  In 2016 Astudillo, Diaz y Diaz and Friedman published sharp lower bounds for
regulators of number fields of all signatures up to degree seven, except for
fields of degree seven having five real places. We deal with this signature,
proving that the field with the first discriminant has minimal regulator. The
new element in the proof is an extension of Pohst's geometric method from the
totally real case to fields having one complex place.
