Minimal Weierstrass equation
\( y^2 + y = x^{3} + x^{2} + x \)
Mordell-Weil group structure
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(0, 0\right) \), \( \left(0, -1\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
|
|||||
| Conductor: | \( 19 \) | = | \(19\) | ||
|
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
|
|||||
| Discriminant: | \(-19 \) | = | \(-1 \cdot 19 \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
|
|||||
| j-invariant: | \( \frac{32768}{19} \) | = | \(2^{15} \cdot 19^{-1}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
|
|||
| Rank: | \(0\) | ||
|
magma: Regulator(E);
sage: E.regulator()
|
|||
| Regulator: | \(1\) | ||
|
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
|
|||
| Real period: | \(4.07927920046\) | ||
|
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
|
|||
| Tamagawa product: | \( 1 \) = \( 1 \) | ||
|
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
|
|||
| Torsion order: | \(3\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 19.2.a.a
|
magma: ModularDegree(E);
sage: E.modular_degree()
|
|||
| Modular degree: | 3 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 3 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.453253244496 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(19\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|
| \(3\) | 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 | 19 |
|---|---|---|---|
| Reduction type | ss | ordinary | split |
| $\lambda$-invariant(s) | 0,3 | 0 | 1 |
| $\mu$-invariant(s) | 0,0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class 19a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 | 3.3.361.1 | \(\Z/9\Z\) | 3.3.361.1-19.1-a3 |
| 3.1.76.1 | \(\Z/6\Z\) | Not in database | |
| 6 | 6.0.9747.1 | \(\Z/9\Z\) | Not in database |
| 6.0.109744.2 | \(\Z/2\Z \times \Z/6\Z\) | Not in database | |
| 6.0.3518667.2 | \(\Z/3\Z \times \Z/3\Z\) | 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.