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We Know that $M_9=Q_8\ltimes C_3^2$ acts symplextically on the K3 surface $X$ which is the double cover of $P_2$ branched along the sextic $X^6+Y^6+Z^6-10(X^3Y^3+Y^3Z^3+X^3Z^3)=0$. My question is how many elliptic fibration does $X$ have?

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This question is highly non-trivial. The usual strategy is too determine the N\'eron-Severi lattice of $X$, determine all effective -2 curves and then determine all possible divisors $F$ consisting of sums of such -2 curves and $F$ is one of the fibers in Kodaira's list. Recently Abhninav Kumar did this programme for Kummer surfaces of jacobians of genus 2 curves.

However if the Picard number of your surface equals 2 then there should not be any elliptic fibration on $X$.

In any case you have first too determine the Picard number of $X$. Upper bounds can be found by considering $X$ modulo several primes $p$, lower bounds can be found by looking for explicit curves on $X$. See Calculations of Pic^0, Pic, NS of surfaces

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