# Symbolic computation of multiples of a point in elliptic curve.

Suppose we have an elliptic curve over the rational field $E(\mathbb{Q}).$ Let $P = (s,t)\in E(\mathbb{Q})$ be a point with $s,t$ being symbolic variables. Then we see that $n*P = (f_{n1}/g_{n1}, f_{n2}/g_{n2})$, where $f_{ni}, g_{ni}$ are polynomials in the variables $s,t.$ In SageMath after defining the curve and the symbolic variables $s$ and $t$ if I write $P = E(s,t)$, it is giving error. I am looking for method by which I can find all these polynomials in the variabls $s,t$ using SageMath.

Could you please provide your current code so that we could understand the issue ?

Note that if $s,t$ are symbolic variables, then $(s,t)$ is not a pair of rational numbers, so it does not lie in $E(\mathbb{Q})$. Furthermore, if $s,t$ are just symbolic variables then $t^2\neq s^3+c_4s+c_6$ identically, since they are just independent variables. You'd want to work in a ring or field where you actually have $s,t$ that satisfy the relation required. The function field of $E$ would do.