Let kk be a real quadratic field and okok, EE the ring of integers and the group of units in kk. Denoting by E(p)E(p) the subgroup represented by EE of (ok/p)×(ok/p)× for a prime ideal pp, we show that prime ideals pp for which the order of E(p)E(p) is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.