# 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 $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## 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.