We generalize Birch's construction of the Heegner points on X0(N) to construct new points X1(N) (and therefore construct points on the abelian varieties associated to J1(N)). Then, we show that these points form an Euler system, and we improve Kolyvagin's Euler system techniques to show that for our point PτK/c and any ring class character χ of the extended ring class field of conductor c satisfying χ=χ‾, if PτK/cχ is non-torsion and GK→AutAf[π] is surjective, then the corank of Sel(Aχ/K) is 1, which implies the rank of Af(K)χ is 1.