/776792641, 1556411640924509010/21649987697311)$ | $24.950459060464979853767082870$ | $\infty$ |
| $(785, 0)$ | $0$ | $2$ |
Integral points
\( \left(785, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 406560 \) | = | $2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-255178779693647040000$ | = | $-1 \cdot 2^{9} \cdot 3^{12} \cdot 5^{4} \cdot 7 \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{5599924283512}{281331579375} \) | = | $2^{3} \cdot 3^{-12} \cdot 5^{-4} \cdot 7^{-1} \cdot 11^{-2} \cdot 13^{3} \cdot 683^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5954307182547393632351562644$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.87662269643559510914126038432$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9705925400978148$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.21444664631561$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $24.950459060464979853767082870$ |
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| Real period: | $\Omega$ | ≈ | $0.083770751434066490721518151660$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.3604748164802559112675175543 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.360474816 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.083771 \cdot 24.950459 \cdot 16}{2^2} \\ & \approx 8.360474816\end{aligned}$$
Modular invariants
Modular form 406560.2.a.cd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14745600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | 1 | 5 | 9 | 0 |
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8711 & 8712 \\ 1562 & 5345 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 3697 & 7568 \\ 4708 & 2553 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3048 & 1793 \\ 2827 & 1134 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9234 & 9235 \end{array}\right),\left(\begin{array}{rr} 6161 & 7568 \\ 5324 & 2553 \end{array}\right),\left(\begin{array}{rr} 4324 & 6721 \\ 8063 & 5886 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$19619905536000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 847 = 7 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 135520 = 2^{5} \cdot 5 \cdot 7 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 81312 = 2^{5} \cdot 3 \cdot 7 \cdot 11^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 58080 = 2^{5} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 406560cd
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 36960bm2, 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.