Properties

Base field 6.6.434581.1
Label 6.6.434581.1-1.1-a1
Conductor \((1)\)
Conductor norm \( 1 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 6.6.434581.1

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

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

Weierstrass equation

\( y^2 + \left(3 a^{5} - 7 a^{4} - 8 a^{3} + 15 a^{2} + a - 3\right) x y + \left(2 a^{5} - 5 a^{4} - 5 a^{3} + 12 a^{2} - 3\right) y = x^{3} + \left(a^{4} - 2 a^{3} - 3 a^{2} + 2 a + 2\right) x^{2} + \left(-2015 a^{5} + 2317 a^{4} + 10491 a^{3} - 2078 a^{2} - 11628 a - 4340\right) x - 87669 a^{5} + 112368 a^{4} + 435832 a^{3} - 134018 a^{2} - 464939 a - 144427 \)
magma: E := ChangeRing(EllipticCurve([3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 3, a^4 - 2*a^3 - 3*a^2 + 2*a + 2, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -2015*a^5 + 2317*a^4 + 10491*a^3 - 2078*a^2 - 11628*a - 4340, -87669*a^5 + 112368*a^4 + 435832*a^3 - 134018*a^2 - 464939*a - 144427]),K);
sage: E = EllipticCurve(K, [3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 3, a^4 - 2*a^3 - 3*a^2 + 2*a + 2, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -2015*a^5 + 2317*a^4 + 10491*a^3 - 2078*a^2 - 11628*a - 4340, -87669*a^5 + 112368*a^4 + 435832*a^3 - 134018*a^2 - 464939*a - 144427])
gp (2.8): E = ellinit([3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 3, a^4 - 2*a^3 - 3*a^2 + 2*a + 2, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -2015*a^5 + 2317*a^4 + 10491*a^3 - 2078*a^2 - 11628*a - 4340, -87669*a^5 + 112368*a^4 + 435832*a^3 - 134018*a^2 - 464939*a - 144427],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)
\(j\) = \( 14273264587780952609125805373809022 a^{5} - 33978664183724880412077066535252082 a^{4} - 44161420314998729136586656312921749 a^{3} + 88173325697661842578861191186611639 a^{2} + 23535955393860563533538162020788064 a - 37503869415369575938251635322944654 \)
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
\(3\) 3B
\(13\) 13B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 13 and 39.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 39.

Base change

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