This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.
Minimal Weierstrass equation
\(y^2+y=x^3-x^2-2x-1\)
Mordell-Weil group structure
trivial
Integral points
None
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 147 \) | = | \(3 \cdot 7^{2}\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(-147 \) | = | \(-1 \cdot 3 \cdot 7^{2} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( -\frac{28672}{3} \) | = | \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7\) |
| 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: | \(0\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(1\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(1.9200537453490419433285021703\) | ||
|
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: | \( 1 \) | ||
|
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
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 6 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.9200537453490419433285021702840752175 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(1\) | \(I_{1}\) | Non-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 |
|---|---|
| \(13\) | B.4.1 |
$p$-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 | ss | nonsplit | ordinary | add | ordinary | ordinary | ss | ordinary | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 0,1 | 0 | 0 | - | 0 | 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 | 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 \(d=\)
13.
Its isogeny class 147.b
consists of 2 curves linked by isogenies of
degree 13.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.588.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.1037232.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $6$ | \(\Q(\zeta_{7})\) | \(\Z/13\Z\) | Not in database |
| $8$ | 8.2.20841167403.1 | \(\Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $18$ | 18.0.14176142707705638912.1 | \(\Z/26\Z\) | 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.