# Properties

 Label 3.3.49.1-49.1-a1 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(2 a^{2} + a - 6)$$ 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: 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

$$y^2+\left(a^{2}-2\right)xy+\left(a+1\right)y=x^{3}+\left(a^{2}-a-3\right)x^{2}+\left(62a^{2}-26a-156\right)x-380a^{2}+192a+886$$
sage: E = EllipticCurve(K, [a^2 - 2, a^2 - a - 3, a + 1, 62*a^2 - 26*a - 156, -380*a^2 + 192*a + 886])

gp: E = ellinit([a^2 - 2, a^2 - a - 3, a + 1, 62*a^2 - 26*a - 156, -380*a^2 + 192*a + 886],K)

magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^2 - a - 3, a + 1, 62*a^2 - 26*a - 156, -380*a^2 + 192*a + 886]),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: $$16581375$$ 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$$(j)_{-}$$
$$\left(-a^{2} - a + 2\right)$$ $$7$$ $$2$$ $$III$$ 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 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.