Admissible tuple data tables

This table contains algebraic data for each inequivalent admissible tuple in dimensions 4–35, comprising 100 total tuples. This list is conjecturally complete for all Weyl-Heisenberg covariant SICs in these dimensions. Each admissible tuple is specified by a dimension $d$ and an integer binary quadratic form $Q$ as follows. First factorize $(d+1)(d-3)=s^2\Delta_0$ where $\Delta_0$ is a fundamental discriminant. Then $ (d,Q) $ gives an admissible tuple if $\mathrm{disc}(Q) = f^2\Delta_0$ where $f$ divides $s$. The other columns can be computed from these data, but they may be difficult to compute, for example requiring integer factoring or finding a fundamental unit. The column $\Delta_0$ contains the fundamental discriminant of $Q$ and $h$ is the order of the class group $\mathrm{Cl}(\mathcal{O}_f)$, given in the next two columns respectively. The Galois group $\mathrm{Gal}(E_t^{(2)}/H)$ of the field containing the overlaps over the class field $H$ of $K=\mathbb{Q}(\sqrt{\Delta_0})$ is given in the next column. As both the class group and the Galois group are finite and abelian, we give the canonical decomposition into cyclic groups $C_k$ of order $k$. For the special case that the tuple has so-called $F_a$ symmetry, we have not yet worked out the Galois groups, so we mark these entries as tbd. The column $L^n$ contains a generator $L$ of the stability group of $Q$ in $\mathrm{GL}_2(\mathbb{Z})$ and its order $n$ in $\mathrm{GL}_2(\mathbb{Z}/\bar{d})$; that is, treating $Q$ as a symmetric matrix we have $L^T Q L = \det(L) Q$ and $L^n = 1\ (\bmod\ \bar{d})$. If the tuple has antiunitary symmetry, we denote this with a Y in the a.u. column. Finally, $\ell$ is the length of the word expansion of $L^n$ using the Hirzebruch-Jung (negative regular) reduction into the standard ($S$ and $T$) generators of $\mathrm{SL}_2(\mathbb{Z})$. This is one measure of the complexity of constructing the actual fiducial vector for that input. The $Q$ in this list were selected among class representatives to minimize this complexity, although this choice is not generally unique. The data here are sufficient to compute a ghost fiducial in each class, but to fully specify a SIC, one must additionally choose a sign-switching Galois automorphism $\sqrt{\Delta_0}\to-\sqrt{\Delta_0}$ over an appropriate field extension of $K$.