Base field 4.4.10304.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, 8, -7, -2, 1]))
gp: K = nfinit(Polrev([8, 8, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 8, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-7/2,0,1/2]),K([1,7/2,0,-1/2]),K([2,-2,-1/2,1/2]),K([-12,20,9/2,-7/2]),K([-18,20,11/2,-7/2])])
gp: E = ellinit([Polrev([0,-7/2,0,1/2]),Polrev([1,7/2,0,-1/2]),Polrev([2,-2,-1/2,1/2]),Polrev([-12,20,9/2,-7/2]),Polrev([-18,20,11/2,-7/2])], K);
magma: E := EllipticCurve([K![0,-7/2,0,1/2],K![1,7/2,0,-1/2],K![2,-2,-1/2,1/2],K![-12,20,9/2,-7/2],K![-18,20,11/2,-7/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+4)\) | = | \((-1/2a^3+a^2+5/2a-2)^{4}\) |
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: | \((a^2-8)\) | = | \((-1/2a^3+a^2+5/2a-2)^{8}\) |
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: | \( -12990 a^{3} + 44327 a^{2} - 2612 a - 22636 \) | ||
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 | $\left(\frac{5}{14} a^{2} - \frac{5}{14} a - \frac{18}{7} : \frac{2}{49} a^{3} + \frac{11}{49} a^{2} - \frac{61}{49} a - \frac{116}{49} : 1\right)$ | |
| Height | \(0.62665398135232240421810214861760466479\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{1}{4} a^{2} - \frac{1}{2} a - 2 : \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - 2 a - 1 : 1\right)$ | $\left(\frac{1}{2} a^{2} - \frac{1}{2} a - 3 : \frac{1}{2} a^{3} - a^{2} - \frac{7}{2} a + 2 : 1\right)$ |
| 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: | \( 0.62665398135232240421810214861760466479 \) | ||
| Period: | \( 2326.7325646963398469544844118947518269 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 1.79548325620151 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/2a^3+a^2+5/2a-2)\) | \(2\) | \(2\) | \(I_0^{*}\) | 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
16.3-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.