This paper investigates the arithmetic of a particular singular K3 surface X over Q, the extremal elliptic fibration with configuration [1,1,1,12,3~*]. First of all, we determine the corresponding weight-3 form (cf. [Li, Ex. 1.6]) explicitly. For this, we calculate the action of Frobenius on the transcendental lattice by counting points and applying the Lefschetz fixed point formula. The proof is based on our previous classification of complex multiplication (CM) forms with rational coefficients in [S1]. In fact, we only have to compute one trace.rnThen we compute the zeta-function of the surface. This is used to study the reductions of X modulo some primes p. We emphasize that we are able to find a model with good reduction at 2. We subsequently verify conjectures of Tate and Shioda. The conjectures will be recalled in Section 4 and verified in Sections 5-7.rnThe final section is devoted to the twists of X.
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