Minimal Weierstrass equation
\( y^2 = x^{3} + x^{2} - 15771488 x - 24113032076 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-\frac{6982245574}{3045025}, -\frac{8712379776}{5313568625}\right) \) |
| \(\hat{h}(P)\) | ≈ | 12.956099304 |
Torsion generators
\( \left(-2293, 0\right) \)
Integral points
\( \left(-2293, 0\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 40432 \) | = | \(2^{4} \cdot 7 \cdot 19^{2}\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(4834455808114688 \) | = | \(2^{21} \cdot 7^{2} \cdot 19^{6} \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{2251439055699625}{25088} \) | = | \(2^{-9} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 2383^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
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: | \(12.956099304\) | ||
|
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.0757585031499\) | ||
|
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: | \( 16 \) = \( 2^{2}\cdot2\cdot2 \) | ||
|
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(2\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 40432.2.a.b
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 1026432 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.92613875973 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \( I_13^{*} \) | Additive | -1 | 4 | 21 | 9 |
| \(7\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(19\) | \(2\) | \( I_0^{*} \) | Additive | -1 | 2 | 6 | 0 |
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 | nonsplit | ss | ordinary | ordinary | add | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | - | 3 | 1,1 | 1 | 3,1 | 1 | 1 | - | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 2 | 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, 6, 9 and 18.
Its isogeny class 40432r
consists of 6 curves linked by isogenies of
degrees dividing 18.
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 | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| \(\Q(\sqrt{-57}) \) | \(\Z/6\Z\) | Not in database | |
| 4 | \(\Q(\sqrt{2}, \sqrt{-57})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| 4.0.141512.1 | \(\Z/4\Z\) | Not in database | |
| 6 | 6.0.20745515423808.8 | \(\Z/18\Z\) | Not in database |
| 6.2.768352423104.2 | \(\Z/6\Z\) | 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.