8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F=Q((-p1p2)) be an imaginary quadratic field with distinct primes p1≡p2≡1 mod8 and the Legendre symbol ((p1)/(p2))=1.Then the 8-rank of the class group of F is equal to 2 if and onlyif the following conditions hold:(1) The quartic residue symbols ((p1)/(p2))=((p2)/(p1))=1;(2) Either bothp1 and p2 are represented by the form a~2+32b~2 over Z and p~(h+(2p1)/4)=x~2-2p1y~2 x,y ∈ Z,or bothp1 and p2 are not represented by the form a~2+32b~2 over Z and p~(h+(2p1)/4)=ε(2x~2-p1y~2),x,y ∈ Z,ε∈{4-1},where h+(2p1) is the narrow class number of Q((2p1)).Moreover,we also generalize theseresults.