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Governing Fields

\(\huge a_{01}\)

Governing field: \[M_{01} = \mathbb{Q}\left(\zeta_8\right)\]

\(G_{01} = C_2 \times C_2\)
\(|S_{01}^1| = 1\)
\(|C_{01}^1| = 1\)
\(|P_{01}| \approx 2.297 \cdot 10^{-303}\)

\(|\{ p \in \mathbb{P} \mid a_{01}(p)=0, p<10^4 \}|=914\)
Ratio: \(\frac{ 914 }{ 1228 } \approx \frac{ 3 }{ 4 }\)
\(|\{ p \in \mathbb{P} \mid a_{01}(p)=1, p<10^4 \}|=314\)
Ratio: \(\frac{ 314 }{ 1228 } \approx \frac{ 1 }{ 4 }\)

\(\huge a_{02}\)

Governing field: \[M_{02} = \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}\right)\]

\(G_{02} = D_8\)
\(|S_{02}^1| = 1\)
\(|C_{02}^1| = 1\)
\(|P_{02}| \approx 4.285 \cdot 10^{-200}\)

\(|\{ p \in \mathbb{P} \mid a_{02}(p)=0, p<10^4 \}|=1076\)
Ratio: \(\frac{ 1076 }{ 1228 } \approx \frac{ 7 }{ 8 }\)
\(|\{ p \in \mathbb{P} \mid a_{02}(p)=1, p<10^4 \}|=152\)
Ratio: \(\frac{ 152 }{ 1228 } \approx \frac{ 1 }{ 8 }\)

\(\huge a_{03}\)

Governing field: \[M_{03} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}\right)\] - or - \[M_{03} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\]

\(G_{03} = D_{16}\)
\(|S_{03}^1| = 2\)
\(|C_{03}^1| = 1\)
\(|P_{03}| \approx 5.099 \cdot 10^{-204}\)

\(|\{ p \in \mathbb{P} \mid a_{03}(p)=0, p<10^4 \}|=1071\)
Ratio: \(\frac{ 1071 }{ 1228 } \approx \frac{ 14 }{ 16 }\)
\(|\{ p \in \mathbb{P} \mid a_{03}(p)=1, p<10^4 \}|=157\)
Ratio: \(\frac{ 157 }{ 1228 } \approx \frac{ 2 }{ 16 }\)

\(\huge a_{04}\)

Same governing field as \(a_{03}\).
Governing field: \[M_{04} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}\right)\] - or - \[M_{04} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\]

\(G_{04} = D_{16}\)
\(|S_{04}^1| = 1\)
\(|C_{04}^1| = 1\)
\(|P_{04}| \approx 1.064 \cdot 10^{-120}\)

\(|\{ p \in \mathbb{P} \mid a_{04}(p)=0, p<10^4 \}|=1155\)
Ratio: \(\frac{ 1155 }{ 1228 } \approx \frac{ 15 }{ 16 }\)
\(|\{ p \in \mathbb{P} \mid a_{04}(p)=1, p<10^4 \}|=73\)
Ratio: \(\frac{ 73 }{ 1228 } \approx \frac{ 1 }{ 16 }\)

\(\huge a_{05}\)

Governing field: \[M_{05} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}, \sqrt{- \frac{8282936156772053 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{6}}{562949953421312} - \frac{336382584949535 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{4}}{140737488355328} - \frac{4638634719581101 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{3}}{8796093022208} + \frac{8240373942248553 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{5}}{562949953421312} + \frac{4981425151744809 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{7}}{18014398509481984} + \frac{1676680829315919 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{15}{2}}}{9007199254740992}}\right)\] - or - \[M_{05} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\] and \[\beta = - \frac{8282936156772053 \alpha^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \alpha^{6}}{562949953421312} - \frac{336382584949535 \alpha^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \alpha^{4}}{140737488355328} - \frac{4638634719581101 \alpha^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \alpha^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{\alpha}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \alpha^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \alpha^{3}}{8796093022208} + \frac{8240373942248553 \alpha^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \alpha^{5}}{562949953421312} + \frac{4981425151744809 \alpha^{7}}{18014398509481984} + \frac{1676680829315919 \alpha^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \alpha^{\frac{15}{2}}}{9007199254740992}\]

