In this work we investigate the effective electron neutrino mass, \(m_\text {e}\) of the neutrino mass matrix where the constrained condition of the vanishing trace and one vanishing minor have been considered. Since there are six textures of one vanishing minor in the case of a Majorana neutrino mass matrix, \(M_\nu \) , therefore we have a total of six cases of neutrino mass matrix with vanishing trace and one vanishing minor. On simultaneously solving both the constrained conditions, we obtained four solutions for the neutrino mass ratios, i.e., \(X_\pm \) and \(Y_\pm \) . Therefore for all the solution pairs, i.e., \((X_+, Y_+)\) , \((X_+, Y_-)\) , \((X_-, Y_+)\) , and \((X_-, Y_-)\) , we checked the viability of the textures using the ratio of solar and atmospheric mass squared difference, \(R_\nu \) and it was observed that only five textures out of six, are allowed according to the latest neutrino oscillation data at 3 \(\sigma \) C.L., for some constrained range of Dirac CP phase \(\delta \) . Now using the allowed range of \(\delta \) , we calculated the two Majorana CP phases \(\alpha \) , \(\beta \) . Finally using the range of the CP phases \(\delta \) , we calculate the effective neutrino mass, \(m_\text {e}\) , for all the viable textures.