Base field 4.4.15952.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -6, 0, 1]))
gp: K = nfinit(Polrev([1, -2, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-5,0,1]),K([0,-11,-1,2]),K([0,-6,0,1]),K([-198,-345,13,58]),K([-1158,-1900,85,320])])
gp: E = ellinit([Polrev([0,-5,0,1]),Polrev([0,-11,-1,2]),Polrev([0,-6,0,1]),Polrev([-198,-345,13,58]),Polrev([-1158,-1900,85,320])], K);
magma: E := EllipticCurve([K![0,-5,0,1],K![0,-11,-1,2],K![0,-6,0,1],K![-198,-345,13,58],K![-1158,-1900,85,320]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+6a+4)\) | = | \((a^2+2a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((7a^3-a^2-34a+1)\) | = | \((a^2+2a)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -19683 \) | = | \(-3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3451351040}{27} a^{3} + \frac{10947869888}{27} a^{2} + \frac{8041774976}{27} a + \frac{1570955776}{27} \) | ||
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: | \(1\) |
| Generator | $\left(-7 a^{3} - a^{2} + 42 a + 23 : -35 a^{3} - 10 a^{2} + 208 a + 129 : 1\right)$ |
| Height | \(0.97090892065314791989986551452145333864\) |
| 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: | $\left(\frac{1}{2} a^{3} + a^{2} - 4 a - 8 : \frac{5}{2} a^{3} - \frac{1}{4} a^{2} - 13 a - \frac{7}{4} : 1\right)$ |
| 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: | \( 0.97090892065314791989986551452145333864 \) | ||
| Period: | \( 150.36210876283264751590439643583574508 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.31174179264793 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+2a)\) | \(3\) | \(2\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(2\) | \(9\) | \(3\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.