Minimal Weierstrass equation
\(y^2+xy=x^3-x^2-198495x+33968461\)
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(215, 986\right) \) | \( \left(270, -79\right) \) |
| \(\hat{h}(P)\) | ≈ | $1.3563702357861320505998273204$ | $3.5559934817309844648419464146$ |
Torsion generators
\( \left(-514, 257\right) \)
Integral points
\( \left(-514, 257\right) \), \( \left(215, 986\right) \), \( \left(215, -1201\right) \), \( \left(270, -79\right) \), \( \left(270, -191\right) \), \( \left(302, 1073\right) \), \( \left(302, -1375\right) \), \( \left(327, 1823\right) \), \( \left(327, -2150\right) \), \( \left(761, 17597\right) \), \( \left(761, -18358\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 338130 \) | = | \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(3519169310142720 \) | = | \(2^{8} \cdot 3^{16} \cdot 5 \cdot 13 \cdot 17^{3} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{241490118740417}{982575360} \) | = | \(2^{-8} \cdot 3^{-10} \cdot 5^{-1} \cdot 13^{-1} \cdot 62273^{3}\) |
| 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: | \(2\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(4.6395132688981239358176779966\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.44681703073520267612988296423\) | ||
|
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: | \( 16 \) = \( 2\cdot2^{2}\cdot1\cdot1\cdot2 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(2\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 338130.2.a.b
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 2785280 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 8.2920541714625347232318701154556733209 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | 1 | 1 | 8 | 8 |
| \(3\) | \(4\) | \(I_{10}^{*}\) | Additive | -1 | 2 | 16 | 10 |
| \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(13\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(17\) | \(2\) | \(III\) | Additive | 1 | 2 | 3 | 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 X6.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.
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 |
$p$-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.
Its isogeny class 338130.b
consists of 2 curves linked by isogenies of
degree 2.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{1105}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $4$ | 4.0.45985680.6 | \(\Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.