Base field 4.4.13068.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,4,2,-1]),K([0,-5,-1,1]),K([-1,4,2,-1]),K([-287,801,252,-159]),K([-4702,13093,4652,-2775])])
gp: E = ellinit([Polrev([-2,4,2,-1]),Polrev([0,-5,-1,1]),Polrev([-1,4,2,-1]),Polrev([-287,801,252,-159]),Polrev([-4702,13093,4652,-2775])], K);
magma: E := EllipticCurve([K![-2,4,2,-1],K![0,-5,-1,1],K![-1,4,2,-1],K![-287,801,252,-159],K![-4702,13093,4652,-2775]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-5a-4)\) | = | \((2a^3-3a^2-10a+2)\cdot(a^2+a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 12 \) | = | \(3\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((81a^3-243a^2-81a+162)\) | = | \((2a^3-3a^2-10a+2)^{16}\cdot(a^2+a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 688747536 \) | = | \(3^{16}\cdot4^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1342804498359466478375}{324} a^{3} + \frac{76231679994737273527}{9} a^{2} + \frac{296226208689188310115}{324} a - \frac{110292388112309740031}{81} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{17}{2} a^{3} - \frac{29}{2} a^{2} - \frac{79}{2} a + \frac{51}{4} : \frac{87}{8} a^{3} - \frac{29}{2} a^{2} - \frac{255}{4} a + \frac{43}{2} : 1\right)$ |
| 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: | \( 8.7216205736662155627241160377950094183 \) | ||
| Tamagawa product: | \( 32 \) = \(2^{4}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.44142059926107 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^3-3a^2-10a+2)\) | \(3\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
| \((a^2+a-1)\) | \(4\) | \(2\) | \(I_{2}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
12.1-d
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.