Let
X
X
(resp.
Y
Y
) be a curve of genus
1
1
(resp.
2
2
) over a base field
k
k
whose characteristic does not equal
2
2
. We give criteria for the existence of a curve
Z
Z
over
k
k
whose Jacobian is up to twist
(
2
,
2
,
2
)
(2,2,2)
-isogenous to the products of the Jacobians of
X
X
and
Y
Y
. Moreover, we give algorithms to construct the curve
Z
Z
once equations for
X
X
and
Y
Y
are given. The first of these is based on interpolation methods involving numerical results over
C
\mathbb {C}
that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of
Y
Y
whose desingularization is isomorphic to
X
X
. As an application, we find a twist of a Jacobian over
Q
\mathbb {Q}
that admits a rational
70
70
-torsion point.