Base field 4.4.5125.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
gp: K = nfinit(Polrev([11, 7, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([4,-4,-2,1]),K([0,0,0,0]),K([-42,-95,-20,22]),K([416,540,34,-107])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([4,-4,-2,1]),Polrev([0,0,0,0]),Polrev([-42,-95,-20,22]),Polrev([416,540,34,-107])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![4,-4,-2,1],K![0,0,0,0],K![-42,-95,-20,22],K![416,540,34,-107]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-4a)\) | = | \((-a^3+a^2+4a+1)\cdot(-a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 55 \) | = | \(5\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((25a^3+50a^2-175a-275)\) | = | \((-a^3+a^2+4a+1)^{8}\cdot(-a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 47265625 \) | = | \(5^{8}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{196392867540579}{3025} a^{3} + \frac{727903424191802}{3025} a^{2} - \frac{12741819493871}{605} a - \frac{1266039682540334}{3025} \) | ||
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 | $\left(a^{3} + 2 a^{2} - 3 a - 2 : 6 a^{3} + 8 a^{2} - 25 a - 36 : 1\right)$ |
| Height | \(0.047398503847179915065632073065345758340\) |
| 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: | $\left(-\frac{3}{2} a^{3} + 2 a^{2} + \frac{23}{4} a + \frac{1}{2} : \frac{3}{8} a^{3} + \frac{1}{2} a^{2} - \frac{25}{8} a - \frac{19}{4} : 1\right)$ |
| 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: | \( 0.047398503847179915065632073065345758340 \) | ||
| Period: | \( 856.40968924317011665992094676002475013 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.26808466293632 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a+1)\) | \(5\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((-a)\) | \(11\) | \(2\) | \(I_{2}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
55.1-e
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.