Minimal Weierstrass equation
\(y^2+xy+y=x^3-10586333x-1489874632\)
Mordell-Weil group structure
\(\Z/{2}\Z \times \Z/{2}\Z\)
Torsion generators
\( \left(-3181, 1590\right) \), \( \left(-141, 70\right) \)
Integral points
\( \left(-3181, 1590\right) \), \( \left(-141, 70\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 400710 \) | = | \(2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(74972767301039235009600 \) | = | \(2^{6} \cdot 3^{12} \cdot 5^{2} \cdot 19^{6} \cdot 37^{4} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{2788936974993502801}{1593609593601600} \) | = | \(2^{-6} \cdot 3^{-12} \cdot 5^{-2} \cdot 13^{6} \cdot 37^{-4} \cdot 8329^{3}\) |
| 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: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.090648903806985863208038654763\) | ||
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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: | \( 384 \) = \( 2\cdot( 2^{2} \cdot 3 )\cdot2\cdot2^{2}\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(4\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 400710.2.a.x
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: | 47775744 | ||
| \( \Gamma_0(N) \)-optimal: | unknown* (one of 4 curves in this isogeny class which might be optimal) | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L(E,1) \) ≈ \( 2.1755736913676607169929277143208785276 \)
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | 1 | 1 | 6 | 6 |
| \(3\) | \(12\) | \(I_{12}\) | Split multiplicative | -1 | 1 | 12 | 12 |
| \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(19\) | \(4\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
| \(37\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 X25.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$ 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 |
|---|---|
| \(2\) | Cs |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 400710.x
consists of 3 curves linked by isogenies of
degrees dividing 8.