Base field 6.6.1397493.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,10,8,-6,-2,1]),K([-1,-7,2,3,-1,0]),K([0,0,0,0,0,0]),K([48,-212,-240,132,80,-32]),K([-75,361,218,-195,-58,31])])
gp: E = ellinit([Polrev([-4,10,8,-6,-2,1]),Polrev([-1,-7,2,3,-1,0]),Polrev([0,0,0,0,0,0]),Polrev([48,-212,-240,132,80,-32]),Polrev([-75,361,218,-195,-58,31])], K);
magma: E := EllipticCurve([K![-4,10,8,-6,-2,1],K![-1,-7,2,3,-1,0],K![0,0,0,0,0,0],K![48,-212,-240,132,80,-32],K![-75,361,218,-195,-58,31]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-3a^4-2a^3+8a^2+a-4)\) | = | \((a^5-3a^4-2a^3+8a^2+a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^5-3a^4-2a^3+8a^2+a-4)\) | = | \((a^5-3a^4-2a^3+8a^2+a-4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 19 \) | = | \(19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{15934214875439431467}{19} a^{5} - \frac{65595149908150910652}{19} a^{4} + \frac{25442385660496971315}{19} a^{3} + \frac{130932604596100200840}{19} a^{2} - \frac{98399667624912226827}{19} a + \frac{14270008618918755090}{19} \) | ||
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 | $\left(4 a^{5} - 12 a^{4} - 10 a^{3} + 32 a^{2} + 12 a - 2 : -2 a^{5} - 9 a^{4} + 60 a^{3} - 13 a^{2} - 129 a + 27 : 1\right)$ |
| Height | \(1.4142683831261262466023743022854746619\) |
| 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: | $\left(-\frac{1}{2} a^{5} + \frac{9}{4} a^{4} - a^{3} - \frac{11}{2} a^{2} + \frac{15}{4} a + 1 : -\frac{11}{8} a^{5} + \frac{35}{8} a^{4} + \frac{21}{8} a^{3} - \frac{91}{8} a^{2} - \frac{5}{2} a + \frac{9}{4} : 1\right)$ |
| 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: | \( 1.4142683831261262466023743022854746619 \) | ||
| Period: | \( 1573.3512463617150873659705523147251804 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.82341 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-4)\) | \(19\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
19.1-b
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.