Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 1]))
gp: K = nfinit(Polrev([1, 0, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-4,1,1]),K([-1,-4,1,1]),K([1,-3,0,1]),K([-9,-19,0,4]),K([-18,-33,4,11])])
gp: E = ellinit([Polrev([-2,-4,1,1]),Polrev([-1,-4,1,1]),Polrev([1,-3,0,1]),Polrev([-9,-19,0,4]),Polrev([-18,-33,4,11])], K);
magma: E := EllipticCurve([K![-2,-4,1,1],K![-1,-4,1,1],K![1,-3,0,1],K![-9,-19,0,4],K![-18,-33,4,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+2a^2-2)\) | = | \((a^3+2a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 47 \) | = | \(47\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3+a^2-5a-4)\) | = | \((a^3+2a^2-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -47 \) | = | \(-47\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{76092676472}{47} a^{3} + \frac{147059917488}{47} a^{2} - \frac{20384319416}{47} a - \frac{39402167512}{47} \) | ||
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 | $\left(2 a^{2} - 2 a - 2 : 4 a^{3} - 6 a^{2} - 4 a - 1 : 1\right)$ |
| Height | \(0.085869761963972778227827393368850946224\) |
| 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: | $\left(a^{3} + \frac{1}{4} a^{2} - \frac{5}{2} a - \frac{7}{4} : \frac{5}{8} a^{3} + \frac{9}{8} a^{2} - \frac{31}{8} a - \frac{27}{8} : 1\right)$ |
| 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: | \( 0.085869761963972778227827393368850946224 \) | ||
| Period: | \( 505.35198929455899636873265538693592881 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.904051146432165 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+2a^2-2)\) | \(47\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 and 4.
Its isogeny class
47.2-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.