The first eight orders are calculated in the high-temperature expansion in powers of $\beta=1/kT$ of the function $\varphi(\alpha , \beta)$ ($\alpha$ is the magnetization), which is the Legendre transform of the specific logarithm of the partition function $w$ with respect to the reduced external field $\alpha\equiv\beta h$. This is equivalent to calculating $w$ in an arbitrary external field in temperature-magnetization variables. The transition from the field to the magnetization enables one to use the high-temperature expansion below the Curie point as well, and, in particular, it enables one to calculate the spontaneous magnetization in zero field below the transition point. The calculations are made for two planar (square and triangular) and three three-dimensional (simple cubic, bcc and fcc) lattices, two variants being considered for the three-dimensional lattices: interaction of only nearest neighbors and interaction of first and second neighbors.