$E_a$ is an ellipse with an equation of the form $x^2+4y^2=4a^2$.
$E_a^{\prime }$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around the origin $O(0,0)$ for $0^{\circ }<\theta <{90}^{\circ }$.

$b$ is the distance to the origin of the two intersection points closest to the origin and $c$ is the distance of the two other intersection points.
We call an ordered triplet $(a,b,c)$ a canonical ellipsoidal triplet if $a,b$ and $c$ are positive integers.
For example, $(209,247,286)$ is a canonical ellipsoidal triplet.

Let $C(N)$ be the number of distinct canonical ellipsoidal triplets $(a,b,c)$ for $a\le N$.
It can be verified that $C({10}^3)=7$, $C({10}^4)=106$ and $C({10}^6)=11845$.

Find $C({10}^{17})$.