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We study a class of elliptic K3 surfaces defined by an explicit Weierstrass equation to find elliptic fibrations of maximal rank on K3 surface in positive characteristic. In particular, we show that the supersingular K3 surface of Artin invariant 1 (unique by Ogus) admits at least one elliptic fibration with maximal rank 20 in every characteristic p > 7, p ̸ = 13, and further that the number, say N (p), of such elliptic fibrations (up to isomorphisms), is unbounded as p → ∞; in fact, we provedoi:10.4213/im8017 fatcat:7isbqnlzfbg63dkn42iscuiasi