  In this paper, we have studied $F(R,T)$ gravity as an arbitrary function of
curvature and torsion scalars in Friedmann--Lema\^{\i}tre--Robertson--Walker
(FLRW) background. Then, we have considered interacting model between $F(R,T)$
gravity and modified Chaplygin gas. The novelty of this model is that the
Universe includes both cases curvature and torsion, and one dominated by a
Chaplygin gas. In order to calculate cosmological solutions, we obtained
Friedmann equations and also equation of state (EoS) parameter of dark energy.
By employing interacting model we considered the total energy density and the
total pressure of Universe as the combination of components of dark energy and
Chaplygin gas. Subsequently, we reconstructed the model by an origin of a
scalar field entitled quintessence model with a field potential. The field
potential has been calculated in terms of free parameters of $F(R,T)$ gravity
and modified Chaplygin gas. In what follows, we used a parametrization, and the
cosmological parameters have been written in terms of redshift $z$. Next, we
plotted cosmological parameters with respect to three variable of cosmic time,
redshift $z$ and $e$-folding number $N=ln(a)$, and the figures showed us an
accelerated expansion of Universe. Also, we have described the scenario in
three status early time, late time and future time by $e$-folding number.
Finally, the stability of scenario has been investigated by a useful function
named sound speed, and the graph of sound speed versus $e$-folding number has
been showed us that there is the stability in late time.
