Number of generators of an elliptic curve Consider the elliptic curve E:$y^2 = x^3 + 3x + 11\,\, mod\,\, 19$. Two questions: 


Let the cardinality of the set of points on the elliptic curve( including
$O$ ) be $|E|  = 25$. How many points are generators and why?
Find an element of this group that has order $5$. The group operation is addition of points.




Answer of (1): First, when the author says points, I assume he is being sloppy and really means group. The number of generators for a cyclic group with $n$ elements in it,
are the number of integers from $[1,n-1]$ that are relatively prime to $n$.
Therefore, in this case, the answer is $19$. Is my answer right? 



Regarding question (2): I am thinking about guessing at the answer and doing a brute force approach (maybe doing the brute force approach in software) but it seems to me that there should be a shorter way to do it. Could you comment on my idea?

I think I just found a trick for solving the second question. Let $a$ be a generator of the group. Then I claim that $a^5$ is an element of order $5$. Is my claim correct?
No, he really means the number of points on $E$. So the number of solutions to the given equation plus 1 (for the point at infinity). He also gives the answer as $25.$

You need to apply your idea about number of generators to the correct group size.