Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2538003x+1556011730\)
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(homogenize, simplify) |
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\(y^2z=x^3-2538003xz^2+1556011730z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2538003x+1556011730\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(910, 0)$ | $0$ | $2$ |
Integral points
\( \left(910, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 413712 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $353820400369532928$ | = | $2^{20} \cdot 3^{12} \cdot 13^{3} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{275602131611533}{53934336} \) | = | $2^{-8} \cdot 3^{-6} \cdot 17^{-2} \cdot 59^{3} \cdot 1103^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3668789736282351900889124114$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48318830936885085096068581109$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9891236446897397$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.3187651526684165$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.29405828108879416829417430324$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.7049324974207066927067888519 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.704932497 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.294058 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 4.704932497\end{aligned}$$
Modular invariants
Modular form 413712.2.a.fc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10616832 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $17$ | $2$ | $I_{2}$ | nonsplit 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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2652 = 2^{2} \cdot 3 \cdot 13 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1873 & 4 \\ 1094 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 665 & 1990 \\ 1988 & 663 \end{array}\right),\left(\begin{array}{rr} 2649 & 4 \\ 2648 & 5 \end{array}\right),\left(\begin{array}{rr} 2452 & 1 \\ 2039 & 0 \end{array}\right),\left(\begin{array}{rr} 1769 & 4 \\ 886 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[2652])$ is a degree-$788363476992$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2652\Z)$.
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 | $2$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 45968 = 2^{4} \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $50$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 24336 = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 413712.fc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 17238.n1, its twist by $12$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.