Properties

Label 5.5.160801.1-27.1-a1
Base field 5.5.160801.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 5.5.160801.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 4 x^{2} + 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(2a^{4}-a^{3}-10a^{2}+4a+4\right){y}={x}^{3}+\left(a^{4}-a^{3}-6a^{2}+5a+3\right){x}^{2}+\left(4a^{4}-3a^{3}-21a^{2}+13a+10\right){x}-2a^{3}+11a^{2}-16a+3\)
sage: E = EllipticCurve([K([-2,0,1,0,0]),K([3,5,-6,-1,1]),K([4,4,-10,-1,2]),K([10,13,-21,-3,4]),K([3,-16,11,-2,0])])
 
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([3,5,-6,-1,1]),Polrev([4,4,-10,-1,2]),Polrev([10,13,-21,-3,4]),Polrev([3,-16,11,-2,0])], K);
 
magma: E := EllipticCurve([K![-2,0,1,0,0],K![3,5,-6,-1,1],K![4,4,-10,-1,2],K![10,13,-21,-3,4],K![3,-16,11,-2,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-6a^2+5)\) = \((-a^4+5a^2+a-3)\cdot(a^4-5a^2+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-23a^4+2a^3+95a^2+2a-59)\) = \((-a^4+5a^2+a-3)^{4}\cdot(a^4-5a^2+3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 387420489 \) = \(3^{4}\cdot9^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{12282248}{2187} a^{4} - \frac{262708}{243} a^{3} + \frac{57820412}{2187} a^{2} + \frac{21681152}{2187} a - \frac{10423585}{2187} \)
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 $\left(-a^{4} + a^{3} + 5 a^{2} - 4 a - 1 : -a^{4} + 6 a^{2} - a - 4 : 1\right)$
Height \(0.021777834249630263428395857711005490764\)
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: $\left(a - 2 : -a^{4} + 6 a^{2} - a - 4 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.021777834249630263428395857711005490764 \)
Period: \( 2307.0901730222572018772779045758699945 \)
Tamagawa product: \( 28 \)  =  \(2^{2}\cdot7\)
Torsion order: \(2\)
Leading coefficient: \( 4.38533656 \)
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\((j)_{-}\)
\((-a^4+5a^2+a-3)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a^4-5a^2+3)\) \(9\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

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 27.1-a 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.