Good reduction of elliptic curves defined over Q(2√3)

The authors prove that if K=Q(α) , where α is the real cube root of 2, then there are no elliptic curves over K with good reduction. This extends to the field Q(α) a well known theorem of J. Tate [Invent. Math. 23 (1974), 179--206; MR0419359 (54 #7380)] for the field Q . C. B. Setzer [Pacific J. Math. 74 (1978), no. 1, 235--250; MR0491710 (58 #10913)] and R. J. Stroeker ["Elliptic curves over complex quadratic fields'', Dissertation, Amsterdam, 1975; per bibl.] have proved similar results for complex quadratic fields. However, there are elliptic curves defined over real quadratic fields, for example Q(29 − − √ ) , with good reduction everywhere.