# Point halving on elliptic curves over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $E(\mathbb{Q})[2]=\{o,T_1,T_2,T_3\}$. Let $P=2R$ be a point in $2E(\mathbb{Q})$, using $2$-division polynomial, we can compute $1/2P$, but it gives the set $\{R,R+T_1,R+T_2,R+T_3\}$. How can find $R$ in this set? In other word, is there an algorithm for point halving in $E(\mathbb{Q})$?Indeed I have $S_1$ and I know there exists $k\in\mathbb{N}$ such that $P=2kS_1$, I need to know $kS_1$.

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How are you supposed to identify $R$? –  Felipe Voloch Oct 8 at 13:13
If you have $P,S_1$ with $P=2kS_1$, then you can compute $k$ by computing the heights of $P$ and $S_1$. –  Felipe Voloch Oct 8 at 14:18
Yes, but I need another method! Without using height function!As I know, we can't compute the exact value of $\hat{h}(P)$, on the other hand, I don't know the complexity of computing the height functions. –  somayeh didari Oct 8 at 15:05
Sage and Magma can compute canonical heights pretty well. –  Daniel Loughran Oct 8 at 20:19
I use Pari/gp, and I know it computes canonical height. On the othere hand since $4k^2=\hat{h}(P)/\hat{h}(S_1)$, actually we don't need to know the exact value of the canonical heights! I'm looking for another algorithm. Without using height functions! –  somayeh didari Oct 9 at 5:26