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([-2,0,1,0,0]),K([-5,-5,9,2,-2]),K([-4,-3,5,1,-1]),K([-3,-1,13,2,-3]),K([10,68,80,-5,-18])])
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([-5,-5,9,2,-2]),Polrev([-4,-3,5,1,-1]),Polrev([-3,-1,13,2,-3]),Polrev([10,68,80,-5,-18])], K);
magma: E := EllipticCurve([K![-2,0,1,0,0],K![-5,-5,9,2,-2],K![-4,-3,5,1,-1],K![-3,-1,13,2,-3],K![10,68,80,-5,-18]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^4-3a^3-9a^2+9a+7)\) | = | \((-a^4+a^3+4a^2-2a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^4+9a^3+21a^2-27a-20)\) | = | \((-a^4+a^3+4a^2-2a-2)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 177147 \) | = | \(3^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -1283318 a^{4} + 1637833 a^{3} + 6106189 a^{2} - 4095058 a - 5583206 \) | ||
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 | $\left(-2 a^{3} + a^{2} + 7 a + 3 : 4 a^{4} + 2 a^{3} - 21 a^{2} - 13 a + 1 : 1\right)$ |
| Height | \(0.070190700896818504483593225094055228925\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.070190700896818504483593225094055228925 \) | ||
| Period: | \( 1906.8653015877971495755733607314665613 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.61173484 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-c
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.