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First case: Complex number. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Robin Chapman denotes below, i.e., it is a complex torus in the form $\mathbb C/\Lambda$ where $\Lambda$ is a lattice in $\mathbb C$.

Let $K$ be an algebraically closed field of char $p$. If $n$ is prime to $p$, the the $n$-torsion is $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$. The $p^e$-torsion could be either trivial for all $e$, or $\mathbb{Z}/p^e\mathbb{Z}$ for all $e$.

Over a non-algebraically closed-field, this is going to be much more complicated. I will try to give just an introduction.

Over $\mathbb Q$ and number fields: Over $\mathbb Q$ or a number field, it is finitely generated by the Mordell-Weil theorem. So it has a torsion part and free part. The free part could be arbitrarily large.

The Mordell-Weil theorem is in fact true for arbitrary finitely generated fields. This is due to Néron.

Over $\mathbb Q$, The torsion part has exactly $15$ possibilities according to the theorem of Mazur. Over number fields, this had been generalized that the torsion part is uniformly bounded.

Over finite fields, the torsion group would be $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$ where $n |m$. There is no free part.

And it could go on like this. Please have a look at Silverman's "Advanced topics in the Arithmetic of Elliptic Curves" for elliptic curves over real numbers, $p$-adic numbers, function fields, etc..

First case: Complex number. Over $\mathbb C$ the structure as an abstract group is $\mathbb C S^1 \oplus \mathbb C$S^1$where$S^1$is the circle, i.e.,$\mathbb R/\mathbb Z$. Let$K$be an algebraically closed field of char$p$. If$n$is prime to$p$, the the$n$-torsion is$\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$. The$p$-torsion p^e$-torsion could be either trivial for all $e$, or $\mathbb{Z}/p^e\mathbb{Z}$ for all $0 \leq e \leq 2$. e$. Over a non-algebraically closed-field, this is going to be much more complicated. I will try to give just an introduction. Over$\mathbb Q$and number fields: Over$\mathbb Q$or a number field, it is finitely generated by the Mordell-Weil theorem. So it has a torsion part and free part. The free part could be arbitrarily large. The Mordell-Weil theorem is in fact true for arbitrary finitely generated fields. This is due to Néron. Over$\mathbb Q$, The torsion part has exactly$15$possibilities according to the theorem of Mazur. Over number fields, this had been generalized that the torsion part is uniformly bounded. Over finite fields, the torsion group would be$\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$where$n |m$. There is no free part. And it could go on like this. Please have a look at Silverman's "Advanced topics in the Arithmetic of Elliptic Curves" for elliptic curves over real numbers,$p$-adic numbers, function fields, etc.. 1 First case: Complex number. Over$\mathbb C$the structure as an abstract group is$\mathbb C \oplus \mathbb C$. Let$K$be an algebraically closed field of char$p$. If$n$is prime to$p$, the the$n$-torsion is$\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$. The$p$-torsion could be$\mathbb{Z}/p^e\mathbb{Z}$for$0 \leq e \leq 2$. Over a non-algebraically closed-field, this is going to be much more complicated. I will try to give just an introduction. Over$\mathbb Q$and number fields: Over$\mathbb Q$or a number field, it is finitely generated by the Mordell-Weil theorem. So it has a torsion part and free part. The free part could be arbitrarily large. The Mordell-Weil theorem is in fact true for arbitrary finitely generated fields. This is due to Néron. Over$\mathbb Q$, The torsion part has exactly$15$possibilities according to the theorem of Mazur. Over number fields, this had been generalized that the torsion part is uniformly bounded. Over finite fields, the torsion group would be$\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$where$n |m$. There is no free part. And it could go on like this. Please have a look at Silverman's "Advanced topics in the Arithmetic of Elliptic Curves" for elliptic curves over real numbers,$p\$-adic numbers, function fields, etc..