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([-10/9,28/9,1/3,-2/9]),K([11/9,61/9,1/3,-5/9]),K([-32/9,14/9,2/3,-1/9]),K([7/3,128/3,-11,-1/3]),K([70/9,677/9,-178/3,140/9])])
gp: E = ellinit([Polrev([-10/9,28/9,1/3,-2/9]),Polrev([11/9,61/9,1/3,-5/9]),Polrev([-32/9,14/9,2/3,-1/9]),Polrev([7/3,128/3,-11,-1/3]),Polrev([70/9,677/9,-178/3,140/9])], K);
magma: E := EllipticCurve([K![-10/9,28/9,1/3,-2/9],K![11/9,61/9,1/3,-5/9],K![-32/9,14/9,2/3,-1/9],K![7/3,128/3,-11,-1/3],K![70/9,677/9,-178/3,140/9]]);
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: | \((-2/9a^3+1/3a^2+28/9a+26/9)\) | = | \((7/9a^3-5/3a^2-71/9a+80/9)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 256 \) | = | \(2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{15319210}{9} a^{3} - \frac{7659605}{3} a^{2} - \frac{214468940}{9} a + \frac{253180334}{9} \) | ||
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: | $\left(\frac{1}{6} a^{3} - \frac{3}{4} a^{2} - \frac{4}{3} a - \frac{7}{6} : -\frac{1}{12} a^{3} - \frac{1}{8} a^{2} + \frac{19}{6} a + \frac{19}{12} : 1\right)$ |
| 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: | \( 289.04308712620672804271601551526200506 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.12531681710446 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((7/9a^3-5/3a^2-71/9a+80/9)\) | \(2\) | \(4\) | \(I_{1}^{*}\) | Additive | \(1\) | \(3\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
8.3-b
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 |
|---|---|
| \(\Q(\sqrt{17}) \) | 2.2.17.1-64.6-a5 |
| \(\Q(\sqrt{17}) \) | 2.2.17.1-512.4-e5 |