  We make effective $l^2 L^p$ decoupling for the parabola in the range $4 < p <
6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove
the conjectural bound for the sixth-order correlation of the integer solutions
of the equation $x^2 + y^2 = m$ in an extremal case. This proves
unconditionally a result that was proven by Bombieri and Bourgain under the
hypotheses of the Birch and Swinnerton-Dyer conjecture and the Riemann
Hypothesis for $L$-functions of elliptic curves over $\mathbb{Q}$.
