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The rank of $E$ is given in terms of an elementary property of the subgroup of $(\\Z/d\\Z)^\\times$ generated by $p$. We show that for many values of $d$ the rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. 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