Properties

Label 5.5.24217.1-59.1-a1
Base field 5.5.24217.1
Conductor norm \( 59 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+\left(2a^{4}-9a^{2}+3\right){x}{y}+\left(3a^{4}-a^{3}-14a^{2}+3a+6\right){y}={x}^{3}+\left(4a^{4}-2a^{3}-19a^{2}+6a+7\right){x}^{2}+\left(-4a^{4}+3a^{3}+18a^{2}-9a-5\right){x}+3a^{4}-2a^{3}-13a^{2}+7a+3\)
sage: E = EllipticCurve([K([3,0,-9,0,2]),K([7,6,-19,-2,4]),K([6,3,-14,-1,3]),K([-5,-9,18,3,-4]),K([3,7,-13,-2,3])])
 
gp: E = ellinit([Polrev([3,0,-9,0,2]),Polrev([7,6,-19,-2,4]),Polrev([6,3,-14,-1,3]),Polrev([-5,-9,18,3,-4]),Polrev([3,7,-13,-2,3])], K);
 
magma: E := EllipticCurve([K![3,0,-9,0,2],K![7,6,-19,-2,4],K![6,3,-14,-1,3],K![-5,-9,18,3,-4],K![3,7,-13,-2,3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-4a^2-1)\) = \((a^4-4a^2-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 59 \) = \(59\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4-2a^3+5a^2+6a-4)\) = \((a^4-4a^2-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3481 \) = \(-59^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4479742}{3481} a^{4} + \frac{1301710}{3481} a^{3} + \frac{19202738}{3481} a^{2} - \frac{677371}{3481} a - \frac{7584461}{3481} \)
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 $\left(-5 a^{4} - a^{3} + 22 a^{2} + 8 a - 1 : 8 a^{4} + 7 a^{3} - 33 a^{2} - 38 a - 10 : 1\right)$
Height \(0.0029772215814875638781359908629100044328\)
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: \( 0.0029772215814875638781359908629100044328 \)
Period: \( 8328.5418546717714081379910769341945120 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 1.59338220 \)
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\((j)_{-}\)
\((a^4-4a^2-1)\) \(59\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 59.1-a consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.