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Saturday, May 24, 2025

Entry 77

Level 6. Define dk=η(kτ) with Dedekind eta function η(kτ) and the McKay-Thompson series of class 6B for the Monster.

j6B(τ)=(d2d3d1d6)12

Example. We select d=12m with class number h(d)=4 and find  

m=10,14,26,34m=7,11,19,31,59 such that the following are special quadratic irrationals j6B(1210/3)=U125=(1+52)12j6B(1214/3)=U214=(15+414)2j6B(1226/3)=U426=(5+26)4j6B(1234/3)=U122=(1+2)12
j6B(1+7/32)=U321=(5+212)3j6B(1+11/32)=U211=(10+311)2j6B(1+19/32)=U63=(2+3)6j6B(1+31/32)=U393=(29+3932)3j6B(1+59/32)=U259=(530+6959)2
since they are fundamental units Un. Note the integer ((530+6959)21/(530+6959)2)2=10602 and similarly for the others as discussed in the previous entry.

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