  The magnetic properties of the pyrochlore iridate material Eu$_2$Ir$_2$O$_7$
(5$d^5$) have been studied based on the first principle calculations, where the
crystal field splitting $\Delta$, spin-orbit coupling (SOC) $\lambda$ and
Coulomb interaction $U$ within Ir 5$d$ orbitals are all playing significant
roles. The ground state phase diagram has been obtained with respect to the
strength of SOC and Coulomb interaction $U$, where a stable anti-ferromagnetic
ground state with all-in/all-out (AIAO) spin structure has been found. Besides,
another anti-ferromagnetic states with close energy to AIAO have also been
found to be stable. The calculated nonlinear magnetization of the two stable
states both have the d-wave pattern but with a $\pi/4$ phase difference, which
can perfectly explain the experimentally observed nonlinear magnetization
pattern. Compared with the results of the non-distorted structure, it turns out
that the trigonal lattice distortion is crucial for stabilizing the AIAO state
in Eu$_2$Ir$_2$O$_7$. Furthermore, besides large dipolar moments, we also find
considerable octupolar moments in the magnetic states.
