BT ships a variety of modems for its VDSL/Infinity product. There are ongoing discussions regarding the merits of each, but what all do have in common is that they are locked-down and inaccessible. The Huawei HG612 is theoretically the easiest to root, as described – but this does require loading a custom firmware. Additionally, mine re-locked itself and I’ve been unable to get into firmware-recovery mode since 🙁
The ECI B-FOCuS V-2FUb/I Rev. 1B can be unlocked via a serial connection without any need to replace the stock firmware – details here.