Base field 4.4.12197.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 3, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1,0]),K([0,6,0,-1]),K([3,-5,-1,1]),K([-8,-6,5,1]),K([-34,-25,6,5])])
gp: E = ellinit([Polrev([-3,1,1,0]),Polrev([0,6,0,-1]),Polrev([3,-5,-1,1]),Polrev([-8,-6,5,1]),Polrev([-34,-25,6,5])], K);
magma: E := EllipticCurve([K![-3,1,1,0],K![0,6,0,-1],K![3,-5,-1,1],K![-8,-6,5,1],K![-34,-25,6,5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-4a-2)\) | = | \((a^3-4a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 37 \) | = | \(37\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1342a^3+301a^2+7815a-2502)\) | = | \((a^3-4a-2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -129961739795077 \) | = | \(-37^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{182667213631522428976}{129961739795077} a^{3} + \frac{50401070034046346530}{129961739795077} a^{2} + \frac{932928939055723203988}{129961739795077} a + \frac{134727608330513111571}{129961739795077} \) | ||
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 | $\left(415 a^{3} - 514 a^{2} - 1952 a + 1713 : 19876 a^{3} - 24681 a^{2} - 93410 a + 82207 : 1\right)$ |
| Height | \(0.62290476149162688104092444824683310210\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{2} + 6 : a^{3} - 5 a - 6 : 1\right)$ |
| 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: | \( 0.62290476149162688104092444824683310210 \) | ||
| Period: | \( 11.845732779074630385684495231195919938 \) | ||
| Tamagawa product: | \( 9 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.40524638681567 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a-2)\) | \(37\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
37.2-a
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.