# Properties

 Base field 3.3.49.1 Label 3.3.49.1-64.1-a3 Conductor $(0,4)$ Conductor norm $64$ CM no base-change yes: 196.b1 Q-curve yes Torsion order $4$ 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

$y^2 = x^{3} - a x^{2} + \left(102 a^{2} - 61 a - 244\right) x + 640 a^{2} - 360 a - 1460$
magma: E := ChangeRing(EllipticCurve([0, -a, 0, 102*a^2 - 61*a - 244, 640*a^2 - 360*a - 1460]),K);
sage: E = EllipticCurve(K, [0, -a, 0, 102*a^2 - 61*a - 244, 640*a^2 - 360*a - 1460])
gp (2.8): E = ellinit([0, -a, 0, 102*a^2 - 61*a - 244, 640*a^2 - 360*a - 1460],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(0,4)$ = $\left(2\right)^{2}$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $64$ = $8^{2}$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(16,16 a,16 a^{2} - 32)$ = $\left(2\right)^{4}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $4096$ = $8^{4}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $406749952$ 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: $\Z/2\Z\times\Z/2\Z$ 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] $\left(5 a^{2} - 10 : 0 : 1\right)$,$\left(4 a^{2} - 4 a - 8 : 0 : 1\right)$ 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$(j)_{-}$
$\left(2\right)$ 8 $1$ $IV$ Additive 2 4 0

## Galois Representations

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

prime Image of Galois Representation
$2$ 2Cs
$3$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 196.b1, defined over $\Q$, so it is also a $\Q$-curve.