Properties

Label 5.5.65657.1-3.1-a1
Base field 5.5.65657.1
Conductor norm \( 3 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
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

\({y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-7a-4\right){y}={x}^{3}+\left(3a^{4}-4a^{3}-13a^{2}+12a+8\right){x}^{2}+\left(2a^{3}-2a^{2}-8a+8\right){x}+4a^{4}-5a^{3}-19a^{2}+16a+11\)
sage: E = EllipticCurve([K([-2,-1,1,0,0]),K([8,12,-13,-4,3]),K([-4,-7,5,2,-1]),K([8,-8,-2,2,0]),K([11,16,-19,-5,4])])
 
gp: E = ellinit([Polrev([-2,-1,1,0,0]),Polrev([8,12,-13,-4,3]),Polrev([-4,-7,5,2,-1]),Polrev([8,-8,-2,2,0]),Polrev([11,16,-19,-5,4])], K);
 
magma: E := EllipticCurve([K![-2,-1,1,0,0],K![8,12,-13,-4,3],K![-4,-7,5,2,-1],K![8,-8,-2,2,0],K![11,16,-19,-5,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+4a^2-2a-2)\) = \((-a^4+a^3+4a^2-2a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3 \) = \(3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4a^4+11a^3+14a^2-31a-11)\) = \((-a^4+a^3+4a^2-2a-2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -59049 \) = \(-3^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{747179612}{59049} a^{4} - \frac{4155930605}{19683} a^{3} - \frac{9128857582}{59049} a^{2} + \frac{3220249090}{6561} a + \frac{7149149617}{59049} \)
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: $\left(-a^{4} + 2 a^{3} + 3 a^{2} - 6 a + 1 : a^{4} - 8 a^{2} + 13 : 1\right)$
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: \( 813.68522079798560460176045794135369887 \)
Tamagawa product: \( 10 \)
Torsion order: \(5\)
Leading coefficient: \( 1.27021109 \)
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\((j)_{-}\)
\((-a^4+a^3+4a^2-2a-2)\) \(3\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

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 3.1-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.