As the definition referred from Silverman's book:
An elliptic curve is a pair $(E,O)$, where $E$ is a nonsingular curve of genus one and $O\in E$. (We generally denote the elliptic curve by $E$, the point $O$ being understand.) The elliptic curve $E$ is defined over $K$, written $E/K$, if $E$ is defined over $K$ as a curve and $O\in E(K)$.
We use Riemann-Roch theorem to prove $E$ has a Weierstrass form(This need us go into the deep part of algebraic geometry). There are some other nice properties, like the dual isogeny, elliptic have CM(there are two simple forms of $End(E)\neq \mathbb{Z}$)...
I wonder why it is so beautiful and what may be the most important part that influenced elliptic curve has this properties.
Thank you for sharing your mind.