Properties

Label 3.3.49.1-49.1-a2
Base field \(\Q(\zeta_{7})^+\)
Conductor \((2 a^{2} + a - 6)\)
Conductor norm \( 49 \)
CM yes (\(-7\))
Base change yes: 49.a2
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: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)
 
gp: K = nfinit(a^3 - a^2 - 2*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a^{2}-2\right)xy+\left(a+1\right)y=x^{3}+\left(a^{2}-a-3\right)x^{2}+\left(2a^{2}-a-6\right)x-9a^{2}+4a+20\)
sage: E = EllipticCurve(K, [a^2 - 2, a^2 - a - 3, a + 1, 2*a^2 - a - 6, -9*a^2 + 4*a + 20])
 
gp: E = ellinit([a^2 - 2, a^2 - a - 3, a + 1, 2*a^2 - a - 6, -9*a^2 + 4*a + 20],K)
 
magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^2 - a - 3, a + 1, 2*a^2 - a - 6, -9*a^2 + 4*a + 20]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2 a^{2} + a - 6)\) = \( \left(-a^{2} - a + 2\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \( 7^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((7)\) = \( \left(-a^{2} - a + 2\right)^{3} \)
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: \( -3375 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-7})/2]\) (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: $\left(-7 a^{2} + 4 a + 16 : -33 a^{2} + 18 a + 73 : 1\right)$
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\((j)_{-}\)
\( \left(-a^{2} - a + 2\right) \) \(7\) \(2\) \(III\) 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 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.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.