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\begin{longtable}{@{}l r l r l c >{\raggedright\arraybackslash}p{5.6cm} >{\raggedright\arraybackslash}p{3.4cm}@{}}
\caption{The $48$ $\Q$-rational smooth rational curves of degree $\le12$ on $V$ that are not sections of a Brudno-type pencil (Section~\ref{sec:newcurves}): name, degree $d$, class $D$ in the basis $\Lambda_1,\dots,\Lambda_{13}$, orbit size, sorted tridegree, the degree $k$ of the form $f_D$, the multiplicity-free expression $kH-D=\sum C_i$ ($C_i$ lines and conics; the zero locus of $f_D$ on $V$ is $C_D\cup\bigcup C_i$), and the solutions of Appendix~\ref{app:table} lying on a symmetric image of $C_D$ (exact test, all placements).}\label{tab:newcurves}\\
\toprule
name & $d$ & class & orbit & tridegree & $k$ & $kH-D$ & solutions on it\\\midrule
\endfirsthead
\toprule
name & $d$ & class & orbit & tridegree & $k$ & $kH-D$ & solutions on it\\\midrule
\endhead
\bottomrule
\endfoot
R7a & 7 & $(-2,0,0,2,0,0,1,1,1,1,0,0,0)$ & 24 & $(3,3,3)$ & 2 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{7}+Q_{7}+Q_{9}+Q_{10}$ & 1, 4, 5 \\
R8a & 8 & $(-3,-1,-1,4,0,2,1,1,2,3,-2,0,-1)$ & 48 & $(2,5,5)$ & 2 & $\Gamma_{1}+\Gamma_{3}+\Gamma_{15}+\Gamma_{16}+Q_{1}+Q_{9}$ & 1, 31, 37, 39, 41, 52, 53 \\
R8b & 8 & $(-3,-1,-1,4,0,2,1,2,2,2,-2,0,-1)$ & 48 & $(3,4,5)$ & 2 & $\Gamma_{1}+\Gamma_{6}+Q_{1}+Q_{3}+Q_{6}$ & 1, 2, 5, 15, 31 \\
R8c & 8 & $(-3,-1,-1,4,0,3,0,1,2,3,-2,0,-1)$ & 48 & $(2,5,5)$ & 2 & $\Gamma_{1}+\Gamma_{4}+\Gamma_{15}+\Gamma_{16}+Q_{3}+Q_{11}$ & 2, 5, 7, 11, 58 \\
R8d & 8 & $(-2,-2,-2,3,1,1,1,2,2,2,-1,0,-1)$ & 24 & $(2,3,3)$ & 2 & $\Gamma_{2}+\Gamma_{3}+\Gamma_{5}+\Gamma_{7}+Q_{3}+Q_{6}$ & 7, 15, 67 \\
R8e & 8 & $(-2,-2,-1,4,0,1,0,1,2,2,0,0,-1)$ & 24 & $(2,3,3)$ & 2 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{7}+\Gamma_{8}+Q_{7}+Q_{11}$ & 2, 21, 36, 50 \\
R9a & 9 & $(-3,-1,-1,5,-1,3,1,1,3,3,-2,0,-2)$ & 48 & $(3,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{29}+\Gamma_{30}+Q_{1}+Q_{2}+Q_{3}+Q_{11}+Q_{23}+Q_{24}$ & 18, 42, 52 \\
R9b & 9 & $(-2,-3,-1,5,0,1,1,2,2,3,-1,0,-2)$ & 48 & $(3,4,4)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{7}+\Gamma_{31}+\Gamma_{32}+Q_{1}+Q_{9}+Q_{11}+Q_{13}+Q_{14}$ & 1, 7, 15, 50 \\
R9c & 9 & $(-2,-2,-2,4,0,3,2,1,3,3,-2,0,-2)$ & 48 & $(3,5,7)$ & 3 & $\Gamma_{2}+\Gamma_{3}+\Gamma_{5}+Q_{1}+Q_{3}+Q_{7}+Q_{8}+Q_{9}+Q_{11}$ & 7, 11, 27 \\
R9d & 9 & $(-2,-1,-1,4,-1,1,1,1,1,2,0,0,0)$ & 48 & $(3,3,5)$ & 2 & $\Gamma_{1}+\Gamma_{3}+\Gamma_{6}+\Gamma_{15}+\Gamma_{16}+\Gamma_{19}+\Gamma_{20}$ & 2, 8 \\
