What is a finite Haken cover of the Seifert–Weber space?

I just checked using SnapPea that there is a cover of the Seifert-Weber dodecahedral space of index 25 (a 5-fold cyclic cover of a 5-fold cyclic cover) which has positive first betti number hence is Haken.

The Seifert-Weber space is a 5-fold cyclic branched cover over the Whitehead link complement. There are two such 5-fold covers (up to homeomorphism), which one may compute using SnapPea (perform $(5,0)$ surgery on each cusp of the Whitehead link, then compute all 5-fold cyclic covers of this orbifold, giving four manifold covers, with two isometry types). One may then compute the 5-fold cyclic covers of these two manifolds. One of them (not Seifert-Weber) has a 5-fold cyclic cover with positive betti number, whereas the 5-fold cyclic covers of the Seifert-Weber space have trivial betti number. However, one of them will be a 5-fold cover of its sibling, and hence will have a 5-fold cyclic cover which has positive betti number.

There are many other ways that we know the Seifert-Weber space to have a finite cover with positive first betti number (Hempel's paper pointed out by Igor shows that there is a 5-fold irregular cover), but it is an interesting question whether given a manifold $M$ with $b_1(M,\mathbb{F}_p)\geq 4$, is there a $p$-cover which has positive first betti number (I asked this as question 5 in a survey paper). Since all 5-fold covers of the Seifert-Weber space have $b_1(*; \mathbb{F}_5)\geq 4$, then this computation shows that a single 5-fold cyclic cover works in some cases.

This is constructed (reasonably explicitly) in John Hempel's 1982 paper.

John Hempel, MR 664329 Orientation reversing involutions and the first Betti number for finite coverings of $3$-manifolds, Invent. Math. 67 (1982), no. 1, 133--142.