This article investigates the effects of ion-channel guiding, axial magnetic fields, helical wiggler magnetic fields (HWMFs), and self-fields on trajectories of the elliptical relativistic electron beam. Two equations for the average axial velocity and the function $\Phi $ are calculated for an electron in the considered fields by solving the motion equation. In the meantime, the effects of elliptical cross section demotions and elliptical electron beam density are also studied. The results are compared in the limit that the elliptical electron beam is converted to a circular electron beam, which have also been graphically presented. It is shown that in the considered conditions, in the case of the elliptical beam, and for spatial cases, there are three types of trajectories. This result is different from the case circular beam because in the case circular beam, there are two types of trajectories. It means that in the case of the elliptical cross section, there are two resonance frequencies of the orbits. Moreover, the resonant frequency of the transverse velocity in the circular beam case is within two resonant frequencies in the elliptical beam case. Furthermore, it has been indicated that for special cases, with increasing $\bar {\omega }_{p} $ , the resonance frequencies of transverse velocity tend toward higher values of $\bar {\omega }_{i} $ . Finally, the conditions of stability of electron orbits in the elliptical electron beam are investigated by considering HWMF. Furthermore, a detailed analysis of the gain equation in an FEL with elliptical electron beam is presented. The obtained gain for an FEL together with a relativistic electron beam with elliptical cross section is compared to the maximum gain for an FEL together with a relativistic electron beam with circular cross section. It is shown that with increasing the ratio of the semimajor to semiminor axes of the beam cross section, the ratio of the gain of the elliptical electron beam increases compare