R10a & 10 & $(-4,-2,-1,6,0,3,0,2,3,4,-2,0,-3)$ & 48 & $(4,5,5)$ & 3 & $\Gamma_{1}+\Gamma_{8}+\Gamma_{15}+\Gamma_{16}+Q_{2}+Q_{5}+Q_{9}+Q_{15}+Q_{16}$ & 1, 7 \\
R10b & 10 & $(-4,-2,-1,6,0,3,0,3,2,4,-2,0,-3)$ & 48 & $(3,5,6)$ & 3 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{7}+\Gamma_{8}+Q_{5}+Q_{6}+Q_{9}+Q_{10}+Q_{11}$ & 2, 22, 27 \\
R10c & 10 & $(-3,-3,-2,5,1,3,1,3,2,4,-2,0,-3)$ & 48 & $(3,5,6)$ & 3 & $\Gamma_{2}+\Gamma_{3}+\Gamma_{5}+\Gamma_{7}+Q_{5}+Q_{6}+Q_{9}+Q_{10}+Q_{11}$ & 3, 7, 58 \\
R10d & 10 & $(-3,-2,-1,5,0,3,0,1,2,3,-1,1,-2)$ & 48 & $(4,5,5)$ & 3 & $\Gamma_{1}+\Gamma_{8}+\Gamma_{9}+\Gamma_{10}+Q_{2}+Q_{6}+Q_{9}+Q_{15}+Q_{16}$ & 64 \\
R10e & 10 & $(-3,-2,-1,5,1,3,-1,2,1,4,-1,0,-2)$ & 48 & $(2,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{7}+\Gamma_{8}+\Gamma_{17}+\Gamma_{18}+\Gamma_{21}+\Gamma_{22}+Q_{8}+Q_{9}+Q_{11}$ & 2, 27, 42 \\
R10f & 10 & $(-3,-1,-1,4,0,0,1,2,2,2,0,0,-1)$ & 48 & $(3,3,4)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{6}+\Gamma_{7}+\Gamma_{21}+\Gamma_{22}+Q_{4}+Q_{7}+Q_{15}+Q_{16}$ & 1, 67 \\
R10g & 10 & $(-3,-1,-1,5,-1,2,1,2,2,2,-1,0,-1)$ & 48 & $(4,5,5)$ & 3 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{21}+\Gamma_{22}+\Gamma_{27}+\Gamma_{28}+Q_{1}+Q_{10}+Q_{23}+Q_{24}$ & 21, 50, 65 \\
R10h & 10 & $(-2,-3,-1,5,0,1,0,2,2,3,-1,0,-1)$ & 48 & $(3,3,4)$ & 3 & $\Gamma_{2}+\Gamma_{3}+\Gamma_{7}+\Gamma_{8}+\Gamma_{15}+\Gamma_{16}+Q_{6}+Q_{8}+Q_{19}+Q_{20}$ & 8, 21, 36 \\
R10i & 10 & $(-2,-3,-1,5,0,1,2,1,3,3,-1,0,-2)$ & 48 & $(3,5,6)$ & 3 & $\Gamma_{2}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{31}+\Gamma_{32}+Q_{9}+Q_{10}+Q_{23}+Q_{24}$ & 21, 74 \\
R10j & 10 & $(-2,-3,-1,5,0,3,0,2,2,4,-1,0,-3)$ & 48 & $(3,5,6)$ & 3 & $\Gamma_{2}+\Gamma_{5}+\Gamma_{7}+\Gamma_{8}+Q_{2}+Q_{5}+Q_{6}+Q_{9}+Q_{11}$ & 1 \\
R10k & 10 & $(-2,0,0,3,-1,1,1,1,1,1,0,1,0)$ & 48 & $(4,5,5)$ & 3 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{19}+\Gamma_{20}+\Gamma_{29}+\Gamma_{30}+Q_{3}+Q_{10}+Q_{23}+Q_{24}$ & 10, 15 \\
R11a & 11 & $(-4,-3,-2,6,1,4,1,2,3,4,-3,0,-2)$ & 48 & $(4,6,7)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{5}+\Gamma_{9}+\Gamma_{10}+Q_{1}+Q_{3}+Q_{6}+Q_{11}$ & 7, 12 \\
R11b & 11 & $(-4,-2,-2,6,0,4,1,2,2,4,-1,0,-3)$ & 48 & $(4,6,7)$ & 3 & $\Gamma_{1}+\Gamma_{3}+\Gamma_{8}+\Gamma_{19}+\Gamma_{20}+Q_{6}+Q_{7}+Q_{9}+Q_{11}$ & 5, 41 \\
