# Properties

 Label 3.3.49.1-27.1-a1 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(3)$$ Conductor norm $$27$$ CM no Base change yes: 147.c1 Q-curve yes Torsion order $$1$$ 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

$${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(652a^{2}-391a-1564\right){x}+10528a^{2}-5979a-24046$$
sage: E = EllipticCurve([K([0,0,0]),K([0,-1,0]),K([0,1,0]),K([-1564,-391,652]),K([-24046,-5979,10528])])

gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([0,-1,0])),Pol(Vecrev([0,1,0])),Pol(Vecrev([-1564,-391,652])),Pol(Vecrev([-24046,-5979,10528]))], K);

magma: E := EllipticCurve([K![0,0,0],K![0,-1,0],K![0,1,0],K![-1564,-391,652],K![-24046,-5979,10528]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3)$$ = $$(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$27$$ = $$27$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-1594323)$$ = $$(3)^{13}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-4052555153018976267$$ = $$27^{13}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1713910976512}{1594323}$$ 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: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.216755369312335$$ Tamagawa product: $$13$$ Torsion order: $$1$$ 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$$(j)_{-}$$
$$(3)$$ $$27$$ $$13$$ $$I_{13}$$ Split multiplicative $$-1$$ $$1$$ $$13$$ $$13$$

## 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.1.2

## 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.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.