  In his previous papers (Math. Res. Letters 7 (2000), 123--13; Progress in
Math. 195 (2001), 473--490; Math. Res. Letters 8 (2001), 429--435; Moscow Math.
J. 2 (2002), issue 2, 403-431; Proc. Amer. Math. Soc. 131 (2003), no. 1,
95--102) the author introduced certain explicit constructions of hyperelliptic
jacobians without nontrivial endomorphisms. In the present paper we discuss
when these jacobians are mutually non-isogenous.
  In addition, a special case ($n=m=3$) of our Theorem 1.2 provides the
following criterion for elliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$ to
be non-isogenous. (Here $f(x)$ and $h(x)$ are cubic polynomials with
coefficients in a field $K$ of characteristic zero.) Suppose that $f(x)$ and
$h(x)$ are irreducible over $K$, their Galois groups over $K$ coincide with the
full symmetric group $S_3$, and their splitting fields are linearly disjoint
over $K$. Then the elliptic curves $C_f$ and $C_h$ are non-isogenous over an
algebraic closure of $K$.
