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Friday, June 13, 2025

Entry 163

There are only seven fundamental d=4p with class number 6, namely p=29,53,61,109,157,277,397.  To prevent clutter, only the first three will be stated. Given the Kronecker symbol (dm), we propose

K(k29)=2π21161(x1)1/3(y21+y41+42)3/4(116m=1[Γ(m116)](116m))1/12K(k53)=2π22121(x2)1/3(y22+y42+42)3/4(212m=1[Γ(m212)](212m))1/12K(k61)=2π22441(x3)1/3(y23+y43+366)3/4(244m=1[Γ(m244)](244m))1/12

where the xn are the real roots of the following cubics x35x23x1=0x315x2x1=0x327x25x1=0
while the yn are also the real roots of cubics y3y24y4=0y37y2+13y11=0y36y227y54=0
and similarly for the other p.

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