8 - Motivating Motives

Let's count the number of solutions of

$y^2 + y = x^3 + x$ over the finite field with q = pⁿ elements, called

$\mathbb{F}_q$ . Take p = 2:

n q = pⁿ number of solutions
1 2 4

2 4 4

3 8 4

4 16 24

5 32 24

6 64 64

7 128 144

8 256 224

9 512 544

10 1024 1024