  The sextic plane curves that are invariant under the standard action of the
icosahedral group on the projective plane make up a pencil of genus ten curves
(spanned by a sum of six lines and a three times a conic). This pencil was
first considered in a note by R.~M.~Winger in 1925 and is nowadays named after
him. The second author recently gave this a modern treatment and proved among
other things that it contains essentially every smooth genus ten curve with
icosahedral symmetry. We here show that the Jacobian of such a curve contains
the tensor product of an elliptic curve with a certain integral representation
of the icosahedral group. We find that the elliptic curve comes with a
distinguished point of order $3$, prove that the monodromy on this part of the
homology is the full congruence subgroup $\Gamma_1(3)\subset \SL_2(\Zds)$ and
subsequently identify the base of the pencil with the associated modular curve.
  We also observe that the Winger pencil `accounts' for the deformation of the
Jacobian of Bring's curve as a principal abelian fourfold with an action of the
icosahedral group.
