The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse goes through, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.