Base field 4.4.17428.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + 4 x + 6 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([6, 4, -6, -1, 1]))
gp: K = nfinit(Polrev([6, 4, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, 4, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1,0]),K([-3,5,1,-1]),K([-2,0,1,0]),K([0,-18,-9,0]),K([-5,10,14,4])])
gp: E = ellinit([Polrev([-2,0,1,0]),Polrev([-3,5,1,-1]),Polrev([-2,0,1,0]),Polrev([0,-18,-9,0]),Polrev([-5,10,14,4])], K);
magma: E := EllipticCurve([K![-2,0,1,0],K![-3,5,1,-1],K![-2,0,1,0],K![0,-18,-9,0],K![-5,10,14,4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-6)\) | = | \((a^3+a^2-3a-2)^{2}\cdot(a^2-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 12 \) | = | \(2^{2}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-9a^3+16a^2+34a-66)\) | = | \((a^3+a^2-3a-2)^{8}\cdot(a^2-a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 186624 \) | = | \(2^{8}\cdot3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{24959305}{729} a^{3} - \frac{44565748}{729} a^{2} - \frac{12666058}{81} a + \frac{192446194}{729} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{3} + a^{2} - 9 a - 5 : -a^{3} - 2 a^{2} + 3 a + 5 : 1\right)$ | $\left(-\frac{1}{4} a^{3} - \frac{1}{2} a^{2} + 2 a + \frac{7}{2} : -\frac{1}{8} a^{3} - a^{2} - \frac{1}{4} a + \frac{9}{4} : 1\right)$ |
| 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: | \( 801.81856467953287408043639021722774131 \) | ||
| Tamagawa product: | \( 6 \) = \(1\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.27763222845421 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-3a-2)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((a^2-a-3)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
12.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.