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For a nonnegative integer $m$, let $\\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\\cdot dL$ on the quadratic space $L^{p^m}\\otimes \\q$ such that for each lattice $T \\in \\mathcal G_{L,p}(m)$, a $\\frac 12\\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. 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For a nonnegative integer $m$, let $\\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\\cdot dL$ on the quadratic space $L^{p^m}\\otimes \\q$ such that for each lattice $T \\in \\mathcal G_{L,p}(m)$, a $\\frac 12\\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. 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