Number of elliptic curves over finite fields

In v.z. Gathen and Gerhard: "Modern Computer Algebra", the number of elliptic curves of orders 3 to 13 over the finite field F7 is given as
1, 4, 3, 6, 4, 6, 4, 6, 3, 4, 1

What is known about the sequence of orders for finite fields Fp in general? Does it always have such remarkable symmetry?


Thanks,
Rolf Bardeli
.

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