DTC P1065: Engine Control Module Power Supply (Back-Up): Description
WARNING: This page is about a different car, the 2003 Infiniti G35. However, it is still accessible from the selected car via links, so may be relevant.
Battery voltage is supplied to ECM even when ignition switch is turned off for ECM memory function of DTC memory, air-fuel ratio feedback compensation value memory, idle air volume learning value memory, etc.