Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-88608x-7045004\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-88608xz^2-7045004z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7177275x-5114276118\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(450, 6664)$ | $0.54739614777366101840557778745$ | $\infty$ |
| $(-145, 1666)$ | $1.3067686186254650872379302186$ | $\infty$ |
| $(331, 0)$ | $0$ | $2$ |
Integral points
\((-236,\pm 882)\), \((-145,\pm 1666)\), \((-110,\pm 1176)\), \((-94,\pm 680)\), \( \left(331, 0\right) \), \((450,\pm 6664)\), \((620,\pm 13294)\), \((1164,\pm 38318)\), \((2354,\pm 113288)\), \((13659,\pm 1596028)\), \((10198274,\pm 32567920952)\)
Invariants
| Conductor: | $N$ | = | \( 13328 \) | = | $2^{4} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $23263320926461952$ | = | $2^{13} \cdot 7^{6} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{159661140625}{48275138} \) | = | $2^{-1} \cdot 5^{6} \cdot 7^{3} \cdot 17^{-6} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8455896531531674757241589944$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.17948739806556551375425050122$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0684849196783666$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.821158900555841$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.70360197521948537541579004151$ |
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| Real period: | $\Omega$ | ≈ | $0.28320016999284475052631354577$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.7822447757470303512173981142 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.782244776 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.283200 \cdot 0.703602 \cdot 96}{2^2} \\ & \approx 4.782244776\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 82944 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 8.6.0.6 |
| $3$ | 3B | 3.4.0.1 |
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 $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2845 & 12 \\ 2844 & 13 \end{array}\right),\left(\begin{array}{rr} 2447 & 0 \\ 0 & 2855 \end{array}\right),\left(\begin{array}{rr} 953 & 420 \\ 2380 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 2689 & 420 \\ 630 & 2521 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2806 & 2847 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2421 & 728 \\ 826 & 1147 \end{array}\right),\left(\begin{array}{rr} 2458 & 819 \\ 2625 & 400 \end{array}\right)$.
The torsion field $K:=\Q(E[2856])$ is a degree-$121286688768$ 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 | $4$ | \( 49 = 7^{2} \) |
| $3$ | good | $2$ | \( 784 = 2^{4} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 272 = 2^{4} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 784 = 2^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 13328w
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 34a4, its twist by $28$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.113288.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.64012032.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.11641525633024.52 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.821386940416.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.66532342173696.41 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.751974647358411618168049092642773711978496.1 | \(\Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ss | add | ord | ord | split | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 2,2 | - | 2 | 2 | 3 | 4 | 2,2 | 2,4 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | - | 1 | 0,0 | - | 0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.