  In this paper, we classify relatively minimal genus-$1$ holomorphic Lefschetz
pencils up to smooth isomorphism. We first show that such a pencil is
isomorphic to either the pencil on $\mathbb{P}^1\times \mathbb{P}^1$ of
bi-degree $(2,2)$ or a blow-up of the pencil on $\mathbb{P}^2$ of degree $3$,
provided that no fiber of a pencil contains an embedded sphere. (Note that one
can easily classify genus-$1$ Lefschetz pencils with an embedded sphere in a
fiber.) We further determine the monodromy factorizations of these pencils and
show that the isomorphism class of a blow-up of the pencil on $\mathbb{P}^2$ of
degree $3$ does not depend on the choice of blown-up base points. We also show
that the genus-$1$ Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine
base points) and Tanaka (with eight base points) are respectively isomorphic to
the pencils on $\mathbb{P}^2$ and $\mathbb{P}^1\times \mathbb{P}^1$ above, in
particular these are both holomorphic.
