Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-2,5,1,-1]),K([5,9,-8,-3,2]),K([-6,-9,9,3,-2]),K([1,-9,4,5,-2]),K([5,10,-11,-2,2])])
gp: E = ellinit([Polrev([-4,-2,5,1,-1]),Polrev([5,9,-8,-3,2]),Polrev([-6,-9,9,3,-2]),Polrev([1,-9,4,5,-2]),Polrev([5,10,-11,-2,2])], K);
magma: E := EllipticCurve([K![-4,-2,5,1,-1],K![5,9,-8,-3,2],K![-6,-9,9,3,-2],K![1,-9,4,5,-2],K![5,10,-11,-2,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-4a)\) | = | \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 45 \) | = | \(3^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^4-a^3-4a^2+4a)\) | = | \((-a^4+a^3+4a^2-2a-2)^{3}\cdot(-a^2+a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -135 \) | = | \(-3^{3}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{34992}{5} a^{4} - \frac{81598}{5} a^{3} - \frac{54171}{5} a^{2} + \frac{89407}{5} a + \frac{46644}{5} \) | ||
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 | $\left(-3 a^{4} + 5 a^{3} + 11 a^{2} - 13 a - 5 : 5 a^{4} - 9 a^{3} - 18 a^{2} + 25 a + 6 : 1\right)$ |
| Height | \(0.14944737562641485818220458685480080612\) |
| 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: | $\left(-a^{4} + a^{3} + 5 a^{2} - 4 a - 3 : a - 1 : 1\right)$ |
| 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: | \( 0.14944737562641485818220458685480080612 \) | ||
| Period: | \( 1827.7802934846516194138399428039640557 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.66508936 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
| \((-a^2+a+2)\) | \(5\) | \(1\) | \(I_{1}\) | Non-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.
Its isogeny class
45.1-a
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.