For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series (equ) This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of M(2, 2k + 1) Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a; n) denotes the number of partitions of n into parts which are not congruent to 0,±a (mod b), then for every positive integer n, we have P27(12; n) = P27(6; n - 1) + P27(3; n - 2). We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k +1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p.