Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-31934948x+7809401031\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(8993, 664725\right) \) |
| \(\hat{h}(P)\) | ≈ | $4.0118524127114224582758913704$ |
Torsion generators
\( \left(245, -123\right) \), \( \left(5525, -2763\right) \)
Integral points
\( \left(-3385, 279387\right) \), \( \left(-3385, -276003\right) \), \( \left(245, -123\right) \), \( \left(5525, -2763\right) \), \( \left(8993, 664725\right) \), \( \left(8993, -673719\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 402930 \) | = | \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 37\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(2058095745227573747150400 \) | = | \(2^{6} \cdot 3^{18} \cdot 5^{2} \cdot 11^{6} \cdot 37^{4} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| 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
|
sage: E.rank()
magma: Rank(E);
|
|||
| Analytic rank: | \(1\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(4.0118524127114224582758913704\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.070859549285813355474315248830\) | ||
|
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
|
|||
| Tamagawa product: | \( 768 \) = \( ( 2 \cdot 3 )\cdot2^{2}\cdot2\cdot2^{2}\cdot2^{2} \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(4\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 402930.2.a.dj
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 70778880 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L'(E,1) \) ≈ \( 13.645346580764844589760642645241040863 \)
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(3\) | \(4\) | \(I_{12}^{*}\) | Additive | -1 | 2 | 18 | 12 |
| \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(11\) | \(4\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
| \(37\) | \(4\) | \(I_{4}\) | 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
\(p\)-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 and 4.
Its isogeny class 402930.dj
consists of 3 curves linked by isogenies of
degrees dividing 8.