On a Linnik problem for elliptic curves

Let S ( Q , B ) S(Q,B) denote the number of moduli q ≤ Q q\leq Q for which a primitive character χ \chi mod q q exists such that n χ > B n_{\chi }>B , where n χ n_{\chi } denotes the smallest natural number such that χ ( n ) ≠ 1 \chi (n) \not =1 . Baier showed that for any β > 2 \beta >2 we have S ( Q , ( log ⁡ Q ) β ) ≪ Q 1 β − 1 + ε S(Q,(\log Q)^{\beta }) \ll Q^{\frac {1}{\beta -1}+\varepsilon } and asked for an analogue of this result for elliptic curves. It is the aim of this note to establish such an analogue.