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,-3,1/3,1/3]),K([-2,-3,1/3,1/3]),K([-3,-1,1,0]),K([1,0,-4/3,-1/3]),K([-2,-2,2/3,2/3])])
gp: E = ellinit([Polrev([0,-3,1/3,1/3]),Polrev([-2,-3,1/3,1/3]),Polrev([-3,-1,1,0]),Polrev([1,0,-4/3,-1/3]),Polrev([-2,-2,2/3,2/3])], K);
magma: E := EllipticCurve([K![0,-3,1/3,1/3],K![-2,-3,1/3,1/3],K![-3,-1,1,0],K![1,0,-4/3,-1/3],K![-2,-2,2/3,2/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/3a^3-1/3a^2+4a-1)\) | = | \((-1/3a^3-1/3a^2+3a+2)\cdot(-1/3a^3-1/3a^2+2a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2/3a^3-1/3a^2-7a-7)\) | = | \((-1/3a^3-1/3a^2+3a+2)^{2}\cdot(-1/3a^3-1/3a^2+2a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3087 \) | = | \(-3^{2}\cdot7^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{45668663}{343} a^{3} - \frac{10461181}{1029} a^{2} + \frac{362255132}{343} a + \frac{367286758}{1029} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(1 : -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a + 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 2775.2341315986191602360511886094125625 \) | ||
| Tamagawa product: | \( 6 \) = \(2\cdot3\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 3.28537990874532 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/3a^3-1/3a^2+3a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((-1/3a^3-1/3a^2+2a)\) | \(7\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.