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Properties

Label 3.1.12.19a
Base 3.1.1.0a1.1
Degree \(12\)
e \(12\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{12} + 3 b_{11} x^{11} + 3 b_{10} x^{10} + 3 a_{8} x^{8} + 3 d_{0} + 9 c_{12}$

Invariants

Residue field characteristic: $3$
Degree: $12$
Base field: $\Q_{3}$
Ramification index $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{2}{3}\rangle$
Rams: $(4)$
Field count: $48$ (complete)
Ambiguity: $6$
Mass: $18$
Absolute Mass: $18$

Diagrams

Varying

Indices of inseparability: $[8,0]$
Associated inertia: $[2,1]$ (show 36), $[2,2]$ (show 12)
Jump Set: undefined (show 24), $[2,6]$ (show 1), $[2,14]$ (show 6), $[2,16]$ (show 12), $[2,17]$ (show 3), $[2,18]$ (show 2)

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 48

  displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
3.1.12.19a1.1 2 $x^{12} + 3 x^{8} + 3$ $D_{12}$ (as 12T12) $24$ $2$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.2 2 $x^{12} + 3 x^{11} + 3 x^{8} + 3$ $C_3^2:D_{12}$ (as 12T118) $216$ $1$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.3 4 $x^{12} + 3 x^{10} + 3 x^{8} + 3$ $C_3:D_{12}$ (as 12T38) $72$ $2$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.4 4 $x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{8} + 3$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.5 4 $x^{12} + 6 x^{10} + 3 x^{8} + 3$ $C_3:D_{12}$ (as 12T38) $72$ $2$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.6 4 $x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{8} + 3$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.7 2 $x^{12} + 6 x^{8} + 6$ $D_{12}$ (as 12T12) $24$ $2$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a1.8 2 $x^{12} + 3 x^{11} + 6 x^{8} + 6$ $C_3^2:D_{12}$ (as 12T118) $216$ $1$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a1.9 4 $x^{12} + 3 x^{10} + 6 x^{8} + 6$ $C_3:D_{12}$ (as 12T38) $72$ $2$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a1.10 4 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 6$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a1.11 4 $x^{12} + 6 x^{10} + 6 x^{8} + 6$ $C_3:D_{12}$ (as 12T38) $72$ $2$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a1.12 4 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 6$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ undefined
3.1.12.19a2.1 6 $x^{12} + 6 x^{8} + 3$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 6]$
3.1.12.19a2.2 6 $x^{12} + 6 x^{8} + 12$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 18]$
3.1.12.19a2.3 6 $x^{12} + 6 x^{8} + 21$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 18]$
3.1.12.19a2.4 6 $x^{12} + 3 x^{11} + 6 x^{8} + 3$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 17]$
3.1.12.19a2.5 6 $x^{12} + 3 x^{11} + 6 x^{8} + 12$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 17]$
3.1.12.19a2.6 6 $x^{12} + 3 x^{11} + 6 x^{8} + 21$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 17]$
3.1.12.19a2.7 12 $x^{12} + 3 x^{10} + 6 x^{8} + 3$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.8 12 $x^{12} + 3 x^{10} + 6 x^{8} + 12$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.9 12 $x^{12} + 3 x^{10} + 6 x^{8} + 21$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.10 12 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 3$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.11 12 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 12$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.12 12 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 21$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.13 12 $x^{12} + 6 x^{10} + 6 x^{8} + 3$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.14 12 $x^{12} + 6 x^{10} + 6 x^{8} + 12$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.15 12 $x^{12} + 6 x^{10} + 6 x^{8} + 21$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.16 12 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 3$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.17 12 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 12$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.18 12 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 21$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.19 6 $x^{12} + 3 x^{8} + 6$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.20 6 $x^{12} + 3 x^{8} + 15$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.21 6 $x^{12} + 3 x^{8} + 24$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2]_{4}^{2}$ $[1]_{4}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.22 6 $x^{12} + 3 x^{11} + 3 x^{8} + 6$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.23 6 $x^{12} + 3 x^{11} + 3 x^{8} + 15$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.24 6 $x^{12} + 3 x^{11} + 3 x^{8} + 24$ $S_3^2:C_6$ (as 12T121) $216$ $3$ $[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]^{2}$ $[\frac{1}{4},\frac{1}{4}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.25 12 $x^{12} + 3 x^{10} + 3 x^{8} + 6$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.26 12 $x^{12} + 3 x^{10} + 3 x^{8} + 15$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.27 12 $x^{12} + 3 x^{10} + 3 x^{8} + 24$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.28 12 $x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{8} + 6$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.29 12 $x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{8} + 15$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.30 12 $x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{8} + 24$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.31 12 $x^{12} + 6 x^{10} + 3 x^{8} + 6$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.32 12 $x^{12} + 6 x^{10} + 3 x^{8} + 15$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.33 12 $x^{12} + 6 x^{10} + 3 x^{8} + 24$ $C_6\wr C_2$ (as 12T42) $72$ $6$ $[\frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{2},1]_{4}^{2}$ $[\frac{3}{2}]^{2}$ $[\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.34 12 $x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{8} + 6$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.35 12 $x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{8} + 15$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
3.1.12.19a2.36 12 $x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{8} + 24$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ undefined
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