An independent cross-check of your field, from the Fermat side. My paper on this is now under review.
1. The quartic field is forced. In the eigenline-block model at level 44, I computed every CM subfield $K\subset\mathbb Q(\zeta_{44})$ exactly, by lattice computation. $\xi_{44}$ is congruent mod $D_{44}+\tilde S_{44}$ to a sum of $K$-Weil classes on Fermat-type CM factors iff $K\supseteq\mathbb Q(i,\sqrt{-11})$. This is the fixed field of the odd part of $(\mathbb Z/44)^\times$. Neither imaginary quadratic subfield works. The same pattern ($K\supseteq$ the 2-primary subfield) holds for all 17 two-factor $q\le95$ I checked. So your twelvefold sits at the minimal possible field. Within Fermat-type factors at level 44 the smallest realisation I found is much larger: degree 20, on a 40-dimensional variety. Your Prym construction is far more economical.
2. A four-curve representative. $\xi_{44}\equiv(1,2,41)+(1,6,37)+(13,33,42)+(17,28,43)$ mod $D+\tilde S$.
- All four characters have trivial stabiliser, so this is a degree-4 Weil class for $K=\mathbb Q(\zeta_{44})$.
- Through Shioda's inductive structure, HC for all degree-44 Fermat varieties is equivalent to algebraicity of this single orbit on $C_{44}^4$.
3. A field-of-definition constraint for any realising cycle. The Frobenius character on that orbit is computed exactly via Stickelberger. Its 2-primary part has order 4 at $p=617,881,1409$ and order 2 at $353$. Empirically, it matched at 80 fitted and 80 further out-of-sample primes the quartic residue symbol of the cyclotomic number $$\alpha=\prod_{k}(1-\zeta_{44}^k)^{a_k},$$ with $(k,a_k)=(31,2),(34,2),(36,3),(37,2),(38,1),(39,2),(40,2),(41,2),(42,1),(43,2)$. This $\alpha$ has odd valuation at the prime above 11. So a determinant cycle on your $A_{\det}\times E_{-11}$ side, transported to $C_{44}^4$, should be defined over a field containing a 4th root of a conjugate of $\alpha$. If you build candidate cycles over number fields, that is a cheap filter to apply. I would be glad to compare with your character certificates.
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