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Friday, June 13, 2025

Entry 164

To summarize, given fundamental discriminant d, class number h(d)=n, complete elliptic integral of the first kind K(kp), and Kronecker symbol (dm)

Conjecture 1. Let even d=4p for prime p1mod4 with even class number h(d)=n and x=1K(kd/4)2π2d(dm=1[Γ(md)](dm))1/(2n)

Conjecture 2. Let odd d=p for prime p3mod4 with odd class number h(d)=n and y=1K(kd)2π2d(dm=1[Γ(md)](dm))1/(2n)
then (x,y) are algebraic numbers as seen in entries 160163 and 165168. They seem to have a closed-form in term of eta quotients but I haven't found it yet.

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