Base field 4.4.17609.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 10 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 10, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, 10, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-11,1,2]),K([-2,-5,1,1]),K([3,-6,0,1]),K([662,-6186,480,986]),K([16845,-157124,12147,25024])])
gp: E = ellinit([Polrev([3,-11,1,2]),Polrev([-2,-5,1,1]),Polrev([3,-6,0,1]),Polrev([662,-6186,480,986]),Polrev([16845,-157124,12147,25024])], K);
magma: E := EllipticCurve([K![3,-11,1,2],K![-2,-5,1,1],K![3,-6,0,1],K![662,-6186,480,986],K![16845,-157124,12147,25024]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-6a)\) | = | \((a^3+a^2-5a+1)\cdot(-a^3+6a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((37a^3-433a^2-282a+2636)\) | = | \((a^3+a^2-5a+1)^{34}\cdot(-a^3+6a-2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -841813590016 \) | = | \(-2^{34}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1402574522628382639}{841813590016} a^{3} - \frac{348633256698689353}{420906795008} a^{2} + \frac{8799534999868774715}{841813590016} a - \frac{844138674551993185}{841813590016} \) | ||
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 | $\left(-10 a^{3} + a^{2} + 78 a - 8 : -39 a^{3} + 15 a^{2} + 335 a - 38 : 1\right)$ |
| Height | \(1.6212225063801513242248252850787733590\) |
| 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: | $\left(\frac{17}{2} a^{3} + 4 a^{2} - \frac{217}{4} a + \frac{25}{4} : \frac{161}{8} a^{3} + \frac{43}{4} a^{2} - \frac{251}{2} a + \frac{95}{8} : 1\right)$ |
| 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: | \( 1.6212225063801513242248252850787733590 \) | ||
| Period: | \( 38.459778976572122310410014254856290836 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.87949880020666 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-5a+1)\) | \(2\) | \(2\) | \(I_{34}\) | Non-split multiplicative | \(1\) | \(1\) | \(34\) | \(34\) |
| \((-a^3+6a-2)\) | \(7\) | \(2\) | \(I_{2}\) | 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.
Its isogeny class
14.1-d
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.