(Not an answer, just examples.)
Here are Rainer's ellipses for $d=5$ and $d=8$:.
The first passes through $6$ lattice points, the second through $10$.
(Not an answer, just examples.)
Here are Rainer's ellipses for $d=5$ and $d=8$:
(Not an answer, just examples.)
Here are Rainer's ellipses for $d=5$ and $d=8$.
The first passes through $6$ lattice points, the second through $10$.
(Not an answer, just examples.)
Here are Rainer's ellipses for $d=5$ and $d=8$: