The conductor of this order equals the ideal "cO""K". Indeed, it is clear that "cO""K" is an ideal of "O""K" contained in "O", so it is contained in the conductor. On the other hand, the ideals of "O" containing "cO""K" are the same as ideals of the quotient ring . The latter ring is isomorphic to by the second isomorphism theorem, so all such ideals of "O" are the sum of "cO""K" with an ideal of Z. Under this isomorphism, the conductor annihilates , so it must be . In this case, the index is also equal to "c", so for orders of quadratic number fields, the index may be identified with the conductor. This identification fails for higher degree number fields.