Base field 5.5.81509.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 5 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 3, -5, -1, 1]))
gp: K = nfinit(Polrev([-2, 5, 3, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 3, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,3,-4,-1,1]),K([0,2,4,0,-1]),K([3,-1,-5,0,1]),K([-3,7,9,-1,-2]),K([-4,13,-1,-4,1])])
gp: E = ellinit([Polrev([2,3,-4,-1,1]),Polrev([0,2,4,0,-1]),Polrev([3,-1,-5,0,1]),Polrev([-3,7,9,-1,-2]),Polrev([-4,13,-1,-4,1])], K);
magma: E := EllipticCurve([K![2,3,-4,-1,1],K![0,2,4,0,-1],K![3,-1,-5,0,1],K![-3,7,9,-1,-2],K![-4,13,-1,-4,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+3a^2-a+1)\) | = | \((-a^4+a^3+3a^2-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(16\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^4+a^3+3a^2-a-1)\) | = | \((-a^4+a^3+3a^2-a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16 \) | = | \(16\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{62067603}{2} a^{4} + \frac{13384031}{2} a^{3} - \frac{294061597}{2} a^{2} - \frac{171274065}{2} a + 51054913 \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{3} - 3 a : -a^{4} + 2 a^{3} + 3 a^{2} - 5 a + 2 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 4269.3218115880405957229954276200504085 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 0.598158255 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+3a^2-a+1)\) | \(16\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.