  We compute the algebraic equation of the universal family over the
Kenyon-Smillie $(2,3,4)$-Teichm\"uller curve and give a nice geometric
description of the torsion map. Moreover, we re-prove independently that the
found algebraic equation describes a Teichm\"uller curve by computing the
Picard-Fuchs equation associated to the absolute cohomology bundle.
