The congruent number curve of $n$ is $E_n : y^2 = x^3 - n^2 x$. For a prime $p$, the trace of Frobenius $a_p(E_n) = p + 1 - \#E_n(\mathbb{F}_p)$ is the character sum $-\sum_{x \in \mathbb{F}_p} \chi(x^3 - n^2 x)$, where $\chi$ is the quadratic character of $\mathbb{F}_p$ with $\chi(0) = 0$: each $x$ contributes $1 + \chi(f(x))$ points. This concept defines $a_p$ by that sum.

The statement is the quadratic twist formula: for every prime $p$ and every integer $n$ not divisible by $p$,

so every $E_n$ is the quadratic twist of $E_1 : y^2 = x^3 - x$ by $n$. The identity is exact at the level of character sums; it is the substitution $x = n t$, a bijection of $\mathbb{F}_p$ because $n$ is invertible, together with $\chi(n^3) = \chi(n)$.