# Properties

 Label 3.3.49.1-27.1-a2 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(3)$$ Conductor norm $$27$$ CM no Base change yes: 147.c2 Q-curve yes Torsion order $$13$$ 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

$$y^2+ay=x^{3}-ax^{2}+\left(2a^{2}-a-4\right)x-2a^{2}+a+4$$
sage: E = EllipticCurve(K, [0, -a, a, 2*a^2 - a - 4, -2*a^2 + a + 4])

gp: E = ellinit([0, -a, a, 2*a^2 - a - 4, -2*a^2 + a + 4],K)

magma: E := ChangeRing(EllipticCurve([0, -a, a, 2*a^2 - a - 4, -2*a^2 + a + 4]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3)$$ = $$\left(3\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$27$$ = $$27$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(3)$$ = $$\left(3\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$27$$ = $$27$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{28672}{3}$$ 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: $$\Z/13\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -1 : 1\right)$ 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: $$476.211546379201$$ Tamagawa product: $$1$$ Torsion order: $$13$$ Leading coefficient: $$0.402545685865766$$ 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$$(j)_{-}$$
$$\left(3\right)$$ $$27$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$13$$ 13B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 13.
Its isogeny class 27.1-a consists of curves linked by isogenies of degree 13.

## Base change

This curve is the base change of elliptic curves 147.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.