  Let E be an elliptic curve over Q, and rho_l: Gal(Q) --> GL_2(Z_l) its l-adic
Galois representation. Serre observed that for l>3 there is no proper closed
subgroup of SL_2(Z_l) that maps surjectively onto SL_2(Z/lZ), and concluded
that if rho_l is surjective mod l then it is surjective onto GL_2(Z_l). We show
that this no longer holds for l=3 by describing a modular curve X of genus 0
parametrizing elliptic curves for which rho_3 is not surjective mod 9 but
generically surjective mod 3. The curve X is defined over Q, and the modular
cover X --> X(1) has degree 27 so X is rational. We exhibit an explicit
rational function of degree 27 that realizes this cover, and use it to exhibit
several elliptic curves with nonzero j-invariant that satisfy this condition on
rho_3, of which the simplest are the curves Y^2 = X^3 - 27X - 42 and Y^2 + Y =
X^3 - 135X - 604 of conductors 1944 = 2^3 3^5 and 6075 = 3^5 5^2 respectively.