\(G_{05} = D_{32}\)
\(|S_{05}^1| = 4\)
\(|C_{05}^1| = 2\)
\(|P_{05}| \approx 7.652 \cdot 10^{-205}\)

\(|\{ p \in \mathbb{P} \mid a_{05}(p)=0, p<10^4 \}|=1069\)
Ratio: \(\frac{ 1069 }{ 1228 } \approx \frac{ 28 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{05}(p)=1, p<10^4 \}|=159\)
Ratio: \(\frac{ 159 }{ 1228 } \approx \frac{ 4 }{ 32 }\)

\(\huge a_{06}\)

Same governing field as \(a_{05}\).
Governing field: \[M_{06} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}, \sqrt{- \frac{8282936156772053 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{6}}{562949953421312} - \frac{336382584949535 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{4}}{140737488355328} - \frac{4638634719581101 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{3}}{8796093022208} + \frac{8240373942248553 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{5}}{562949953421312} + \frac{4981425151744809 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{7}}{18014398509481984} + \frac{1676680829315919 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{15}{2}}}{9007199254740992}}\right)\] - or - \[M_{06} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\] and \[\beta = - \frac{8282936156772053 \alpha^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \alpha^{6}}{562949953421312} - \frac{336382584949535 \alpha^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \alpha^{4}}{140737488355328} - \frac{4638634719581101 \alpha^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \alpha^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{\alpha}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \alpha^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \alpha^{3}}{8796093022208} + \frac{8240373942248553 \alpha^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \alpha^{5}}{562949953421312} + \frac{4981425151744809 \alpha^{7}}{18014398509481984} + \frac{1676680829315919 \alpha^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \alpha^{\frac{15}{2}}}{9007199254740992}\]

\(G_{06} = D_{32}\)
\(|S_{06}^1| = 2\)
\(|C_{06}^1| = 1\)
\(|P_{06}| \approx 1.668 \cdot 10^{-129}\)

\(|\{ p \in \mathbb{P} \mid a_{06}(p)=0, p<10^4 \}|=1147\)
Ratio: \(\frac{ 1147 }{ 1228 } \approx \frac{ 30 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{06}(p)=1, p<10^4 \}|=81\)
Ratio: \(\frac{ 81 }{ 1228 } \approx \frac{ 2 }{ 32 }\)

\(\huge a_{07}\)

Same governing field as \(a_{05}\).
Governing field: \[M_{07} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}, \sqrt{- \frac{8282936156772053 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{6}}{562949953421312} - \frac{336382584949535 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{4}}{140737488355328} - \frac{4638634719581101 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{3}}{8796093022208} + \frac{8240373942248553 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{5}}{562949953421312} + \frac{4981425151744809 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{7}}{18014398509481984} + \frac{1676680829315919 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{15}{2}}}{9007199254740992}}\right)\] - or - \[M_{07} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\] and \[\beta = - \frac{8282936156772053 \alpha^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \alpha^{6}}{562949953421312} - \frac{336382584949535 \alpha^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \alpha^{4}}{140737488355328} - \frac{4638634719581101 \alpha^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \alpha^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{\alpha}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \alpha^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \alpha^{3}}{8796093022208} + \frac{8240373942248553 \alpha^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \alpha^{5}}{562949953421312} + \frac{4981425151744809 \alpha^{7}}{18014398509481984} + \frac{1676680829315919 \alpha^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \alpha^{\frac{15}{2}}}{9007199254740992}\]

\(G_{07} = D_{32}\)
\(|S_{07}^1| = 2\)
\(|C_{07}^1| = 1\)
\(|P_{07}| \approx 5.612 \cdot 10^{-126}\)

\(|\{ p \in \mathbb{P} \mid a_{07}(p)=0, p<10^4 \}|=1150\)
Ratio: \(\frac{ 1150 }{ 1228 } \approx \frac{ 30 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{07}(p)=1, p<10^4 \}|=78\)
Ratio: \(\frac{ 78 }{ 1228 } \approx \frac{ 2 }{ 32 }\)

\(\huge a_{08}\)

