The question below comes to my mind when I am trying to explore something related to the formulas found by Jesus Guillera:
a)Generalized hypergeometric function
$${}_3 F_2\left(\begin{matrix}1/4& 3/4& 1/2\\1& 1\end{matrix};\cdots\right)$$
is related to the Dwork family
$x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$ through Picard-Fuchs equation;
b)Certain $\lambda$ corresponds to singular K3 surfaces, which has been given in the note of N. Elkies and M. Schutt. They calculated all possible rational $\mu,\mu=\lambda^4$ that the corresponding surface is a singular K3 surface(with highest possible Picard number 20), and almost all the numerical values appear in Ramanujan's 1914 paper on $1/\pi$(formulas (35)-(44));
c)J. Guillera has already given certain formulas(formula (92)(93)(94) in W. Zudilin's paper) on $1/\pi^2$ related to
$${}_5 F_4\left(\begin{matrix}1/6& 2/6& 3/6& 4/6& 5/6\\1& 1&1&1\end{matrix};\cdots\right)$$
and their Dwork family should be
$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$. Similarly, certain rational $\mu,\mu=\lambda^6$ correspond to Guillera's formulas;
Question: It might be a vague question, but is there any "singular" member in the family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$ having similar properties as whose analog is treated in the note of N. Elkies and M. Schutt? It is said that singular K3 surfaces share similar properties with elliptic curves with complex multiplication. Is there an analog of "complex multiplication" for $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6?$$
