Level 7. This level is special since we have to consider the curve x3−1=7y2. Define dk=η(kτ) with the Dedekind eta function η(τ) and the McKay-Thompson series of class 7A for the Monster. j7A(τ)=((d1d7)2+7(d7d1)2)2 Examples. We select d=7m with class number h(−d)=2 and find m=5,13,61 such that the following are well-behaved integers j7A(1+√−5/72)=−7×12=−23+1j7A(1+√−13/72)=−7×32=−43+1j7A(1+√−61/72)=−7×392=−223+1 A WolframAlpha search for positive integer solutions to x3−1=7y2 reveals only these three. However, if we allow (x,y) to be higher algebraic integers with the same odd degree and a solvable Galois group, then it seems there are infinitely many. For example, we select d=7m with class number h(−d)=6 and find m=101 and others so j7A(1+√−101/72)=−x3+1=−7y2 where (x,y) are the real roots of cubics x3−46x2−380x−800=0y3−145y2−357y−1235=0 For class number h(−d)=10, we find m=17 and others so j7A(1+√−17/72)=−x3+1=−7y2 where (x,y) are the real roots of solvable quintics x5−5x4+4x3−15x2−23x−11=0y5−5y4+5y3−7y2−1=0 and so on.
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