Three important points make LaN an attractive topic for theoretical investigations: (i) it is a transitionmetal nitride, (ii) it can be a superconductor [1], and (iii) it presents many similarities with both ScN[2] and YN [3]. It is widely accepted that LaN crystallizes in the rocksalt (B1) phase with a possiblephase transition to the CsCl (B2) under pressure [1] but has been found recently to crystallize also inother phases [4]. On the other hand, both ScN and YN have been predicted to have a second localminimum in the hexagonal A3 phase [3, 5]The exchange correlation energy of electrons is described inboth the local density approximation (LDA) [8, 9] and the generalized gradient approximation(GGA96) using the parameterisation of Perdew et al [10, 11]. FLAPW and FPLMTO have beencarried out using the WIEN2K [12] and the lmtART [7, 13] codes respectivelyThe primitive unit cell contains two La atoms at r 1 =0 and r 2 =(2/3,1/3, c 0 /2.a 0 ) a 0 , and two N atoms at r 3 =(0, 0, u. c 0 /a 0 ) a 0 and r 4 =(2/3,1/3, c 0 (u+1/2)/a 0 ) a 0 , u being the internal parameter (dimensionless)\cite{Ghezali_2008}elastic stability of GaN and AlN in wurzite and zincblende have been established by [] using Tight-binding linearized muffin tin orbital(TB-LMTO) method with LDA exchange functional.