TITLE: Point addition on an elliptic curve
QUESTION [6 upvotes]: I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$. 
I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive that answer. 
I have $6\cdot(5, 1) = (16,13)$ and $6\cdot(0,6)=(0, 11)$ however when I use point addition to add them together I get $(16,13)+(0, 11)=(14,11)$ which isn't even a point on the curve... 
Could someone help me identify where and why I've gone wrong? 
For further information here's each of the points:

And here's the curve plotted out:

REPLY [2 votes]: You can aid (read: double-check) your calculations with the computer, for instance using Sage or Magma. For example, you can use for free the Magma online calculator. The following code defines your curve $E$ over $\mathbb{Z}/17\mathbb{Z}$, and calculates $6\cdot (0,6)$. 

g:=GF(17).1;
E:=EllipticCurve([0,0,0,2*g,2*g]);
P:=E![0,6];
6*P; 

The output is $[3:1:1]$ which are projective coordinates for the point $(3,1)$, as it should be.