Properties

Label 6.6.434581.1-71.2-a2
Base field 6.6.434581.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-2a^{4}-4a^{3}+5a^{2}+4a-2\right){x}{y}+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+5a-3\right){y}={x}^{3}+\left(a^{5}-4a^{4}+10a^{2}-2a-4\right){x}^{2}+\left(11a^{5}-15a^{4}-45a^{3}+17a^{2}+18a-6\right){x}-14a^{5}+9a^{4}+66a^{3}+31a^{2}-27a-17\)
sage: E = EllipticCurve([K([-2,4,5,-4,-2,1]),K([-4,-2,10,0,-4,1]),K([-3,5,12,-10,-6,3]),K([-6,18,17,-45,-15,11]),K([-17,-27,31,66,9,-14])])
 
gp: E = ellinit([Polrev([-2,4,5,-4,-2,1]),Polrev([-4,-2,10,0,-4,1]),Polrev([-3,5,12,-10,-6,3]),Polrev([-6,18,17,-45,-15,11]),Polrev([-17,-27,31,66,9,-14])], K);
 
magma: E := EllipticCurve([K![-2,4,5,-4,-2,1],K![-4,-2,10,0,-4,1],K![-3,5,12,-10,-6,3],K![-6,18,17,-45,-15,11],K![-17,-27,31,66,9,-14]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-3a)\) = \((a^3-3a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71 \) = \(71\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^5-6a^4+17a^3+28a^2-29a-21)\) = \((a^3-3a)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -357911 \) = \(-71^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1334490057051325}{357911} a^{5} - \frac{3683030066630924}{357911} a^{4} - \frac{2539374775289438}{357911} a^{3} + \frac{8601694812985271}{357911} a^{2} - \frac{1197153818502630}{357911} a - \frac{1757930401343265}{357911} \)
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: $\left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 3 : -3 a^{5} + 7 a^{4} + 9 a^{3} - 17 a^{2} - 4 a + 6 : 1\right)$
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: \( 3952.2701846318698643749894959727111098 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.49883 \)
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\((j)_{-}\)
\((a^3-3a)\) \(71\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 71.2-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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