R11c & 11 & $(-4,-2,-1,6,0,2,0,2,3,3,-1,0,-2)$ & 48 & $(4,4,5)$ & 4 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{8}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+Q_{3}+Q_{6}+Q_{12}+Q_{13}+Q_{14}+Q_{15}+Q_{16}$ & 1, 2 \\
R11d & 11 & $(-4,-2,-1,6,0,4,0,2,2,4,-1,0,-3)$ & 48 & $(4,6,7)$ & 4 & $\Gamma_{1}+\Gamma_{15}+\Gamma_{16}+\Gamma_{27}+\Gamma_{28}+Q_{5}+Q_{7}+Q_{9}+Q_{11}+Q_{13}+Q_{14}+Q_{19}+Q_{20}$ & 2, 10, 22 \\
R11e & 11 & $(-4,-2,-1,6,1,3,0,2,3,3,-2,0,-2)$ & 48 & $(5,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{8}+\Gamma_{9}+\Gamma_{10}+\Gamma_{29}+\Gamma_{30}+Q_{3}+Q_{6}+Q_{11}$ & 1, 5, 7 \\
R11f & 11 & $(-4,-2,-1,6,1,4,-1,2,2,4,-2,0,-2)$ & 48 & $(5,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{7}+\Gamma_{8}+\Gamma_{15}+\Gamma_{16}+\Gamma_{17}+\Gamma_{18}+\Gamma_{27}+\Gamma_{28}+Q_{9}+Q_{11}$ & 10 \\
R11g & 11 & $(-4,-1,-1,5,0,3,1,1,3,4,-2,0,-2)$ & 48 & $(3,7,7)$ & 3 & $\Gamma_{1}+\Gamma_{15}+\Gamma_{16}+Q_{3}+Q_{9}+Q_{12}+Q_{23}+Q_{24}$ & 1, 5, 71 \\
R11h & 11 & $(-4,-1,-1,6,-1,3,1,1,3,4,-2,0,-2)$ & 48 & $(3,7,7)$ & 3 & $\Gamma_{1}+\Gamma_{3}+\Gamma_{6}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+\Gamma_{29}+\Gamma_{30}+Q_{9}+Q_{11}$ & 10 \\
R11i & 11 & $(-4,-1,-1,6,-1,3,1,2,2,4,-2,0,-2)$ & 48 & $(5,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{3}+\Gamma_{6}+\Gamma_{15}+\Gamma_{16}+\Gamma_{19}+\Gamma_{20}+\Gamma_{29}+\Gamma_{30}+Q_{9}+Q_{11}$ & 10, 21, 50, 53, 58 \\
R11j & 11 & $(-4,-1,-1,6,0,2,1,3,2,3,-2,0,-2)$ & 48 & $(5,5,7)$ & 3 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{7}+\Gamma_{19}+\Gamma_{20}+\Gamma_{29}+\Gamma_{30}+Q_{3}+Q_{6}+Q_{11}$ & 1, 5, 11 \\
R11k & 11 & $(-3,-3,-2,5,1,4,1,2,3,4,-3,0,-2)$ & 48 & $(4,6,7)$ & 4 & $\Gamma_{2}+\Gamma_{5}+\Gamma_{8}+\Gamma_{25}+\Gamma_{26}+Q_{1}+Q_{2}+Q_{3}+Q_{9}+Q_{10}+Q_{11}+Q_{23}+Q_{24}$ & 1, 37, 58 \\
R11l & 11 & $(-3,-2,-2,6,0,2,2,1,3,4,-2,0,-2)$ & 48 & $(3,7,7)$ & 3 & $\Gamma_{3}+\Gamma_{6}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+\Gamma_{25}+\Gamma_{26}+Q_{9}+Q_{11}$ & 21 \\
R11m & 11 & $(-3,-2,-2,6,0,4,0,1,3,4,-2,0,-2)$ & 48 & $(3,7,7)$ & 3 & $\Gamma_{8}+\Gamma_{9}+\Gamma_{10}+Q_{3}+Q_{10}+Q_{11}+Q_{23}+Q_{24}$ & 2, 39, 54, 74 \\
R11n & 11 & $(-3,-1,0,4,0,0,1,1,2,2,0,0,0)$ & 48 & $(3,5,5)$ & 4 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+Q_{6}+Q_{7}+Q_{9}+Q_{19}+Q_{20}+Q_{21}+Q_{22}$ & 8, 50 \\
R11o & 11 & $(-2,-3,-2,4,2,2,0,3,2,3,-2,0,-1)$ & 48 & $(4,4,5)$ & 4 & $\Gamma_{2}+\Gamma_{5}+\Gamma_{8}+\Gamma_{25}+\Gamma_{26}+\Gamma_{31}+\Gamma_{32}+Q_{2}+Q_{9}+Q_{10}+Q_{19}+Q_{20}+Q_{23}+Q_{24}$ & 2, 3, 5, 21 \\
