Minimal Weierstrass equation
\(y^2=x^3+x^2-88608x-7045004\)
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(450, 6664\right) \) | \( \left(-145, 1666\right) \) |
| \(\hat{h}(P)\) | ≈ | $0.54739614777366101840557778745$ | $1.3067686186254650872379302186$ |
Torsion generators
\( \left(331, 0\right) \)
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
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 13328 \) | = | \(2^{4} \cdot 7^{2} \cdot 17\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(23263320926461952 \) | = | \(2^{13} \cdot 7^{6} \cdot 17^{6} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{159661140625}{48275138} \) | = | \(2^{-1} \cdot 5^{6} \cdot 7^{3} \cdot 17^{-6} \cdot 31^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(2\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(0.70360197521948537541579004151\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.28320016999284475052631354577\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 96 \) = \( 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 ) \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 13328.2.a.e
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 82944 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 4.7822447757470303512173981142333130491 \)
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_5^{*}\) | 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 image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X17.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ and has index 6.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
| \(3\) | B |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ordinary | ss | add | ordinary | ordinary | split | ordinary | ss | ss | ordinary | ordinary | ordinary | ordinary | 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.
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.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | 4.4.113288.1 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-21})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $6$ | 6.2.64012032.1 | \(\Z/6\Z\) | Not in database |
| $8$ | 8.0.11641525633024.52 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.8.821386940416.2 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.66532342173696.41 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | 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.