For \(a_{08}\), it is a little bit special: We found 3 governing fields. This is possible, since governing fields aren't unique. Note, for this case, we did computation only up to 10³, which doesn't give a result as strong as for other \(a_{ij}\) discussed. First possibility

Same governing field as \(a_{05}\).
Governing field: \[M_{08} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}, \sqrt{- \frac{8282936156772053 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{6}}{562949953421312} - \frac{336382584949535 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{4}}{140737488355328} - \frac{4638634719581101 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{3}}{8796093022208} + \frac{8240373942248553 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{5}}{562949953421312} + \frac{4981425151744809 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{7}}{18014398509481984} + \frac{1676680829315919 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{15}{2}}}{9007199254740992}}\right)\] - or - \[M_{08} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\] and \[\beta = - \frac{8282936156772053 \alpha^{\frac{13}{2}}}{1125899906842624} - \frac{1240182980093567 \alpha^{6}}{562949953421312} - \frac{336382584949535 \alpha^{\frac{9}{2}}}{2199023255552} - \frac{6445823996745319 \alpha^{4}}{140737488355328} - \frac{4638634719581101 \alpha^{\frac{5}{2}}}{35184372088832} - \frac{2954723016803317 \alpha^{2}}{70368744177664} - \frac{5142889464378747 \sqrt[4]{2}}{140737488355328} - \frac{4198844765367981 \sqrt{\alpha}}{1125899906842624} + \frac{3450571136356681 \sqrt{2}}{281474976710656} + \frac{6022015868546055 \cdot \sqrt[4]{2}^3}{140737488355328} + \frac{5763554133419461}{70368744177664} + \frac{7633450872164841 \alpha^{\frac{3}{2}}}{281474976710656} + \frac{615248862392953 \alpha^{3}}{8796093022208} + \frac{8240373942248553 \alpha^{\frac{7}{2}}}{35184372088832} + \frac{8030384673908857 \alpha^{5}}{562949953421312} + \frac{4981425151744809 \alpha^{7}}{18014398509481984} + \frac{1676680829315919 \alpha^{\frac{11}{2}}}{35184372088832} + \frac{8299866982438859 \alpha^{\frac{15}{2}}}{9007199254740992}\]

\(G_{08} = D_{32}\)
\(|S_{08}^1| = 1\)
\(|C_{08}^1| = 1\)
\(|P_{08}| \approx 5.800 \cdot 10^{-10}\)

\(|\{ p \in \mathbb{P} \mid a_{08}(p)=0, p<10^4 \}|=162\)
Ratio: \(\frac{ 162 }{ 167 } \approx \frac{ 31 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{08}(p)=1, p<10^4 \}|=5\)
Ratio: \(\frac{ 5 }{ 167 } \approx \frac{ 1 }{ 32 }\)

Second possibility

\(G_{08} = (C_8 \times C_2) : C_2\)
\(|S_{08}^1| = 2\)
\(|C_{08}^1| = 2\)
\(|P_{08}| \approx 1.050 \cdot 10^{-09}\)

\(|\{ p \in \mathbb{P} \mid a_{08}(p)=0, p<10^4 \}|=162\)
Ratio: \(\frac{ 162 }{ 167 } \approx \frac{ 31 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{08}(p)=1, p<10^4 \}|=5\)
Ratio: \(\frac{ 5 }{ 167 } \approx \frac{ 1 }{ 32 }\)

