Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} + x^{2} - 875 x - 8440 \)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-22, 38\right) \) |
| \(\hat{h}(P)\) | ≈ | $0.8500903028810441$ |
Torsion generators
\( \left(-11, 5\right) \), \( \left(33, -17\right) \)
Integral points
\( \left(-22, 38\right) \), \( \left(-22, -17\right) \), \( \left(-12, 28\right) \), \( \left(-12, -17\right) \), \( \left(-11, 5\right) \), \( \left(33, -17\right) \), \( \left(38, 103\right) \), \( \left(38, -142\right) \), \( \left(133, 1433\right) \), \( \left(133, -1567\right) \), \( \left(528, 11863\right) \), \( \left(528, -12392\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 1155 \) | = | \(3 \cdot 5 \cdot 7 \cdot 11\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(14707625625 \) | = | \(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{2} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{74093292126001}{14707625625} \) | = | \(3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 11^{-2} \cdot 97^{3} \cdot 433^{3}\) |
| 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);
|
|||
| Rank: | \(1\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(0.850090302881\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.889973049328\) | ||
|
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: | \( 32 \) = \( 2\cdot2^{2}\cdot2\cdot2 \) | ||
|
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
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 1024 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 1.51311491812 \)
Local data
This elliptic curve is semistable.
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(2\) | \( I_{4} \) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(5\) | \(4\) | \( I_{4} \) | Split multiplicative | -1 | 1 | 4 | 4 |
| \(7\) | \(2\) | \( I_{4} \) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(11\) | \(2\) | \( I_{2} \) | 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 X25h.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$ and has index 24.
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 | split | nonsplit | split | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 1 | 1 | 2 | 3 | 2 | 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 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 1155.e
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(i, \sqrt{11})\) | \(\Z/4\Z \times \Z/4\Z\) | 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.