There are many in the set d=4m with class number 8. When m is a prime or a semiprime (a product of two primes) like m=3p, then it may be well-behaved. For this set, there are only three, namely p=23,47,71, thus m=3p=69,141,213. Ramanujan found the radicals below and the G-function have a common form
G3p=U1/24pU1/163px1/2p
with fundamental unit Un and where x2p is a root of a unit quartic
G69=(5+√23√2)1/12(3√3+√232)1/8(√2+3√34+√6+3√34)1/2G141=(7+√47√2)1/12(4√3+√47)1/8(√14+9√34+√18+9√34)1/2G213=(59+7√71√2)1/12(5√3+√712)1/8(√19+12√32+√21+12√32)1/2
P.S. Checking h(−3d)=16, one finds the only primes are d=167,191,239,383,311 which are the smallest d with class number h(−d)=11,13,15,17,19, respectively. Makes me wonder if their G3d would be analogous.
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