Third possibility

Governing field: \[M_{08} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}, \sqrt{- \frac{8834128592369193 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{15}{2}}}{18014398509481984} - \frac{5764005538121637 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{7}}{18014398509481984} - \frac{6525657542771441 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{11}{2}}}{281474976710656} - \frac{8271178638947881 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{5}}{562949953421312} - \frac{6081987607676345 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{7}{2}}}{70368744177664} - \frac{6598671728459543 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{3}}{140737488355328} - \frac{1208437681968305 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{3}{2}}}{140737488355328} - \frac{183920253122025}{4398046511104} - \frac{6079543723614321 \cdot \sqrt[4]{2}^3}{281474976710656} - \frac{6967068354798893 \sqrt{2}}{1125899906842624} - \frac{1068523219860669 \sqrt{- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}}}{2251799813685248} + \frac{1298004773107437 \sqrt[4]{2}}{70368744177664} + \frac{1958567319150421 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{2}}{281474976710656} + \frac{463099233299539 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{5}{2}}}{17592186044416} + \frac{1458843417828589 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{4}}{35184372088832} + \frac{4742512389557653 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{9}{2}}}{70368744177664} + \frac{5197763479709557 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{6}}{2251799813685248} + \frac{507219318934741 \left(- \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\right)^{\frac{13}{2}}}{140737488355328}}\right)\] - or - \[M_{08} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt[4]{2}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = - \frac{3136435454775881 \sqrt[4]{2}}{562949953421312} + \frac{4208721080340285 \sqrt{2}}{2251799813685248} + \frac{3672578267558083 \cdot \sqrt[4]{2}^3}{562949953421312} + \frac{3582104167901087}{281474976710656}\] and \[\beta = - \frac{8834128592369193 \alpha^{\frac{15}{2}}}{18014398509481984} - \frac{5764005538121637 \alpha^{7}}{18014398509481984} - \frac{6525657542771441 \alpha^{\frac{11}{2}}}{281474976710656} - \frac{8271178638947881 \alpha^{5}}{562949953421312} - \frac{6081987607676345 \alpha^{\frac{7}{2}}}{70368744177664} - \frac{6598671728459543 \alpha^{3}}{140737488355328} - \frac{1208437681968305 \alpha^{\frac{3}{2}}}{140737488355328} - \frac{183920253122025}{4398046511104} - \frac{6079543723614321 \cdot \sqrt[4]{2}^3}{281474976710656} - \frac{6967068354798893 \sqrt{2}}{1125899906842624} - \frac{1068523219860669 \sqrt{\alpha}}{2251799813685248} + \frac{1298004773107437 \sqrt[4]{2}}{70368744177664} + \frac{1958567319150421 \alpha^{2}}{281474976710656} + \frac{463099233299539 \alpha^{\frac{5}{2}}}{17592186044416} + \frac{1458843417828589 \alpha^{4}}{35184372088832} + \frac{4742512389557653 \alpha^{\frac{9}{2}}}{70368744177664} + \frac{5197763479709557 \alpha^{6}}{2251799813685248} + \frac{507219318934741 \alpha^{\frac{13}{2}}}{140737488355328}\]

\(G_{08} = QD_{32}\)
\(|S_{08}^1| = 2\)
\(|C_{08}^1| = 2\)
\(|P_{08}| \approx 5.800 \cdot 10^{-10}\)

\(|\{ p \in \mathbb{P} \mid a_{08}(p)=0, p<10^4 \}|=162\)
Ratio: \(\frac{ 162 }{ 167 } \approx \frac{ 31 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{08}(p)=1, p<10^4 \}|=5\)
Ratio: \(\frac{ 5 }{ 167 } \approx \frac{ 1 }{ 32 }\)

\(\huge a_{11}\)

Governing field: \[M_{11} = \mathbb{Q}\left(\zeta_{16}\right)\]

\(G_{11} = C_4 \times C_2\)
\(|S_{11}^1| = 2\)
\(|C_{11}^1| = 2\)
\(|P_{11}| \approx 1.172 \cdot 10^{-299}\)

\(|\{ p \in \mathbb{P} \mid a_{11}(p)=0, p<10^4 \}|=920\)
Ratio: \(\frac{ 920 }{ 1228 } \approx \frac{ 6 }{ 8 }\)
\(|\{ p \in \mathbb{P} \mid a_{11}(p)=1, p<10^4 \}|=308\)
Ratio: \(\frac{ 308 }{ 1228 } \approx \frac{ 2 }{ 8 }\)

\(\huge a_{10}\)

Governing field: \[M_{10} = \mathbb{Q}\left(\zeta_8\right)\]

\(G_{10} = C_2 \times C_2\)
\(|S_{10}^1| = 1\)
\(|C_{10}^1| = 1\)
\(|P_{10}| \approx 6.202 \cdot 10^{-302}\)

