↑

Properties

Label 5.5.65657.1-45.1-c2
Base field 5.5.65657.1
Conductor norm \( 45 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

\({y}^2+\left(-a^{4}+2a^{3}+5a^{2}-7a-5\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-4\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-11a-9\right){x}^{2}+\left(24a^{4}-31a^{3}-110a^{2}+79a+89\right){x}+93a^{4}-125a^{3}-415a^{2}+318a+327\)
sage: E = EllipticCurve([K([-5,-7,5,2,-1]),K([-9,-11,13,4,-3]),K([-4,-6,5,2,-1]),K([89,79,-110,-31,24]),K([327,318,-415,-125,93])])
 
gp: E = ellinit([Polrev([-5,-7,5,2,-1]),Polrev([-9,-11,13,4,-3]),Polrev([-4,-6,5,2,-1]),Polrev([89,79,-110,-31,24]),Polrev([327,318,-415,-125,93])], K);
 
magma: E := EllipticCurve([K![-5,-7,5,2,-1],K![-9,-11,13,4,-3],K![-4,-6,5,2,-1],K![89,79,-110,-31,24],K![327,318,-415,-125,93]]);
 

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: \((-13a^4+21a^3+43a^2-52a-8)\) = \((-a^4+a^3+4a^2-2a-2)^{9}\cdot(-a^2+a+2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -492075 \) = \(-3^{9}\cdot5^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{11314724189}{25} a^{4} - \frac{30736776391}{25} a^{3} - \frac{3562664832}{25} a^{2} + \frac{29192288229}{25} a + \frac{6597394298}{25} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{3}{4} a^{4} - \frac{5}{4} a^{3} - 2 a^{2} + \frac{11}{4} a - 2 : -\frac{7}{4} a^{4} + \frac{13}{4} a^{3} + \frac{53}{8} a^{2} - \frac{19}{2} a - \frac{45}{8} : 1\right)$
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: \( 443.50585652389200261293061850472300683 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.73084764 \)
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 \(1\) \(2\) \(9\) \(0\)
\((-a^2+a+2)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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-c 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.