Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([-3,-1,1,0]),K([-3,0,1,0]),K([-641,-602,58,104]),K([2117,2035,-185,-354])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([-3,-1,1,0]),Polrev([-3,0,1,0]),Polrev([-641,-602,58,104]),Polrev([2117,2035,-185,-354])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![-3,-1,1,0],K![-3,0,1,0],K![-641,-602,58,104],K![2117,2035,-185,-354]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 16 \) | = | \(16\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((2)\) | = | \((2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 16 \) | = | \(16\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -720868474875 a^{3} + \frac{2726844170253}{2} a^{2} + \frac{3499385683905}{2} a - 1144450579761 \) | ||
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: | \(1\) |
Generator | $\left(-5 a^{3} - 3 a^{2} + 31 a + 35 : -39 a^{3} - 20 a^{2} + 227 a + 237 : 1\right)$ |
Height | \(0.39007934142347511091294045184392270062\) |
Torsion structure: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-a^{2} + \frac{1}{4} a + \frac{7}{2} : -\frac{1}{8} a^{3} - \frac{3}{4} a^{2} + \frac{15}{8} a + \frac{21}{4} : 1\right)$ |
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: | \(1\) | ||
Regulator: | \( 0.39007934142347511091294045184392270062 \) | ||
Period: | \( 529.19409989574803846919345764551501513 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.07373391299568 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(16\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.