Base field 4.4.19821.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-2,1/3,1/3]),K([0,1,0,0]),K([-4,0,1,0]),K([9,22,2/3,-7/3]),K([7,32,5/3,-10/3])])
gp: E = ellinit([Polrev([0,-2,1/3,1/3]),Polrev([0,1,0,0]),Polrev([-4,0,1,0]),Polrev([9,22,2/3,-7/3]),Polrev([7,32,5/3,-10/3])], K);
magma: E := EllipticCurve([K![0,-2,1/3,1/3],K![0,1,0,0],K![-4,0,1,0],K![9,22,2/3,-7/3],K![7,32,5/3,-10/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/3a^3-1/3a^2+3a+2)\) | = | \((-1/3a^3-1/3a^2+3a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 3 \) | = | \(3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-25/3a^3+23/3a^2+65a-34)\) | = | \((-1/3a^3-1/3a^2+3a+2)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 177147 \) | = | \(3^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{73343}{729} a^{3} + \frac{24131}{729} a^{2} + \frac{615892}{729} a + \frac{18719}{27} \) | ||
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: | \(0 \le r \le 1\) |
| Torsion structure: | \(\Z/11\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a : \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 3 a + 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0 \le r \le 1\) | ||
| Regulator: | not available | ||
| Period: | \( 317.92834724060537881845757007132405132 \) | ||
| Tamagawa product: | \( 11 \) | ||
| Torsion order: | \(11\) | ||
| Leading coefficient: | \( 2.78022165091790 \) | ||
| Analytic order of Ш: | not available | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/3a^3-1/3a^2+3a+2)\) | \(3\) | \(11\) | \(I_{11}\) | Split multiplicative | \(-1\) | \(1\) | \(11\) | \(11\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
3.1-a
consists of curves linked by isogenies of
degree 11.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.