  In his previous papers the author proved that in characteristic different
from 2 the jacobian J(C) of a hyperelliptic curve C: y^2=f(x) has only trivial
endomorphisms over an algebraic closure K_a of the ground field
  K if the Galois group of the irreducible polynomial f(x) in K[x] is either
the full symmetric group S_n or the alternating group A_n. Here n > 8 is the
degree of f. The goal of this paper is to extend this result to the case when
either n=7,8 or n=5,6 and char(K)>3.
