Minimal Weierstrass equation
\(y^2+xy=x^3-1428840x+639791100\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(\frac{9964}{9}, \frac{535214}{27}\right) \) |
| \(\hat{h}(P)\) | ≈ | $5.6602325109968378290139373723$ |
Torsion generators
\( \left(596, -298\right) \), \( \left(780, -390\right) \)
Integral points
\( \left(596, -298\right) \), \( \left(780, -390\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 111090 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(9796654296540562500 \) | = | \(2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{2179252305146449}{66177562500} \) | = | \(2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{3} \cdot 9973^{3}\) |
| 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: | \(5.6602325109968378290139373723\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.22852871313605142913442681039\) | ||
|
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: | \( 192 \) = \( 2\cdot2\cdot( 2 \cdot 3 )\cdot2\cdot2^{2} \) | ||
|
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 111090.2.a.di
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 3649536 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 15.522307821467381021862410044433794341 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(5\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(7\) | \(2\) | \(I_{6}\) | Non-split multiplicative | 1 | 1 | 6 | 6 |
| \(23\) | \(4\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 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 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 |
| \(3\) | B |
$p$-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 | split | split | split | nonsplit | ss | ordinary | ordinary | ordinary | add | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 4 | 4 | 4 | 1 | 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, 3 and 6.
Its isogeny class 111090dd
consists of 4 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{69}) \) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{46})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{-46})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{105}, \sqrt{-161})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $6$ | 6.0.11495186928.5 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.0.222919710806016.102 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.8707801203360000.124 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.8.58027829760000.1 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $18$ | 18.6.962422833243339066733532293686914603812500000000.1 | \(\Z/2\Z \times \Z/18\Z\) | 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.