Properties

Label 5.5.81509.1-16.1-c2
Base field 5.5.81509.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+\left(a+1\right){x}{y}+\left(a^{4}-5a^{2}+4\right){y}={x}^{3}+\left(-a^{4}+4a^{2}+2a-1\right){x}^{2}+\left(a^{3}-a^{2}-5a-3\right){x}+12a^{4}+4a^{3}-57a^{2}-40a+14\)
sage: E = EllipticCurve([K([1,1,0,0,0]),K([-1,2,4,0,-1]),K([4,0,-5,0,1]),K([-3,-5,-1,1,0]),K([14,-40,-57,4,12])])
 
gp: E = ellinit([Polrev([1,1,0,0,0]),Polrev([-1,2,4,0,-1]),Polrev([4,0,-5,0,1]),Polrev([-3,-5,-1,1,0]),Polrev([14,-40,-57,4,12])], K);
 
magma: E := EllipticCurve([K![1,1,0,0,0],K![-1,2,4,0,-1],K![4,0,-5,0,1],K![-3,-5,-1,1,0],K![14,-40,-57,4,12]]);
 

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: \((-a^4-a^3+7a^2+3a-7)\) = \((-a^4+a^3+3a^2-a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -256 \) = \(-16^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1867107}{2} a^{4} - \frac{5796783}{2} a^{3} + 220121 a^{2} + \frac{16606357}{4} a - \frac{2880145}{2} \)
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: $\left(-\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{4} a^{2} + \frac{7}{4} a + \frac{5}{4} : -\frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{7}{8} a^{2} - \frac{17}{8} a - \frac{19}{8} : 1\right)$
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: \( 818.52806396576207639305891660439953024 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.43351023 \)
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\((j)_{-}\)
\((-a^4+a^3+3a^2-a+1)\) \(16\) \(2\) \(I_{2}\) 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 16.1-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.