Base field \(\Q(\sqrt{2}, \sqrt{17})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 12, -11, -2, 1]))
gp: K = nfinit(Polrev([2, 12, -11, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 12, -11, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-20/9,47/9,2/3,-4/9]),K([-32/9,14/9,2/3,-1/9]),K([-22/9,-14/9,1/3,1/9]),K([-781/9,430/9,37/3,-41/9]),K([286,-318,-53,29])])
gp: E = ellinit([Polrev([-20/9,47/9,2/3,-4/9]),Polrev([-32/9,14/9,2/3,-1/9]),Polrev([-22/9,-14/9,1/3,1/9]),Polrev([-781/9,430/9,37/3,-41/9]),Polrev([286,-318,-53,29])], K);
magma: E := EllipticCurve([K![-20/9,47/9,2/3,-4/9],K![-32/9,14/9,2/3,-1/9],K![-22/9,-14/9,1/3,1/9],K![-781/9,430/9,37/3,-41/9],K![286,-318,-53,29]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2/3a^3-a^2-22/3a+10/3)\) | = | \((7/9a^3-5/3a^2-71/9a+80/9)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2/9a^3-1/3a^2-28/9a+10/9)\) | = | \((7/9a^3-5/3a^2-71/9a+80/9)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{16150}{3} a^{3} + 8075 a^{2} + \frac{226100}{3} a + \frac{51934}{3} \) | ||
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: | $\left(\frac{7}{9} a^{3} - \frac{17}{12} a^{2} - \frac{80}{9} a + \frac{115}{18} : \frac{1}{8} a^{3} - \frac{5}{4} a^{2} - \frac{1}{4} a + \frac{25}{2} : 1\right)$ | $\left(-\frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{14}{9} a + \frac{40}{9} : \frac{10}{9} a^{3} - \frac{5}{3} a^{2} - \frac{113}{9} a + \frac{50}{9} : 1\right)$ |
| 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: | \( 1288.4054170634759410615901637171884334 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.18419615538922 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((7/9a^3-5/3a^2-71/9a+80/9)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(3\) | \(4\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
8.3-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{17}) \) | 2.2.17.1-8.3-a3 |
| \(\Q(\sqrt{17}) \) | a curve with conductor norm 4096 (not in the database) |