joint with Mark Watkins, Stephen Donnelly, Noam D. Elkies, Andrew Granville and Nicholas F. Rogers.
We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-a-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.