For nonzero k let \(E_k\) be the “Mordell curve” \(y^2 = x^3 + k\) . Let \(\begin{aligned} D = 72513834653847828539450325493 = 41p \quad \qquad \qquad (1) \end{aligned}\) where p is the prime 1768630113508483622913422573. Then the elliptic curve \(E_{16D}\) has rank \(r=16\) over \({\textbf {Q}}\) . Because \(E_k\) is always 3-isogenous with \(E_{-27k}\) , it follows that \(E_{-432D}\) has rank 16 as well. This was the first pair of Mordell curves known to have rank at least 16; we now prove that it has rank exactly 16. Having shown \(r \ge 16\) by exhibiting 16 independent points, we must prove \(r \le 16\) by descent. This leads us to compute the 3-torsion in the class group of \({\textbf {Q}}(\sqrt{-3D}\,)\) . The discriminant of this field has absolute value \(|\Delta | = 3D > 2 \cdot 10^{29}\) , so large that it is not routine to compute the class group without a GRH assumption. We compute it unconditionally using the Burgess bounds on short character sums, which reduce the calculation from \({\,\widetilde{\!O\!}\,}(|\Delta |^{1/2})\) to \(|\Delta |^{1/4+\epsilon }\) , and Treviño and Booker’s explicit bounds on the constants in the Burgess bounds, to make the factor \(|\Delta |^\epsilon \) explicit as well. Along the way we compute unconditionally the class group of \({\textbf {Q}}(\sqrt{-3D})\) , whose 3-rank of 8 is the current record for the class group of a quadratic number field.
