Properties

Label 3.3.49.1-49.1-a1
Base field \(\Q(\zeta_{7})^+\)
Conductor \((2a^2+a-6)\)
Conductor norm \( 49 \)
CM yes (\(-28\))
Base change yes: 49.a1
Q-curve yes
Torsion order \( 14 \)
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

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(62a^{2}-26a-156\right){x}-380a^{2}+192a+886\)
sage: E = EllipticCurve([K([-2,0,1]),K([-3,-1,1]),K([1,1,0]),K([-156,-26,62]),K([886,192,-380])])
 
gp: E = ellinit([Pol(Vecrev([-2,0,1])),Pol(Vecrev([-3,-1,1])),Pol(Vecrev([1,1,0])),Pol(Vecrev([-156,-26,62])),Pol(Vecrev([886,192,-380]))], K);
 
magma: E := EllipticCurve([K![-2,0,1],K![-3,-1,1],K![1,1,0],K![-156,-26,62],K![886,192,-380]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^2+a-6)\) = \((-a^2-a+2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((7)\) = \((-a^2-a+2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 343 \) = \(7^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 16581375 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-7}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/14\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(4 a^{2} - 3 a - 7 : 20 a^{2} - 12 a - 45 : 1\right)$
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: \( 354.080264860149 \)
Tamagawa product: \( 2 \)
Torsion order: \(14\)
Leading coefficient: \( 0.516151989592054 \)
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\((j)_{-}\)
\((-a^2-a+2)\) \(7\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.1[3]

For all other primes \(p\), the image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 49.1-a consists of curves linked by isogenies of degrees dividing 14.

Base change

This curve is the base change of elliptic curves 49.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.