Base field 4.4.9248.1
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-3,1,1]),K([2,1,-1,0]),K([0,1,0,0]),K([-7,7,4,-1]),K([3,-8,1,3])])
gp: E = ellinit([Polrev([-2,-3,1,1]),Polrev([2,1,-1,0]),Polrev([0,1,0,0]),Polrev([-7,7,4,-1]),Polrev([3,-8,1,3])], K);
magma: E := EllipticCurve([K![-2,-3,1,1],K![2,1,-1,0],K![0,1,0,0],K![-7,7,4,-1],K![3,-8,1,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2)\) | = | \((a)^{4}\) |
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: | \((a^2+2)\) | = | \((a)^{8}\) |
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: | \( -1995 a^{2} + 11006 \) | ||
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 | $\left(-a : a^{3} + a^{2} - 4 a - 1 : 1\right)$ | |
| Height | \(0.26989964619191992421094359459057857236\) | |
| 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: | $\left(a^{2} - a - 5 : 2 a^{3} + 2 a^{2} - 8 a - 5 : 1\right)$ | $\left(-\frac{1}{4} a^{2} - a + \frac{1}{2} : \frac{1}{2} a^{3} + \frac{9}{8} a^{2} - a - \frac{3}{4} : 1\right)$ |
| 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: | \( 0.26989964619191992421094359459057857236 \) | ||
| Period: | \( 2312.3446970096538243417281241220960405 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.24489750201335 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
16.4-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{17}) \) | 2.2.17.1-256.1-f3 |
| \(\Q(\sqrt{17}) \) | 2.2.17.1-64.7-a3 |