Base field 5.5.195829.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 6, -5, -2, 1]))
gp: K = nfinit(Polrev([1, 7, 6, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 6, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-8,3,3,-1]),K([2,8,-3,-3,1]),K([-2,-9,3,3,-1]),K([-5,2,4,-2,0]),K([5,15,-5,-6,2])])
gp: E = ellinit([Polrev([-3,-8,3,3,-1]),Polrev([2,8,-3,-3,1]),Polrev([-2,-9,3,3,-1]),Polrev([-5,2,4,-2,0]),Polrev([5,15,-5,-6,2])], K);
magma: E := EllipticCurve([K![-3,-8,3,3,-1],K![2,8,-3,-3,1],K![-2,-9,3,3,-1],K![-5,2,4,-2,0],K![5,15,-5,-6,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-1)\) | = | \((a^4-2a^3-5a^2+7a+5)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-5a^4+8a^3+22a^2-9a-24)\) | = | \((a^4-2a^3-5a^2+7a+5)^{21}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2097152 \) | = | \(2^{21}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{28609729}{512} a^{4} + \frac{14983671}{512} a^{3} - \frac{48563491}{256} a^{2} - \frac{4370945}{32} a - \frac{3686985}{512} \) | ||
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: | $\left(-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 4 : a - 2 : 1\right)$ |
| 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: | \( 1749.8900632765635695802944528716422059 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.97716205 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-5a^2+7a+5)\) | \(2\) | \(2\) | \(I_{13}^{*}\) | Additive | \(-1\) | \(4\) | \(21\) | \(9\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.2-d
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.