Finding subgroup in elliptic curve over finite field $ \mathbb{F}_{11}$
In elliptic curve algebra, operations are computed over a field.
In your case, the field is $\mathbb{F}_{11}$.

$\mathbb{F}_{11}$ is a finite field containing 11 elements ($\{0,1,2,...,10\}$), and like every field there is and addition and multiplicative law.
Theses two laws work other the ring of integers modulo 11 (often noted $\mathbb{Z}/11\mathbb{Z}$).

In $\mathbb{F}_{11}$, when we write ${(\frac{39}{2})}^2 - 10 = \frac{1481}{4} = \frac{7}{4}$ it is in fact 7 times the inverse of 4 modulo 11, which is 3 because $3*4 = 1$ modulo 11.

Calculating an inverse modulo an integer can be done with the Extended Euclidian Algorithm. The below is the inverse table;

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
x \in \mathbb{F}_{11}&amp;1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10\\\hline
x^{-1} &amp;1&amp; 6 &amp; 4 &amp; 3 &amp; 9 &amp; 2 &amp; 8 &amp; 7 &amp; 5 &amp; 10\\\hline
\end{array}

In your example, the x-coordinate of $[2]P$ is then $7*3 = 10$, an element of $\mathbb{F}_{11}$.