Improved methods for finding imaginary quadratic fields with high 𝑛-rank

Abstract

We describe a generalization and improvement of Diaz y Diaz’s search technique for imaginary quadratic fields with 3 3 -rank at least 2, one of the most successful algorithms for generating many examples with relatively small discriminants, to find quadratic fields with large n n -ranks for odd n ≥ 3 n \geq 3 . An extensive search using our new algorithm in conjunction with a variety of further practical improvements produced billions of fields with non-trivial p p -rank for the primes p = 3 , 5 , 7 , 11 p = 3, 5, 7, 11 and 13 13 , and a large volume of fields with high p p -ranks and unusual class group structures. Our numerical results include a field with 5 5 -rank at least 4 with the smallest absolute discriminant discovered to date and the first known examples of imaginary quadratic fields with 7 7 -rank at least 4.