Properties

 Base field $$\Q(\zeta_{7})^+$$ Label 3.3.49.1-64.1-a7 Conductor $$(0,4)$$ Conductor norm $$64$$ CM no base-change yes: 196.b2 Q-curve yes Torsion order $$12$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\zeta_{7})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 2 x + 1$$; class number $$1$$.

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

Weierstrass equation

$$y^2 = x^{3} - a x^{2} + \left(2 a^{2} - a - 4\right) x$$
magma: E := ChangeRing(EllipticCurve([0, -a, 0, 2*a^2 - a - 4, 0]),K);
sage: E = EllipticCurve(K, [0, -a, 0, 2*a^2 - a - 4, 0])
gp (2.8): E = ellinit([0, -a, 0, 2*a^2 - a - 4, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(0,4)$$ = $$\left(2\right)^{2}$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$64$$ = $$8^{2}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(16,16 a,16 a^{2} - 32)$$ = $$\left(2\right)^{4}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$4096$$ = $$8^{4}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$1792$$ 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: $$\Z/2\Z\times\Z/6\Z$$ 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] $\left(a^{2} - a - 1 : a - 2 : 1\right)$,$\left(a^{2} - 2 : 0 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(2\right)$$ $$8$$ $$3$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 196.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.