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([1,-11,-3,2]),K([0,0,0,0]),K([10,-39,-19,5]),K([3,-28,32,31])])
gp: E = ellinit([Polrev([-2,4,2,-1]),Polrev([1,-11,-3,2]),Polrev([0,0,0,0]),Polrev([10,-39,-19,5]),Polrev([3,-28,32,31])], K);
magma: E := EllipticCurve([K![-2,4,2,-1],K![1,-11,-3,2],K![0,0,0,0],K![10,-39,-19,5],K![3,-28,32,31]]);
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: | \((15a^3-15a^2-105a-72)\) | = | \((2a^3-3a^2-10a+2)^{4}\cdot(a^2+a-1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5308416 \) | = | \(3^{4}\cdot4^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{75343375421525}{48} a^{3} + \frac{3404376659285269}{768} a^{2} + \frac{42633632439585}{32} a - \frac{660888366058603}{768} \) | ||
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: | $\left(-\frac{7}{4} a^{3} + \frac{3}{4} a^{2} + \frac{37}{4} a + 1 : -2 a^{3} + \frac{29}{8} a^{2} + \frac{75}{8} a - \frac{19}{8} : 1\right)$ |
| 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: | \( 34.886482294664862250896464151180037673 \) | ||
| Tamagawa product: | \( 32 \) = \(2^{2}\cdot2^{3}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.44142059926107 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^3-3a^2-10a+2)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a^2+a-1)\) | \(4\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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-e
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.