Minimal Weierstrass equation
\( y^2 = x^{3} - x^{2} - 134153345 x + 575935024257 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(3488, 387855\right) \) |
| \(\hat{h}(P)\) | ≈ | 2.93068023455 |
Torsion generators
\( \left(7713, 0\right) \)
Integral points
\((3488,\pm 387855)\), \( \left(7713, 0\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 212160 \) | = | \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(11247280550133363847987200 \) | = | \(2^{42} \cdot 3^{6} \cdot 5^{2} \cdot 13^{4} \cdot 17^{3} \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{1018563973439611524445729}{42904970360310988800} \) | = | \(2^{-24} \cdot 3^{-6} \cdot 5^{-2} \cdot 11^{6} \cdot 13^{-4} \cdot 17^{-3} \cdot 831529^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(2.93068023455\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.0710882755374\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 96 \) = \( 2^{2}\cdot2\cdot2\cdot2\cdot3 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(2\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 212160.2.a.cc
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 53084160 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 5.00008809662 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \( I_32^{*} \) | Additive | 1 | 6 | 42 | 24 |
| \(3\) | \(2\) | \( I_{6} \) | Non-split multiplicative | 1 | 1 | 6 | 6 |
| \(5\) | \(2\) | \( I_{2} \) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(13\) | \(2\) | \( I_{4} \) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(17\) | \(3\) | \( I_{3} \) | Split multiplicative | -1 | 1 | 3 | 3 |
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 X13e.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.
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.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class 212160gf
consists of 8 curves linked by isogenies of
degrees dividing 12.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 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 | \(\Q(\sqrt{2}) \) | \(\Z/12\Z\) | Not in database |
| \(\Q(\sqrt{34}) \) | \(\Z/4\Z\) | Not in database | |
| \(\Q(\sqrt{17}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database | |
| 4 | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| 6 | 6.0.246767040000.3 | \(\Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.