Base field 4.4.7168.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, 0, -6, 0, 1]))
gp: K = nfinit(Polrev([7, 0, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 0, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-3,1,1]),K([2,-3,-1,1]),K([0,0,0,0]),K([39,-21,-24,13]),K([-266,125,168,-79])])
gp: E = ellinit([Polrev([-3,-3,1,1]),Polrev([2,-3,-1,1]),Polrev([0,0,0,0]),Polrev([39,-21,-24,13]),Polrev([-266,125,168,-79])], K);
magma: E := EllipticCurve([K![-3,-3,1,1],K![2,-3,-1,1],K![0,0,0,0],K![39,-21,-24,13],K![-266,125,168,-79]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2-2a-4)\) | = | \((a^2-a-2)^{5}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 32 \) | = | \(2^{5}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((8)\) | = | \((a^2-a-2)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4096 \) | = | \(2^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 43136 a^{2} - 68416 \) | ||
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 | $\left(-a^{3} + 2 a^{2} + 2 a - 4 : a^{2} - 2 : 1\right)$ |
| Height | \(0.048120886805243357844181384246571592620\) |
| 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: | $\left(-a^{3} + \frac{3}{2} a^{2} + 2 a - \frac{7}{2} : 2 a^{2} - \frac{1}{2} a - \frac{7}{2} : 1\right)$ |
| 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: | \( 0.048120886805243357844181384246571592620 \) | ||
| Period: | \( 1092.5081306133141136952259452255813222 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.48381527145012 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-a-2)\) | \(2\) | \(4\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(5\) | \(12\) | \(0\) |
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.
Its isogeny class
32.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.