Base field 3.3.985.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-1,1]),K([0,0,0]),K([-4,-1,1]),K([4,1,-1]),K([5,1,-1])])
gp: E = ellinit([Polrev([-4,-1,1]),Polrev([0,0,0]),Polrev([-4,-1,1]),Polrev([4,1,-1]),Polrev([5,1,-1])], K);
magma: E := EllipticCurve([K![-4,-1,1],K![0,0,0],K![-4,-1,1],K![4,1,-1],K![5,1,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a+3)\) | = | \((-a-1)\cdot(a^2-2a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 55 \) | = | \(5\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^2+12a-9)\) | = | \((-a-1)^{2}\cdot(a^2-2a-2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3025 \) | = | \(5^{2}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3616326}{3025} a^{2} - \frac{9298503}{3025} a + \frac{5981869}{3025} \) | ||
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 | $\left(-\frac{2}{9} a^{2} + \frac{2}{9} a + \frac{4}{9} : \frac{5}{27} a^{2} - \frac{1}{9} a - 1 : 1\right)$ |
| Height | \(2.1412865109972386156366404812162751420\) |
| 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: | $\left(-a^{2} + a + 6 : -6 a^{2} + 5 a + 36 : 1\right)$ |
| 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: | \( 2.1412865109972386156366404812162751420 \) | ||
| Period: | \( 179.13148461925611399666666449018914566 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 4.07387160 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((a^2-2a-2)\) | \(11\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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
55.2-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.