Properties

Label 6.6.703493.1-71.1-c2
Base field 6.6.703493.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 0 \)

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Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{5}-2a^{4}-11a^{3}+9a^{2}+10a-2\right){x}^{2}+\left(-a^{4}+3a^{2}+1\right){x}-7a^{5}+5a^{4}+39a^{3}-20a^{2}-41a+4\)
sage: E = EllipticCurve([K([1,1,0,0,0,0]),K([-2,10,9,-11,-2,2]),K([-2,0,1,0,0,0]),K([1,0,3,0,-1,0]),K([4,-41,-20,39,5,-7])])
 
gp: E = ellinit([Polrev([1,1,0,0,0,0]),Polrev([-2,10,9,-11,-2,2]),Polrev([-2,0,1,0,0,0]),Polrev([1,0,3,0,-1,0]),Polrev([4,-41,-20,39,5,-7])], K);
 
magma: E := EllipticCurve([K![1,1,0,0,0,0],K![-2,10,9,-11,-2,2],K![-2,0,1,0,0,0],K![1,0,3,0,-1,0],K![4,-41,-20,39,5,-7]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+a^4+12a^3-4a^2-12a-1)\) = \((-2a^5+a^4+12a^3-4a^2-12a-1)\)
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: \((3a^5-a^4-17a^3+5a^2+17a+8)\) = \((-2a^5+a^4+12a^3-4a^2-12a-1)^{3}\)
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: \( -\frac{29169505440}{357911} a^{5} + \frac{151117918400}{357911} a^{4} - \frac{25425543508}{357911} a^{3} - \frac{654677096029}{357911} a^{2} + \frac{727216594590}{357911} a - \frac{74279894665}{357911} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-3 a^{5} + 2 a^{4} + 17 a^{3} - 9 a^{2} - 17 a + 2 : 6 a^{5} - 3 a^{4} - 35 a^{3} + 15 a^{2} + 35 a - 3 : 1\right)$
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: \( 55543.943665711012889444769701669017801 \)
Tamagawa product: \( 1 \)
Torsion order: \(6\)
Leading coefficient: \( 1.83952 \)
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\((j)_{-}\)
\((-2a^5+a^4+12a^3-4a^2-12a-1)\) \(71\) \(1\) \(I_{3}\) 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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 71.1-c 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.