  We consider the scalar sector of a general renormalizable theory and evaluate
the effective potential through three loops analytically. We encounter
three-loop vacuum bubble diagrams with up to two masses and six lines, which we
solve using differential equations transformed into the favorable $\epsilon$
form of dimensional regularization. The master integrals of the canonical basis
thus obtained are expressed in terms of cyclotomic polylogarithms up to weight
four. We also introduce an algorithm for the numerical evaluation of cyclotomic
polylogarithms with multiple-precision arithmetic, which is implemented in the
Mathematica package cyclogpl.m supplied here.
