\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Qp}[1]{\mathbb{Q}_{#1}}
\newcommand{\Zp}[1]{\mathbb{Z}_{#1}}
\newcommand{\Fq}[1]{\mathbb{F}_{#1}}
\newcommand{\PP}{\mathbb{P}}

\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\Nm}{N}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\rk}{rank}
\DeclareMathOperator{\type}{type}
\newcommand{\Norm}[2]{\Nm_{#1}\!\left(#2\right)}
\newcommand{\isom}{\simeq}
\newcommand{\eqdef}{\coloneqq}

\newcommand{\Aeta}{A_{\eta}}
\newcommand{\feta}{f_{\eta}}
\newcommand{\Lo}{L_{8}}
\newcommand{\Lt}{L_{24}}
\newcommand{\vt}{\theta}

\newcommand{\Sel}{\Lt(S,5)}
\newcommand{\OS}{\mathcal{O}_{S}}
\newcommand{\PT}{\mathfrak{P}}
\newcommand{\pe}{\mathfrak{p}}
\newcommand{\qQ}{\mathfrak{Q}}
\newcommand{\qq}{\mathfrak{q}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\Iq}{I(q)}
\newcommand{\Mq}{M_{q}}
\newcommand{\NM}{\mathsf{N}}
\newcommand{\TT}{\mathsf{T}}
\newcommand{\sym}{\chi}
\newcommand{\Lop}[1]{L_{8,#1}}
\newcommand{\Ltp}[1]{L_{24,#1}}