We study the properties of an advanced superconducting circuit, the 0-π circuit. This device, previously introduced by Brooks et al. in Phys. Rev. A 87, 052306 (2013), has multiple intriguing characteristics. With the right choice of parameters – it can exhibit a (near) degeneracy of the ground state, be robust with respect to dephasing, and relaxation. We study in detail the extent of the suitable parameter regime is and investigate how the degeneracy is affected by different types of disorder.