Properties

Label 3.3.49.1-64.1-a7
Base field \(\Q(\zeta_{7})^+\)
Conductor \((4)\)
Conductor norm \( 64 \)
CM no
Base change yes: 196.b2
Q-curve yes
Torsion order \( 12 \)
Rank \( 0 \)

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-a{x}^{2}+\left(2a^{2}-a-4\right){x}\)
sage: E = EllipticCurve([K([0,0,0]),K([0,-1,0]),K([0,0,0]),K([-4,-1,2]),K([0,0,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([0,-1,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([-4,-1,2])),Pol(Vecrev([0,0,0]))], K);
 
magma: E := EllipticCurve([K![0,0,0],K![0,-1,0],K![0,0,0],K![-4,-1,2],K![0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4)\) = \((2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(8^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((16)\) = \((2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4096 \) = \(8^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 1792 \)
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: \(\Z/2\Z\times\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(a^{2} - 2 : 0 : 1\right)$ $\left(a^{2} - a - 1 : a - 2 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 188.639082378515 \)
Tamagawa product: \( 3 \)
Torsion order: \(12\)
Leading coefficient: \( 0.561425840412247 \)
Analytic order of Ш: \( 1 \) (rounded)

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\((j)_{-}\)
\((2)\) \(8\) \(3\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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.