Let p ≡ 1 mod 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 Cl(−4p) of the imaginary quadratic number field Q(
√
−4p);
(ii) 8-rank of the ordinary class group Cl(8p) of the real quadratic field Q(
√
8p); (iii) the
solvability of the negative Pell equation x
2 − 2py2 = −1 over the integers; (iv) 2-part
of the Tate-Šafarevič group X(Ep) of the congruent number elliptic curve Ep : y
2 =
x
3 − p
2x. Our results are conditional on a standard conjecture about short character
sums