\(|\{ p \in \mathbb{P} \mid a_{10}(p)=0, p<10^4 \}|=917\)
Ratio: \(\frac{ 917 }{ 1228 } \approx \frac{ 3 }{ 4 }\)
\(|\{ p \in \mathbb{P} \mid a_{10}(p)=1, p<10^4 \}|=311\)
Ratio: \(\frac{ 311 }{ 1228 } \approx \frac{ 1 }{ 4 }\)

\(\huge a_{20}\)

Governing field: \[M_{20} = \mathbb{Q}\left(\zeta_8, \sqrt{1+i}\right)\]

\(G_{20} = D_8\)
\(|S_{20}^1| = 1\)
\(|C_{20}^1| = 1\)
\(|P_{20}| \approx 1.469 \cdot 10^{-197}\)

\(|\{ p \in \mathbb{P} \mid a_{20}(p)=0, p<10^4 \}|=1079\)
Ratio: \(\frac{ 1079 }{ 1228 } \approx \frac{ 7 }{ 8 }\)
\(|\{ p \in \mathbb{P} \mid a_{20}(p)=1, p<10^4 \}|=149\)
Ratio: \(\frac{ 149 }{ 1228 } \approx \frac{ 1 }{ 8 }\)

\(\huge a_{30}\)

Governing field: \[M_{30} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}\right)\] - or - \[M_{30} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\]

\(G_{30} = D_{16}\)
\(|S_{30}^1| = 2\)
\(|C_{30}^1| = 1\)
\(|P_{30}| \approx 7.285 \cdot 10^{-205}\)

\(|\{ p \in \mathbb{P} \mid a_{30}(p)=0, p<10^4 \}|=1070\)
Ratio: \(\frac{ 1070 }{ 1228 } \approx \frac{ 14 }{ 16 }\)
\(|\{ p \in \mathbb{P} \mid a_{30}(p)=1, p<10^4 \}|=158\)
Ratio: \(\frac{ 158 }{ 1228 } \approx \frac{ 2 }{ 16 }\)

\(\huge a_{40}\)

Same governing field as \(a_{30}\).
Governing field: \[M_{40} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}\right)\] - or - \[M_{40} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\]

\(G_{40} = D_{16}\)
\(|S_{40}^1| = 1\)
\(|C_{40}^1| = 1\)
\(|P_{40}| \approx 1.401 \cdot 10^{-126}\)

\(|\{ p \in \mathbb{P} \mid a_{40}(p)=0, p<10^4 \}|=1150\)
Ratio: \(\frac{ 1150 }{ 1228 } \approx \frac{ 15 }{ 16 }\)
\(|\{ p \in \mathbb{P} \mid a_{40}(p)=1, p<10^4 \}|=78\)
Ratio: \(\frac{ 78 }{ 1228 } \approx \frac{ 1 }{ 16 }\)

\(\huge a_{50}\)

Governing field: \[M_{50} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}, \sqrt{\frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{11}{2}}}{4} - \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{7}{2}}}{4} - \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}} + \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{3}{2}}}{4} - \frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{15}{2}}}{4}}\right)\] - or - \[M_{50} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\] and \[\beta = \frac{\alpha^{\frac{11}{2}}}{4} - \frac{7 \alpha^{\frac{7}{2}}}{4} - \sqrt{\alpha} + \frac{7 \alpha^{\frac{3}{2}}}{4} - \frac{\alpha^{\frac{15}{2}}}{4}\]

\(G_{50} = D_{32}\)
\(|S_{50}^1| = 4\)
\(|C_{50}^1| = 2\)
\(|P_{50}| \approx 1.242 \cdot 10^{-200}\)

\(|\{ p \in \mathbb{P} \mid a_{50}(p)=0, p<10^4 \}|=1074\)
Ratio: \(\frac{ 1074 }{ 1228 } \approx \frac{ 28 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{50}(p)=1, p<10^4 \}|=154\)
Ratio: \(\frac{ 154 }{ 1228 } \approx \frac{ 4 }{ 32 }\)

\(\huge a_{60}\)

