Base field 3.3.361.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -6, -1, 1]))
gp: K = nfinit(Polrev([7, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([0,0,0]),K([1,0,0]),K([-16,0,0]),K([22,0,0])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([0,0,0]),Polrev([1,0,0]),Polrev([-16,0,0]),Polrev([22,0,0])], K);
magma: E := EllipticCurve([K![1,0,0],K![0,0,0],K![1,0,0],K![-16,0,0],K![22,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2-2a-8)\) | = | \((2)\cdot(a^2-a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 152 \) | = | \(8\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-152)\) | = | \((2)^{3}\cdot(a^2-a-4)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3511808 \) | = | \(-8^{3}\cdot19^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{413493625}{152} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(2\) | |
| Generators | $\left(\frac{73}{49} a^{2} + \frac{19}{49} a - \frac{44}{7} : \frac{226}{343} a^{2} - \frac{13}{343} a - \frac{272}{49} : 1\right)$ | $\left(8 a^{2} + 10 a - 24 : 48 a^{2} + 60 a - 150 : 1\right)$ |
| Heights | \(1.1770068430849075450129949620794194453\) | \(1.1770068430849075450129949620794194453\) |
| Torsion structure: | \(\Z/9\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generator: | $\left(a + 1 : a^{2} - a - 2 : 1\right)$ | |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 1.0390088315015251290093453598381036192 \) | ||
| Period: | \( 182.46725367387427957523565450721944330 \) | ||
| Tamagawa product: | \( 3 \) = \(1\cdot3\) | ||
| Torsion order: | \(9\) | ||
| Leading coefficient: | \( 3.3260541759120084807228687381942275689 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(8\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((a^2-a-4)\) | \(19\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
152.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
| Base field | Curve |
|---|---|
| \(\Q\) | 38.a2 |