  We study the cubic fixed point for $N=3$ and $4$ by using finite size scaling
applied to data obtained from Monte Carlo simulations of the $N$-component
$\phi^4$ model on the simple cubic lattice. We generalize the idea of improved
models to a two-parameter family of models. The two-parameter space is scanned
for the point, where the amplitudes of the two leading corrections to scaling
vanish. To this end, a dimensionless quantity is introduced that monitors the
breaking of the $O(N)$-invariance. For $N=4$, we determine the correction
exponents $\omega_1=0.763(24)$ and $\omega_2=0.082(5)$. In the case of $N=3$,
we obtain $Y_4=0.0142(6)$ for the RG-exponent of the cubic perturbation at the
$O(3)$-invariant fixed point, while the correction exponent
$\omega_2=0.0133(8)$ at the cubic fixed point. Simulations close to the
improved point result in the estimates $\nu=0.7202(7)$ and $\eta=0.0371(2)$ of
the critical exponents of the cubic fixed point for $N=4$. For $N=3$, at the
cubic fixed point, the $O(3)$-symmetry is only mildly broken and the critical
exponents differ only by little from those of the $O(3)$-invariant fixed point.
We find $-0.00001 \lessapprox \eta_{cubic}- \eta_{O(3)} \lessapprox 0.00007$
and $\nu_{cubic}-\nu_{O(3)} =-0.00061(10)$.
