Properties

Label 5.5.65657.1-15.1-a2
Base field 5.5.65657.1
Conductor norm \( 15 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}^{2}+\left(10a^{4}-10a^{3}-40a^{2}+16a+5\right){x}-25a^{4}+18a^{3}+100a^{2}-13a-9\)
sage: E = EllipticCurve([K([-2,0,1,0,0]),K([2,2,-4,-1,1]),K([-2,0,1,0,0]),K([5,16,-40,-10,10]),K([-9,-13,100,18,-25])])
 
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([2,2,-4,-1,1]),Polrev([-2,0,1,0,0]),Polrev([5,16,-40,-10,10]),Polrev([-9,-13,100,18,-25])], K);
 
magma: E := EllipticCurve([K![-2,0,1,0,0],K![2,2,-4,-1,1],K![-2,0,1,0,0],K![5,16,-40,-10,10],K![-9,-13,100,18,-25]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^4-3a^3-8a^2+7a+5)\) = \((-a^4+a^3+4a^2-2a-2)\cdot(-a^2+a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 15 \) = \(3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((94a^4-101a^3-392a^2+276a+150)\) = \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)^{14}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 54931640625 \) = \(3^{2}\cdot5^{14}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{810387038438023516}{54931640625} a^{4} + \frac{363395431073753132}{18310546875} a^{3} - \frac{1491237718490098883}{54931640625} a^{2} - \frac{207634903981355686}{6103515625} a - \frac{338342690307372463}{54931640625} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a^{4} - 2 a^{3} - 8 a^{2} + 4 a + 1 : a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 2 : 1\right)$ $\left(-\frac{5}{4} a^{4} + a^{3} + 5 a^{2} - 2 a - 1 : -\frac{1}{2} a^{4} + \frac{11}{8} a^{3} + \frac{13}{8} a^{2} - \frac{13}{4} a - \frac{1}{8} : 1\right)$
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: \( 1587.3957610643180385564014192988980181 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.54876208 \)
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\((j)_{-}\)
\((-a^4+a^3+4a^2-2a-2)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a^2+a+2)\) \(5\) \(2\) \(I_{14}\) Non-split multiplicative \(1\) \(1\) \(14\) \(14\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 15.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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