Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} + x^{2} - 14113 x - 208594 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
\(P\) | = | \( \left(-64, 693\right) \) |
\(\hat{h}(P)\) | ≈ | 1.30634595814 |
Torsion generators
\( \left(-15, 7\right) \), \( \left(125, -63\right) \)
Integral points
\( \left(-64, 693\right) \), \( \left(-50, 637\right) \), \( \left(-15, 7\right) \), \( \left(125, -63\right) \), \( \left(129, 307\right) \), \( \left(260, 3582\right) \), \( \left(370, 6552\right) \)
Invariants
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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Conductor: | \( 40425 \) | = | \(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\) | ||
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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Discriminant: | \(162151572515625 \) | = | \(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{2} \) | ||
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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j-invariant: | \( \frac{169112377}{88209} \) | = | \(3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}\) | ||
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: | \(1\) | ||
magma: Regulator(E);
sage: E.regulator()
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Regulator: | \(1.30634595814\) | ||
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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Real period: | \(0.463927711946\) | ||
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: | \( 64 \) = \( 2\cdot2^{2}\cdot2^{2}\cdot2 \) | ||
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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Torsion order: | \(4\) | ||
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 40425.2.a.p
magma: ModularDegree(E);
sage: E.modular_degree()
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Modular degree: | 110592 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L'(E,1) \) ≈ \( 2.42420036547 \)
Local data
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(3\) | \(2\) | \( I_{6} \) | Non-split multiplicative | 1 | 1 | 6 | 6 |
\(5\) | \(4\) | \( I_0^{*} \) | Additive | 1 | 2 | 6 | 0 |
\(7\) | \(4\) | \( I_0^{*} \) | Additive | -1 | 2 | 6 | 0 |
\(11\) | \(2\) | \( I_{2} \) | Split multiplicative | -1 | 1 | 2 | 2 |
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 X8.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ 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 |
---|---|
\(2\) | Cs |
$p$-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 | ordinary | nonsplit | add | add | split | ordinary | ordinary | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ss | ordinary |
$\lambda$-invariant(s) | 8 | 1 | - | - | 2 | 3 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 |
$\mu$-invariant(s) | 1 | 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.
Its isogeny class 40425.p
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
---|---|---|---|
4 | \(\Q(\sqrt{3}, \sqrt{35})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
\(\Q(\sqrt{33}, \sqrt{-105})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database | |
\(\Q(\sqrt{-11}, \sqrt{-35})\) | \(\Z/2\Z \times \Z/4\Z\) | 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.