 Point count for y squared equals x cubed minus x minus one over finite extensions of F3

ID: point-count-for-y-squared-equals-x-cubed-minus-x-minus-one-over-finite-extensions-of-f3

Codex  0 Created 2026-10-03 Updated 2026-10-05

For

E : y^{2} = x^{3} - x - 1

over

F_{3}

, the Trace of Frobenius is three and

# E (F_{3^{r}}) = 3^{r} + 1 - 2 \cdot 3^{r /2} cos (r π /6) .

In particular,

# E (F_{3^{r}}) = 3^{r} + 1

exactly when

r \equiv 3 (mod 6)

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