Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2560594764x-12533759058800\)
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(homogenize, simplify) |
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\(y^2z=x^3-2560594764xz^2-12533759058800z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2560594764x-12533759058800\)
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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 |
|---|---|---|
| $(-6286784898392387511360868/135812517252222543121, 4129391899021541365959549483259132560/1582740437283346339762538617081)$ | $51.509283049752786219389613506$ | $\infty$ |
| $(-4942, 0)$ | $0$ | $2$ |
| $(52892, 0)$ | $0$ | $2$ |
Integral points
\( \left(-47950, 0\right) \), \( \left(-4942, 0\right) \), \( \left(52892, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $1006625294018868876956575727616$ | = | $2^{30} \cdot 3^{14} \cdot 7^{14} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{82582985847542515777}{44772582831427584} \) | = | $2^{-12} \cdot 3^{-8} \cdot 7^{-8} \cdot 13^{3} \cdot 17^{-2} \cdot 457^{3} \cdot 733^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4479637784647019154937221720$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8859817887630724531175749996$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0972067382550121$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.856073063776351$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $51.509283049752786219389613506$ |
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| Real period: | $\Omega$ | ≈ | $0.022627173936815808722729652090$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\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)$ | ≈ | $9.3240760554194764640023326362 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.324076055 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.022627 \cdot 51.509283 \cdot 128}{4^2} \\ & \approx 9.324076055\end{aligned}$$
Modular invariants
Modular form 479808.2.a.ni
For more coefficients, see the Downloads section to the right.
| Modular degree: | 566231040 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{20}^{*}$ | additive | -1 | 6 | 30 | 12 |
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $7$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $17$ | $2$ | $I_{2}$ | 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 | 8.48.0.97 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2856 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 2852 & 2853 \end{array}\right),\left(\begin{array}{rr} 2447 & 0 \\ 0 & 2855 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2309 & 2478 \\ 1092 & 923 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1903 & 0 \\ 0 & 2855 \end{array}\right),\left(\begin{array}{rr} 2849 & 8 \\ 2848 & 9 \end{array}\right),\left(\begin{array}{rr} 463 & 1638 \\ 2226 & 1219 \end{array}\right),\left(\begin{array}{rr} 335 & 1596 \\ 1596 & 1427 \end{array}\right)$.
The torsion field $K:=\Q(E[2856])$ is a degree-$60643344384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2856\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 479808.ni
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714.f4, its twist by $-168$.
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.