Base field 4.4.12725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 10 x^{2} + 11 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 11, -10, -2, 1]))
gp: K = nfinit(Polrev([29, 11, -10, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 11, -10, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,-1,1,0]),K([-6,-8,0,1]),K([-5,-1,1,0]),K([36,24,-3,-3]),K([-12,-10,0,1])])
gp: E = ellinit([Polrev([-5,-1,1,0]),Polrev([-6,-8,0,1]),Polrev([-5,-1,1,0]),Polrev([36,24,-3,-3]),Polrev([-12,-10,0,1])], K);
magma: E := EllipticCurve([K![-5,-1,1,0],K![-6,-8,0,1],K![-5,-1,1,0],K![36,24,-3,-3],K![-12,-10,0,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2a-4)\) | = | \((a^2-2a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^2-12a-36)\) | = | \((a^2-2a-4)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1771561 \) | = | \(11^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{24388973560}{1771561} a^{3} - \frac{17420952534}{1771561} a^{2} + \frac{202430910278}{1771561} a + \frac{275090744229}{1771561} \) | ||
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 | $\left(-\frac{3}{121} a^{3} + \frac{97}{121} a^{2} + \frac{32}{121} a - \frac{439}{121} : \frac{62}{1331} a^{3} - \frac{109}{1331} a^{2} + \frac{105}{1331} a + \frac{643}{1331} : 1\right)$ |
| Height | \(0.84876069482007184374641674344274040296\) |
| 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: | $\left(a + 2 : a + 1 : 1\right)$ |
| 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: | \( 0.84876069482007184374641674344274040296 \) | ||
| Period: | \( 142.58731998596709580305780908003578812 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.86092190317445 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2a-4)\) | \(11\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
11.3-d
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.