  This paper is to investigate an elliptic fibration over $\mathbb{CP}^2$
arising from the Lagrange top from the viewpoint of complex algebraic geometry.
The description of the discriminant locus of this elliptic fibration is given
in detail. Moreover, the concrete description of the discriminant locus and the
complete classification of singular fibres of the elliptic fibration are
obtained according to Miranda's theory of elliptic threefolds after suitable
modifications of the base and total spaces. Furthermore, the monodromy of the
elliptic fibration is described.
