# Properties

 Base field 3.3.49.1 Label 3.3.49.1-27.1-a2 Conductor $(0,3)$ Conductor norm $27$ CM no base-change yes: 147.c2 Q-curve yes Torsion order $13$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field 3.3.49.1

Generator $a$, with minimal polynomial $x^{3} - x^{2} - 2 x + 1$; class number $1$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)
gp (2.8): K = nfinit(a^3 - a^2 - 2*a + 1);

## Weierstrass equation

$y^2 + a y = x^{3} - a x^{2} + \left(2 a^{2} - a - 4\right) x - 2 a^{2} + a + 4$
magma: E := ChangeRing(EllipticCurve([0, -a, a, 2*a^2 - a - 4, -2*a^2 + a + 4]),K);
sage: E = EllipticCurve(K, [0, -a, a, 2*a^2 - a - 4, -2*a^2 + a + 4])
gp (2.8): E = ellinit([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

 $\mathfrak{N}$ = $(0,3)$ = $\left(3\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $27$ = $27$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(3,3 a,3 a^{2} - 6)$ = $\left(3\right)$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $27$ = $27$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $-\frac{28672}{3}$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $\text{End} (E)$ = $\Z$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $\text{ST} (E)$ = $\mathrm{SU}(2)$

## Mordell-Weil rank and generators

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $\Z/13\Z$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(1 : -1 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(3\right)$ 27 $1$ $I_{1}$ Split multiplicative 1 1 1

## Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p$ 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 degree13.

## Base change

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