Properties

Label 6.6.300125.1-71.1-a2
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

\({y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-28a-7\right){x}{y}+\left(-a^{5}+8a^{3}+5a^{2}-7a-2\right){y}={x}^{3}+\left(-7a^{5}+2a^{4}+50a^{3}+22a^{2}-30a-10\right){x}^{2}+\left(2a^{5}+7a^{4}-24a^{3}-51a^{2}+29a+8\right){x}-327a^{5}+677a^{4}+1538a^{3}-2267a^{2}+286a+277\)
sage: E = EllipticCurve([K([-7,-28,17,43,2,-6]),K([-10,-30,22,50,2,-7]),K([-2,-7,5,8,0,-1]),K([8,29,-51,-24,7,2]),K([277,286,-2267,1538,677,-327])])
 
gp: E = ellinit([Polrev([-7,-28,17,43,2,-6]),Polrev([-10,-30,22,50,2,-7]),Polrev([-2,-7,5,8,0,-1]),Polrev([8,29,-51,-24,7,2]),Polrev([277,286,-2267,1538,677,-327])], K);
 
magma: E := EllipticCurve([K![-7,-28,17,43,2,-6],K![-10,-30,22,50,2,-7],K![-2,-7,5,8,0,-1],K![8,29,-51,-24,7,2],K![277,286,-2267,1538,677,-327]]);
 

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: \((322a^5+209a^4-2714a^3-2861a^2+2996a+1612)\) = \((4a^5-a^4-29a^3-13a^2+18a+3)^{10}\)
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: \( \frac{182367117100063826098201858}{3255243551009881201} a^{5} - \frac{50495473559193738950995130}{3255243551009881201} a^{4} - \frac{1314604738469846691446620746}{3255243551009881201} a^{3} - \frac{582379993113800706094818270}{3255243551009881201} a^{2} + \frac{859358590365859284077183254}{3255243551009881201} a + \frac{253657973068162907887516563}{3255243551009881201} \)
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: $\left(\frac{5}{4} a^{5} - 3 a^{4} - 5 a^{3} + \frac{41}{4} a^{2} - \frac{7}{2} a + \frac{1}{2} : \frac{31}{8} a^{5} - \frac{19}{8} a^{4} - \frac{53}{2} a^{3} - \frac{39}{8} a^{2} + 16 a + \frac{23}{8} : 1\right)$
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: \( 1348.5051518367487080175587843280379234 \)
Tamagawa product: \( 2 \)
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\((j)_{-}\)
\((4a^5-a^4-29a^3-13a^2+18a+3)\) \(71\) \(2\) \(I_{10}\) 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 71.1-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.