Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds

For nonzero k let(Ek) be the "Mordell curve"y(2)=x(3)+k.Let D=72513834653847828539450325493=41p 1)where p is the prime 1768630113508483622913422573. Then the elliptic curve E(16D)hasrankr=16 over Q. Because E(k )is always 3-isogenous withE(-27k), it follows thatE(-432D )has rank 16 as well. This was the first pair of Mordell curves known to have rank atleast 16; we now prove that it has rank exactly 16. Having shownr >= 16 by exhibiting16 independent points, we must prover <= 16 by descent. This leads us to compute the3-torsion in the class group of Q(root-3D). The discriminant of this field has absolute value |Delta|=3D >2<middle dot>10(29), so large that it is not routine to compute the class groupwithout a GRH assumption. We compute it unconditionally using the Burgess boundson short character sums, which reduce the calculation from O(|Delta|(1/2)) (|Delta|(1/4), and Trevino and Booker's explicit bounds on the constants in the Burgess bounds, to make the factor |Delta|(is an element of) explicit as well. Along the way we compute unconditionally the class group of Q (root-3D), whose 3-rank of 8 is the current record for the class group of a quadratic number field.