Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,3,-9,-1,2]),K([4,-1,-10,0,2]),K([-2,1,1,0,0]),K([9,6,-20,-2,4]),K([4,9,-7,-4,2])])
gp: E = ellinit([Polrev([3,3,-9,-1,2]),Polrev([4,-1,-10,0,2]),Polrev([-2,1,1,0,0]),Polrev([9,6,-20,-2,4]),Polrev([4,9,-7,-4,2])], K);
magma: E := EllipticCurve([K![3,3,-9,-1,2],K![4,-1,-10,0,2],K![-2,1,1,0,0],K![9,6,-20,-2,4],K![4,9,-7,-4,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-3a)\) | = | \((a^3+a^2-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 61 \) | = | \(61\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^4-a^3-15a^2+2a+7)\) | = | \((a^3+a^2-3a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 61 \) | = | \(61\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{115152428}{61} a^{4} - \frac{248328657}{61} a^{3} + \frac{38281223}{61} a^{2} + \frac{189872228}{61} a + \frac{54566560}{61} \) | ||
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: | $\left(-a^{4} + 5 a^{2} - 1 : -a^{2} + 2 : 1\right)$ |
| 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: | \( 5294.1962819544817580505716150778592893 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 1.36081742 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-3a)\) | \(61\) | \(1\) | \(I_{1}\) | 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
61.2-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.