Base field 4.4.7488.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -4, -2, 1]))
gp: K = nfinit(Polrev([1, 2, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-2,1]),K([1,6,1,-1]),K([1,0,0,0]),K([-3,-9,-9,4]),K([3,8,-6,1])])
gp: E = ellinit([Polrev([2,-3,-2,1]),Polrev([1,6,1,-1]),Polrev([1,0,0,0]),Polrev([-3,-9,-9,4]),Polrev([3,8,-6,1])], K);
magma: E := EllipticCurve([K![2,-3,-2,1],K![1,6,1,-1],K![1,0,0,0],K![-3,-9,-9,4],K![3,8,-6,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-2a^2-3a+1)\) | = | \((a^3-2a^2-3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^3+6a^2+9a-3)\) | = | \((a^3-2a^2-3a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 729 \) | = | \(9^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{12413440}{9} a^{3} + \frac{24826880}{9} a^{2} + \frac{12413440}{3} a + \frac{9047744}{9} \) | ||
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 | $\left(-2 a^{3} + 6 a^{2} : 5 a^{3} - 19 a^{2} + 10 a + 5 : 1\right)$ |
| Height | \(0.97137802925084081290577493246976424037\) |
| 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: | $\left(-a^{2} + 2 a + 3 : 2 a^{3} - 3 a^{2} - 9 a - 2 : 1\right)$ |
| 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: | \( 0.97137802925084081290577493246976424037 \) | ||
| Period: | \( 445.03967444381771234628142985218807474 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.66526300107125 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-2a^2-3a+1)\) | \(9\) | \(3\) | \(I_{3}\) | 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 |
| \(3\) | 3Cs.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
9.1-a
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.