Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-64.1-a8
Conductor \((0,4)\)
Conductor norm \( 64 \)
CM no
base-change no
Q-curve yes
Torsion order \( 6 \)
Rank not available

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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

\( y^2 = x^{3} - a x^{2} + \left(-8 a^{2} + 4 a + 16\right) x + 4 a^{2} - 8 \)
magma: E := ChangeRing(EllipticCurve([0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8]),K);
 
sage: E = EllipticCurve(K, [0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8])
 
gp (2.8): E = ellinit([0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8],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()
 
\(\mathfrak{D}\) = \((256,256 a,256 a^{2} - 512)\) = \( \left(2\right)^{8} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 16777216 \) = \( 8^{8} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( 143392 a^{2} - 208912 a + 66240 \)
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: \(\Z/6\Z\)
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]
 
Generator: $\left(-4 a^{2} + 4 a + 10 : -18 a^{2} + 8 a + 38 : 1\right)$
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 \(-1\) \(2\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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