R11p & 11 & $(0,0,0,1,-1,-1,0,0,0,0,2,2,2)$ & 24 & $(3,3,3)$ & 4 & $\Gamma_{1}+\Gamma_{6}+\Gamma_{7}+Q_{7}+Q_{9}+Q_{10}+Q_{13}+Q_{14}+Q_{15}+Q_{16}+Q_{17}+Q_{18}$ & 3, 8 \\
R12a & 12 & $(-4,-3,-3,7,0,3,2,2,4,5,-3,0,-3)$ & 48 & $(3,6,7)$ & 4 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\Gamma_{7}+\Gamma_{31}+\Gamma_{32}+Q_{1}+Q_{3}+Q_{4}+Q_{9}+Q_{11}+Q_{23}+Q_{24}$ & -- \\
R12b & 12 & $(-4,-3,-1,7,0,1,2,2,4,3,-1,0,-3)$ & 48 & $(4,5,7)$ & 4 & $\Gamma_{1}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+Q_{5}+Q_{6}+Q_{11}+Q_{15}+Q_{16}+Q_{21}+Q_{22}$ & 3, 7, 9 \\
R12c & 12 & $(-4,-2,-1,6,0,2,1,3,2,4,-1,0,-3)$ & 48 & $(3,6,7)$ & 3 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{6}+\Gamma_{7}+\Gamma_{21}+\Gamma_{22}+Q_{5}+Q_{6}+Q_{9}$ & 1 \\
R12d & 12 & $(-4,-2,-1,6,1,3,-1,2,2,4,-1,0,-2)$ & 48 & $(4,5,7)$ & 4 & $\Gamma_{1}+\Gamma_{8}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+Q_{3}+Q_{5}+Q_{6}+Q_{13}+Q_{14}+Q_{19}+Q_{20}$ & 37 \\
R12e & 12 & $(-4,-1,-1,6,-1,2,1,2,2,3,-1,0,-1)$ & 48 & $(5,5,6)$ & 4 & $\Gamma_{1}+\Gamma_{4}+\Gamma_{6}+\Gamma_{7}+\Gamma_{15}+\Gamma_{16}+\Gamma_{19}+\Gamma_{20}+\Gamma_{21}+\Gamma_{22}+\Gamma_{27}+\Gamma_{28}+Q_{3}+Q_{10}+Q_{23}+Q_{24}$ & 1, 3, 5, 10 \\
R12f & 12 & $(-3,-3,-2,7,0,3,0,2,3,5,-3,0,-2)$ & 48 & $(3,6,7)$ & 3 & $\Gamma_{2}+\Gamma_{3}+\Gamma_{7}+\Gamma_{8}+\Gamma_{15}+\Gamma_{16}+Q_{1}+Q_{9}+Q_{11}$ & 27 \\
R12g & 12 & $(-3,-3,-1,6,0,0,3,2,4,2,0,0,-3)$ & 48 & $(3,6,7)$ & 4 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\Gamma_{7}+\Gamma_{15}+\Gamma_{16}+Q_{5}+Q_{6}+Q_{7}+Q_{8}+Q_{12}+Q_{15}+Q_{16}$ & 42 \\
R12h & 12 & $(-3,-3,-1,6,1,0,2,2,4,2,-1,0,-2)$ & 48 & $(3,5,8)$ & 4 & $\Gamma_{2}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+Q_{3}+Q_{5}+Q_{6}+Q_{15}+Q_{16}+Q_{21}+Q_{22}$ & 18, 42 \\
R12i & 12 & $(-3,-1,-1,5,-1,4,1,0,3,4,-1,0,-2)$ & 48 & $(3,8,9)$ & 4 & $\Gamma_{1}+\Gamma_{8}+Q_{1}+Q_{2}+Q_{3}+Q_{4}+Q_{7}+Q_{9}+Q_{11}+Q_{23}+Q_{24}$ & 2, 39 \\
R12j & 12 & $(-2,-3,-1,6,1,0,1,2,2,3,-1,0,-1)$ & 48 & $(5,5,6)$ & 4 & $\Gamma_{1}+\Gamma_{2}+\Gamma_{5}+\Gamma_{7}+\Gamma_{9}+\Gamma_{10}+\Gamma_{15}+\Gamma_{16}+\Gamma_{19}+\Gamma_{20}+\Gamma_{31}+\Gamma_{32}+Q_{5}+Q_{8}+Q_{19}+Q_{20}$ & 2, 4, 65 \\
R12k & 12 & $(-2,-1,-1,5,2,-1,2,3,3,0,-1,0,-1)$ & 24 & $(2,9,9)$ & 3 & $\Gamma_{1}+\Gamma_{7}+\Gamma_{25}+\Gamma_{26}+\Gamma_{27}+\Gamma_{28}+\Gamma_{29}+\Gamma_{30}+\Gamma_{31}+\Gamma_{32}+Q_{11}$ & 2 \\
\end{longtable}
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