Base field \(\Q(\zeta_{16})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1,0]),K([0,4,0,-1]),K([-1,1,1,0]),K([-22,-34,-5,6]),K([-90,-138,-9,24])])
gp: E = ellinit([Polrev([-2,0,1,0]),Polrev([0,4,0,-1]),Polrev([-1,1,1,0]),Polrev([-22,-34,-5,6]),Polrev([-90,-138,-9,24])], K);
magma: E := EllipticCurve([K![-2,0,1,0],K![0,4,0,-1],K![-1,1,1,0],K![-22,-34,-5,6],K![-90,-138,-9,24]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3-a^2+3a+1)\) | = | \((-a^3-a^2+3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-20a^3-33a^2+81a+17)\) | = | \((-a^3-a^2+3a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 24137569 \) | = | \(17^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{6050416701310208}{24137569} a^{3} + \frac{9306221858122368}{24137569} a^{2} - \frac{5580466233834496}{24137569} a - \frac{3892474815436352}{24137569} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{3} + 2 a^{2} - 4 a - \frac{7}{2} : -\frac{3}{4} a^{2} - \frac{5}{2} a - 1 : 1\right)$ | $\left(a^{3} + \frac{1}{2} a^{2} - 6 a - 2 : 2 a^{3} - \frac{11}{2} a - 1 : 1\right)$ |
| 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: | \( 29.334435989700441028394029949434142924 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.729231279278903 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3-a^2+3a+1)\) | \(17\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 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
17.2-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.