Properties

Label 6.6.300125.1-71.5-a2
Base field 6.6.300125.1
Conductor \((-7a^5+2a^4+50a^3+22a^2-30a-10)\)
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(Pol(Vecrev([-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

\({y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-12\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-6\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-30a-6\right){x}^{2}+\left(10a^{5}-2a^{4}-72a^{3}-41a^{2}+36a+12\right){x}+15a^{5}-27a^{4}-174a^{3}-76a^{2}+101a+30\)
sage: E = EllipticCurve([K([-12,-38,27,58,2,-8]),K([-6,-30,17,43,2,-6]),K([-6,-17,14,23,0,-3]),K([12,36,-41,-72,-2,10]),K([30,101,-76,-174,-27,15])])
 
gp: E = ellinit([Pol(Vecrev([-12,-38,27,58,2,-8])),Pol(Vecrev([-6,-30,17,43,2,-6])),Pol(Vecrev([-6,-17,14,23,0,-3])),Pol(Vecrev([12,36,-41,-72,-2,10])),Pol(Vecrev([30,101,-76,-174,-27,15]))], K);
 
magma: E := EllipticCurve([K![-12,-38,27,58,2,-8],K![-6,-30,17,43,2,-6],K![-6,-17,14,23,0,-3],K![12,36,-41,-72,-2,10],K![30,101,-76,-174,-27,15]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-7a^5+2a^4+50a^3+22a^2-30a-10)\) = \((-7a^5+2a^4+50a^3+22a^2-30a-10)\)
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: \((-2277a^5+869a^4+16168a^3+5386a^2-9540a-969)\) = \((-7a^5+2a^4+50a^3+22a^2-30a-10)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3255243551009881201 \) = \(-71^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{41014241389245672911233616}{3255243551009881201} a^{5} + \frac{31251665362953723702641564}{3255243551009881201} a^{4} + \frac{283323882933070565416889280}{3255243551009881201} a^{3} - \frac{8921536262849968349901544}{3255243551009881201} a^{2} - \frac{210325636400488563887402742}{3255243551009881201} a + \frac{49837787027906807246579723}{3255243551009881201} \)
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: $\left(6 a^{5} - \frac{5}{4} a^{4} - 43 a^{3} - 21 a^{2} + 27 a + 8 : \frac{59}{8} a^{5} - \frac{15}{8} a^{4} - \frac{217}{4} a^{3} - \frac{205}{8} a^{2} + \frac{145}{4} a + \frac{89}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: not available
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\((-7a^5+2a^4+50a^3+22a^2-30a-10)\) \(71\) \(2\) \(I_{10}\) Non-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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 71.5-a consists of curves linked by isogenies of degree 2.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.