Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - x^{2} - 132 x - 549 \)
Mordell-Weil group structure
Integral points
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||||
| Conductor: | \( 490 \) | = | \(2 \cdot 5 \cdot 7^{2}\) | ||
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||||
| Discriminant: | \(-980 \) | = | \(-1 \cdot 2^{2} \cdot 5 \cdot 7^{2} \) | ||
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||||
| j-invariant: | \( -\frac{5154200289}{20} \) | = | \(-1 \cdot 2^{-2} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{4} \cdot 43^{3}\) | ||
| Endomorphism Ring: | \(\Z\) | ||||
| Geometric Endomorphism Ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
sage: E.rank()
magma: Rank(E);
|
|||
| Rank: | \(0\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(1\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.704666003711\) | ||
|
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: | \( 2 \) = \( 2\cdot1\cdot1 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(1\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 490.2.a.e
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 120 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.40933200742 \)
Local data
This elliptic curve is not semistable.
| 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 |
| \(5\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(7\) | \(1\) | \( II \) | Additive | -1 | 2 | 2 | 0 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(7\) | B.1.3 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ss | split | add | ordinary | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 1 | 0,0 | 1 | - | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0,0 | 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 \(d=\)
7.
Its isogeny class 490k
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.1.980.1 | \(\Z/2\Z\) | Not in database |
| 6 | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | Not in database |
| 6.0.19208000.2 | \(\Z/2\Z \times \Z/2\Z\) | Not in database | |
| 7 | 7.1.16807000000.2 | \(\Z/7\Z\) | 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.