show/hide this revision's text 2 fixed some typos

As Xandi Tuni said, most of the answers to your questions can be found in standard references.

  1. Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google ) for elliptic curves).

  2. Yes

  3. The number of points is bounded by the Hasse-bound. Within that bound, the exact number depends a lot on the elliptic curve itself. The $l^n$-torsion for prime $l\neq \text{char }K$ is isomorphic to $\mathbb{Z}/l^n\mathbb{Z} \times \mathbb{Z}/l^n\mathbb{Z}$ over the algebraic closure of $K$. For $p= \text{char }K$, the $p$-primary torsion over the algebraic closure is either cyclic (then $E$ is called ordinary) or 0 (then $E$ is called supersingular). So this again depends on the curve.

  4. I don't know what you mean by that question. You can always compute all the $K$-rational points by hand, since there are only finitely many values you have to try.
show/hide this revision's text 1

As Xandi Tuni said, most of the answers to your questions can be found in standard references.

  1. Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google) for elliptic curves).

  2. Yes

  3. The number of points is bounded by the Hasse-bound. Within that bound, the exact number depends a lot on the elliptic curve itself. The $l^n$-torsion for prime $l\neq \text{char }K$ is isomorphic to $\mathbb{Z}/l^n\mathbb{Z} \times \mathbb{Z}/l^n\mathbb{Z}$ over the algebraic closure of $K$. For $p= \text{char }K$, the $p$-primary torsion over the algebraic closure is either cyclic (then $E$ is called ordinary) or 0 (then $E$ is called supersingular). So this again depends on the curve.

  4. I don't know what you mean by that question. You can always compute all the $K$-rational points by hand, since there are only finitely values you have to try.