  Let $p\equiv 1\bmod 4$ be a prime number. We use a number field variant of
Vinogradov's method to prove density results about the following four
arithmetic invariants: (i) $16$-rank of the class group $\mathrm{Cl}(-4p)$ of
the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of
the ordinary class group $\mathrm{Cl}(8p)$ of the real quadratic field
$\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation
$x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the
Tate-\v{S}afarevi\v{c} group of the congruent number elliptic curve $E_p: y^2 =
x^3-p^2x$. Our results are conditional on a standard conjecture about short
character sums.
