The FastECPP algorithm is currently the fastest approach to prove the primality of general numbers, and has the additional benefit of creating certificates that can be checked independently and with a lower complexity. This article shows how by parallelising over a linear number of cores, its quartic time complexity becomes a cubic wallclock time complexity; and it presents the algorithmic choices of the FastECPP implementation in the author's \cm\ software \\ \centerline {\url {https://www.multiprecision.org/cm/}} which has been written with massive parallelisation over MPI in mind, and which has been used to establish a new primality record for the ``repunit'' $(10^{86453} - 1) / 9$.