Same governing field as \(a_{50}\).
Governing field: \[M_{60} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}, \sqrt{\frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{11}{2}}}{4} - \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{7}{2}}}{4} - \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}} + \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{3}{2}}}{4} - \frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{15}{2}}}{4}}\right)\] - or - \[M_{60} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\] and \[\beta = \frac{\alpha^{\frac{11}{2}}}{4} - \frac{7 \alpha^{\frac{7}{2}}}{4} - \sqrt{\alpha} + \frac{7 \alpha^{\frac{3}{2}}}{4} - \frac{\alpha^{\frac{15}{2}}}{4}\]

\(G_{60} = D_{32}\)
\(|S_{60}^1| = 2\)
\(|C_{60}^1| = 1\)
\(|P_{60}| \approx 1.892 \cdot 10^{-122}\)

\(|\{ p \in \mathbb{P} \mid a_{60}(p)=0, p<10^4 \}|=1153\)
Ratio: \(\frac{ 1153 }{ 1228 } \approx \frac{ 30 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{60}(p)=1, p<10^4 \}|=75\)
Ratio: \(\frac{ 75 }{ 1228 } \approx \frac{ 2 }{ 32 }\)

\(\huge a_{70}\)

Same governing field as \(a_{50}\).
Governing field: \[M_{70} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}, \sqrt{\frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{11}{2}}}{4} - \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{7}{2}}}{4} - \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}} + \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{3}{2}}}{4} - \frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{15}{2}}}{4}}\right)\] - or - \[M_{70} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\] and \[\beta = \frac{\alpha^{\frac{11}{2}}}{4} - \frac{7 \alpha^{\frac{7}{2}}}{4} - \sqrt{\alpha} + \frac{7 \alpha^{\frac{3}{2}}}{4} - \frac{\alpha^{\frac{15}{2}}}{4}\]

\(G_{70} = D_{32}\)
\(|S_{70}^1| = 2\)
\(|C_{70}^1| = 1\)
\(|P_{70}| \approx 1.892 \cdot 10^{-122}\)

\(|\{ p \in \mathbb{P} \mid a_{70}(p)=0, p<10^4 \}|=1153\)
Ratio: \(\frac{ 1153 }{ 1228 } \approx \frac{ 30 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{70}(p)=1, p<10^4 \}|=75\)
Ratio: \(\frac{ 75 }{ 1228 } \approx \frac{ 2 }{ 32 }\)

\(\huge a_{80}\)

Same governing field as \(a_{50}\).
Governing field: \[M_{80} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}}, \sqrt{\frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{11}{2}}}{4} - \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{7}{2}}}{4} - \sqrt{-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}} + \frac{7 \left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{3}{2}}}{4} - \frac{\left(-4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\right)^{\frac{15}{2}}}{4}}\right)\] - or - \[M_{80} \stackrel{?}{=} \mathbb{Q}\left(\zeta_8, \sqrt{1+i}, \sqrt{\alpha}, \sqrt{\beta}\right)\] where: \[\alpha = -4 - \frac{65 i}{16} - \frac{31 \sqrt{1 + i}}{16} - \frac{5 \left(1 + i\right)^{\frac{5}{2}}}{16} - \frac{3 \left(1 + i\right)^{3}}{16} + \frac{\left(1 + i\right)^{\frac{7}{2}}}{16} + \frac{11 \left(1 + i\right)^{2}}{16} + \frac{27 \left(1 + i\right)^{\frac{3}{2}}}{16}\] and \[\beta = \frac{\alpha^{\frac{11}{2}}}{4} - \frac{7 \alpha^{\frac{7}{2}}}{4} - \sqrt{\alpha} + \frac{7 \alpha^{\frac{3}{2}}}{4} - \frac{\alpha^{\frac{15}{2}}}{4}\]

\(G_{80} = D_{32}\)
\(|S_{80}^1| = 1\)
\(|C_{80}^1| = 1\)
\(|P_{80}| \approx 1.420 \cdot 10^{-08}\)

\(|\{ p \in \mathbb{P} \mid a_{80}(p)=0, p<10^4 \}|=163\)
Ratio: \(\frac{ 163 }{ 167 } \approx \frac{ 31 }{ 32 }\)
\(|\{ p \in \mathbb{P} \mid a_{80}(p)=1, p<10^4 \}|=4\)
Ratio: \(\frac{ 4 }{ 167 } \approx \frac{ 1 }{ 32 }\)