Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - x^{2} + 4 x - 1 \)
Mordell-Weil group structure
Torsion generators
\( \left(1, 1\right) \)
Integral points
\( \left(1, 1\right) \), \( \left(1, -3\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||||
| Conductor: | \( 162 \) | = | \(2 \cdot 3^{4}\) | ||
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||||
| Discriminant: | \(-5184 \) | = | \(-1 \cdot 2^{6} \cdot 3^{4} \) | ||
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||||
| j-invariant: | \( \frac{109503}{64} \) | = | \(2^{-6} \cdot 3^{2} \cdot 23^{3}\) | ||
| 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: | \(2.60517025042\) | ||
|
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: | \( 6 \) = \( ( 2 \cdot 3 )\cdot1 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(3\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 162.2.a.d
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 12 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.73678016694 \)
Local data
This elliptic curve is not semistable.
| 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 |
| \(3\) | \(1\) | \( II \) | Additive | 1 | 4 | 4 | 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 X20b.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 1 & 0 \end{array}\right)$ and has index 16.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(3\) | B.1.1 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 |
|---|---|---|
| Reduction type | split | add |
| $\lambda$-invariant(s) | 4 | - |
| $\mu$-invariant(s) | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 \(d=\)
3.
Its isogeny class 162.d
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.1.324.1 | \(\Z/12\Z\) | Not in database |
| 6 | 6.0.177147.2 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| 6.0.419904.2 | \(\Z/4\Z \times \Z/12\Z\) | 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.