Let $E$ be a real elliptic curve defined by $y^2+a_1xy+a_3y=f(x)$ where $f$ is a polynomial of degree 3 s.t. $E$ is non-singular with the point at infinite added.
(7.1) Assertion: An abelian, connected, compact Lie group is isomorphic to a product of $S^1$. The number of factors is equal to dimension of locally euclidean space.
There are 2 cases. $f(x)$ contains 1 real root and $f(x)$ contains $3$ real roots.
I do not understand the argument of $3$ real roots.
If $f$ has 3 real roots, then $E$ has 2 components. Then $E\cong S^1\times Z_2$.
$\textbf{Q:}$ How do I see $Z_2$ part of $E$ here? Where does $Z_2$ part entering the picture? I knew there is 2 torsion part for $E(Q)$. This had better be maintained by $E(Q)\subset E(R)$.
Ref: Elliptic Curves by Dale Husemoller Chapter on Introduction to Rational Points on Plane Curves, Sec 7.