Properties

Label 6.6.300125.1-71.1-a1
Base field 6.6.300125.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 6.6.300125.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-12\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-5\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a^{5}+a^{4}-130a^{3}-87a^{2}+99a+31\right){x}+13a^{5}+10a^{4}-112a^{3}-119a^{2}+104a+35\)
sage: E = EllipticCurve([K([-12,-38,27,58,2,-8]),K([1,1,0,0,0,0]),K([-5,-17,14,23,0,-3]),K([31,99,-87,-130,1,17]),K([35,104,-119,-112,10,13])])
 
gp: E = ellinit([Polrev([-12,-38,27,58,2,-8]),Polrev([1,1,0,0,0,0]),Polrev([-5,-17,14,23,0,-3]),Polrev([31,99,-87,-130,1,17]),Polrev([35,104,-119,-112,10,13])], K);
 
magma: E := EllipticCurve([K![-12,-38,27,58,2,-8],K![1,1,0,0,0,0],K![-5,-17,14,23,0,-3],K![31,99,-87,-130,1,17],K![35,104,-119,-112,10,13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4a^5-a^4-29a^3-13a^2+18a+3)\) = \((4a^5-a^4-29a^3-13a^2+18a+3)\)
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: \((171a^5-38a^4-1230a^3-617a^2+734a+258)\) = \((4a^5-a^4-29a^3-13a^2+18a+3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1804229351 \) = \(-71^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2098710915405748}{1804229351} a^{5} + \frac{2683537051449484}{1804229351} a^{4} + \frac{13974658584141464}{1804229351} a^{3} - \frac{8128924082191888}{1804229351} a^{2} - \frac{12613327929279580}{1804229351} a + \frac{7776135079307907}{1804229351} \)
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: $\left(-5 a^{5} + 2 a^{4} + 35 a^{3} + 13 a^{2} - 22 a - 7 : -13 a^{5} + 4 a^{4} + 93 a^{3} + 40 a^{2} - 60 a - 18 : 1\right)$
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: \( 2697.0103036734974160351175686560758469 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.23075 \)
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\((j)_{-}\)
\((4a^5-a^4-29a^3-13a^2+18a+3)\) \(71\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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 71.1-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.