Base field 4.4.10512.1
Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-5,-5,0,1]),K([-4,-1,1,0]),K([-83,-80,4,12]),K([-254,-252,7,37])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-5,-5,0,1]),Polrev([-4,-1,1,0]),Polrev([-83,-80,4,12]),Polrev([-254,-252,7,37])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-5,-5,0,1],K![-4,-1,1,0],K![-83,-80,4,12],K![-254,-252,7,37]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a+3)\) | = | \((a+3)\) |
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: | \((-a^3+2a^2+4a+4)\) | = | \((a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1369 \) | = | \(37^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{5509092}{1369} a^{3} - \frac{16694553}{1369} a^{2} - \frac{33114084}{1369} a - \frac{4668088}{1369} \) | ||
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 | $\left(-\frac{454}{121} a^{3} - \frac{1}{121} a^{2} + \frac{3295}{121} a + \frac{3320}{121} : -\frac{21348}{1331} a^{3} - \frac{1781}{1331} a^{2} + \frac{154654}{1331} a + \frac{154893}{1331} : 1\right)$ |
| Height | \(2.4995414840673097166761132201186242664\) |
| 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: | $\left(-2 a^{3} + 13 a + 12 : a^{3} - 6 a - 5 : 1\right)$ |
| 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: | \( 2.4995414840673097166761132201186242664 \) | ||
| Period: | \( 80.571256622091386383036520423502520111 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.92850948168808 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(37\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
37.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.