Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-6775008x-3518431488\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-6775008xz^2-3518431488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-548775675x-2566582881750\)
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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 |
|---|---|---|
| \( \left(-\frac{215008624}{395641}, \frac{402275182592}{248858189}\right) \) | $17.216919148586732822300855380$ | $\infty$ |
| \( \left(-543, 0\right) \) | $0$ | $2$ |
| \( \left(2832, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-135240424496:402275182592:248858189]\) | $17.216919148586732822300855380$ | $\infty$ |
| \([-543:0:1]\) | $0$ | $2$ |
| \([2832:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{1936264539}{395641}, \frac{10861429929984}{248858189}\right) \) | $17.216919148586732822300855380$ | $\infty$ |
| \( \left(-4890, 0\right) \) | $0$ | $2$ |
| \( \left(25485, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-2288, 0\right) \), \( \left(-543, 0\right) \), \( \left(2832, 0\right) \)
\([-2288:0:1]\), \([-543:0:1]\), \([2832:0:1]\)
\( \left(-2288, 0\right) \), \( \left(-543, 0\right) \), \( \left(2832, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 418800 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 349$ |
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| Minimal Discriminant: | $\Delta$ | = | $14547833487360000000000$ | = | $2^{24} \cdot 3^{6} \cdot 5^{10} \cdot 349^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{537369779909439001}{227309898240000} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-4} \cdot 7^{3} \cdot 37^{3} \cdot 43^{3} \cdot 73^{3} \cdot 349^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9494201743525981514772992097$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4515540375756026547596874216$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9578707573611972$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.5422310393079215$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.216919148586732822300855380$ |
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| Real period: | $\Omega$ | ≈ | $0.097188894306959100170666390770$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6931733416938249425368715999 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.693173342 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.097189 \cdot 17.216919 \cdot 64}{4^2} \\ & \approx 6.693173342\end{aligned}$$
Modular invariants
Modular form 418800.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35831808 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 4 | 24 | 12 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $349$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20940 = 2^{2} \cdot 3 \cdot 5 \cdot 349 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 20937 & 4 \\ 20936 & 5 \end{array}\right),\left(\begin{array}{rr} 10469 & 12560 \\ 16750 & 4179 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13961 & 8380 \\ 11170 & 16761 \end{array}\right),\left(\begin{array}{rr} 3841 & 4190 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12563 & 0 \\ 0 & 20939 \end{array}\right)$.
The torsion field $K:=\Q(E[20940])$ is a degree-$681654693888000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20940\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 139600 = 2^{4} \cdot 5^{2} \cdot 349 \) |
| $5$ | additive | $18$ | \( 16752 = 2^{4} \cdot 3 \cdot 349 \) |
| $349$ | split multiplicative | $350$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 418800c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 10470a2, its twist by $-20$.
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.