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.