Suppose $E$ is an elliptic curve over $\mathbb{C}$ given by the equation $$ y^2=x^3+ax+b,\quad a,b\in\mathbb{R}. $$
I know that depending on whether $x^3+ax+b$ has one or three real roots, then $E(\mathbb{R})$ is either $\mathbb{R}/\mathbb{Z}$ or $\mathbb{R}/\mathbb{Z}\times C_2$, respectively. I know a proof from the latter parts of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.
However, this problem of classifying the subgroup of real points also is Exercise 1 of Section VI.1 of Koblitz's text Number Theory and Cryptography, which assumes very little mathematical prerequisites, and briefly states that $E\simeq\mathbb{C}/L$, where $L$ is some lattice, without really giving any details of the Weierstrass function, and then sketches that this is isomorphic to a torus.
My question is, is there a "very elementary" way to classify the group of real points $E(\mathbb{R})$, in this case where the coefficients of the elliptic curve are real, using only tools from a one semester undergrad algebra course, as this is roughly the mathematical requirements of said book?