Properties

Label 6.6.300125.1-71.5-a1
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

\({y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-7\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-10\right){y}={x}^{3}+\left(a^{5}-8a^{3}-4a^{2}+7a+2\right){x}^{2}+\left(108a^{5}-25a^{4}-776a^{3}-378a^{2}+466a+140\right){x}+536a^{5}-123a^{4}-3847a^{3}-1892a^{2}+2295a+696\)
sage: E = EllipticCurve([K([-7,-29,17,43,2,-6]),K([2,7,-4,-8,0,1]),K([-10,-28,23,44,1,-6]),K([140,466,-378,-776,-25,108]),K([696,2295,-1892,-3847,-123,536])])
 
gp: E = ellinit([Pol(Vecrev([-7,-29,17,43,2,-6])),Pol(Vecrev([2,7,-4,-8,0,1])),Pol(Vecrev([-10,-28,23,44,1,-6])),Pol(Vecrev([140,466,-378,-776,-25,108])),Pol(Vecrev([696,2295,-1892,-3847,-123,536]))], K);
 
magma: E := EllipticCurve([K![-7,-29,17,43,2,-6],K![2,7,-4,-8,0,1],K![-10,-28,23,44,1,-6],K![140,466,-378,-776,-25,108],K![696,2295,-1892,-3847,-123,536]]);
 

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: \((18a^5+5a^4-138a^3-130a^2+112a+68)\) = \((-7a^5+2a^4+50a^3+22a^2-30a-10)^{5}\)
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: \( \frac{18671790400300000}{1804229351} a^{5} - \frac{5223136808131864}{1804229351} a^{4} - \frac{134463355085824784}{1804229351} a^{3} - \frac{59498874802185092}{1804229351} a^{2} + \frac{87859926083703700}{1804229351} a + \frac{25940020939985535}{1804229351} \)
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: $\left(-19 a^{5} + 4 a^{4} + 137 a^{3} + 69 a^{2} - 83 a - 27 : -9 a^{5} + 3 a^{4} + 64 a^{3} + 26 a^{2} - 41 a - 10 : 1\right)$
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: \( 1 \)
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\((j)_{-}\)
\((-7a^5+2a^4+50a^3+22a^2-30a-10)\) \(71\) \(1\) \(I_{5}\) 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 71.5-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.