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Properties

Base field 6.6.300125.1
Label 6.6.300125.1-1.1-a1
Conductor \((1)\)
Conductor norm \( 1 \)
CM no
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

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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 \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
 
gp (2.8): K = nfinit(a^6 - a^5 - 7*a^4 + 2*a^3 + 7*a^2 - 2*a - 1);
 

Weierstrass equation

\( y^2 + \left(-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 9 a - 2\right) x y + \left(-5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 26 a - 7\right) y = x^{3} + \left(-3 a^{5} + 2 a^{4} + 20 a^{3} + 3 a^{2} - 12 a\right) x^{2} + \left(321037 a^{5} - 77286 a^{4} - 2300766 a^{3} - 1111743 a^{2} + 1379267 a + 398326\right) x + 54665451 a^{5} - 12782935 a^{4} - 392075036 a^{3} - 191607310 a^{2} + 234159226 a + 69935775 \)
magma: E := ChangeRing(EllipticCurve([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 321037*a^5 - 77286*a^4 - 2300766*a^3 - 1111743*a^2 + 1379267*a + 398326, 54665451*a^5 - 12782935*a^4 - 392075036*a^3 - 191607310*a^2 + 234159226*a + 69935775]),K);
 
sage: E = EllipticCurve(K, [-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 321037*a^5 - 77286*a^4 - 2300766*a^3 - 1111743*a^2 + 1379267*a + 398326, 54665451*a^5 - 12782935*a^4 - 392075036*a^3 - 191607310*a^2 + 234159226*a + 69935775])
 
gp (2.8): E = ellinit([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 321037*a^5 - 77286*a^4 - 2300766*a^3 - 1111743*a^2 + 1379267*a + 398326, 54665451*a^5 - 12782935*a^4 - 392075036*a^3 - 191607310*a^2 + 234159226*a + 69935775],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((1)\) = \((1)\)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1 \) = 1
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1)\) = \((1)\)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1 \) = 1
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -162677523113838677 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: Trivial
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
No primes of bad reduction.

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(37\) 37B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 37.
Its isogeny class 1.1-a consists of curves linked by isogenies of degree 37.

Base change

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