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Degree Residue characteristic Galois group Galois unramified degree Assoc. Inertia
Ramification index Discriminant exponent Inertia subgroup Galois tame degree Indices
Residue field degree Top slope Wild Visible Wild inertia subgroup
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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
3.6.0.1 $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ $3$ $1$ $6$ $0$ $C_6$ (as 6T1) $[\ ]$ $[\ ]^{6}$ $t^{6} + 2 t^{4} + t^{2} + 2 t + 2$ $x - 3$
3.6.3.1 $x^{6} + 18 x^{2} - 27$ $3$ $2$ $3$ $3$ $C_6$ (as 6T1) $[\ ]$ $[\ ]_{2}^{3}$ $t^{3} + 2 t + 1$ $x^{2} + 3 t$
3.6.3.2 $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ $3$ $2$ $3$ $3$ $C_6$ (as 6T1) $[\ ]$ $[\ ]_{2}^{3}$ $t^{3} + 2 t + 1$ $x^{2} + 3$
3.6.6.1 $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ $3$ $3$ $2$ $6$ $C_3^2:C_4$ (as 6T10) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 3 t x^{2} + \left(3 t + 6\right) x + 3$
3.6.6.2 $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ $3$ $3$ $2$ $6$ $C_3^2:C_4$ (as 6T10) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(6 t + 3\right) x^{2} + 3 t x + 3$
3.6.6.3 $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ $3$ $3$ $2$ $6$ $D_{6}$ (as 6T3) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 3 x + 3$
3.6.6.4 $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ $3$ $3$ $2$ $6$ $D_{6}$ (as 6T3) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(6 t + 6\right) x^{2} + 6 x + 3$
3.6.6.5 $x^{6} + 6 x^{3} + 36 x^{2} + 9$ $3$ $3$ $2$ $6$ $S_3^2$ (as 6T9) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(6 t + 6\right) x + 3$
3.6.6.6 $x^{6} + 3 x + 3$ $3$ $6$ $1$ $6$ $C_3^2:D_4$ (as 6T13) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t + 1$ $x^{6} + 3 x + 3$
3.6.6.7 $x^{6} + 3 x + 6$ $3$ $6$ $1$ $6$ $C_3^2:D_4$ (as 6T13) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t + 1$ $x^{6} + 3 x + 6$
3.6.7.1 $x^{6} + 6 x^{2} + 6$ $3$ $6$ $1$ $7$ $S_3$ (as 6T2) $[3/2]$ $[3/2]_{2}$ $t + 1$ $x^{6} + 6 x^{2} + 6$
3.6.7.2 $x^{6} + 3 x^{2} + 6$ $3$ $6$ $1$ $7$ $D_{6}$ (as 6T3) $[3/2]$ $[3/2]_{2}^{2}$ $t + 1$ $x^{6} + 3 x^{2} + 6$
3.6.7.3 $x^{6} + 3 x^{3} + 6 x^{2} + 6$ $3$ $6$ $1$ $7$ $S_3\times C_3$ (as 6T5) $[3/2]$ $[3/2]_{2}^{3}$ $t + 1$ $x^{6} + 3 x^{3} + 6 x^{2} + 6$
3.6.7.4 $x^{6} + 3 x^{2} + 3$ $3$ $6$ $1$ $7$ $S_3$ (as 6T2) $[3/2]$ $[3/2]_{2}$ $t + 1$ $x^{6} + 3 x^{2} + 3$
3.6.7.5 $x^{6} + 6 x^{2} + 3$ $3$ $6$ $1$ $7$ $D_{6}$ (as 6T3) $[3/2]$ $[3/2]_{2}^{2}$ $t + 1$ $x^{6} + 6 x^{2} + 3$
3.6.7.6 $x^{6} + 3 x^{3} + 3 x^{2} + 3$ $3$ $6$ $1$ $7$ $S_3\times C_3$ (as 6T5) $[3/2]$ $[3/2]_{2}^{3}$ $t + 1$ $x^{6} + 3 x^{3} + 3 x^{2} + 3$
3.6.8.1 $x^{6} - 6 x^{5} + 45 x^{4} + 6 x^{3} - 18 x^{2} + 9$ $3$ $3$ $2$ $8$ $C_3^2:C_4$ (as 6T10) $[2]$ $[2, 2]^{4}$ $t^{2} + 2 t + 2$ $x^{3} + \left(6 t + 3\right) x^{2} + 3$
3.6.8.10 $x^{6} + 9 x^{4} + 42 x^{3} + 441$ $3$ $3$ $2$ $8$ $S_3\times C_3$ (as 6T5) $[2]$ $[2, 2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(3 t + 3\right) x^{2} + 21$
3.6.8.2 $x^{6} - 12 x^{5} + 72 x^{4} + 6 x^{3} + 72 x^{2} + 90$ $3$ $3$ $2$ $8$ $C_3^2:C_4$ (as 6T10) $[2]$ $[2, 2]^{4}$ $t^{2} + 2 t + 2$ $x^{3} + 6 t x^{2} + 9 t + 12$
3.6.8.3 $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ $3$ $3$ $2$ $8$ $C_6$ (as 6T1) $[2]$ $[2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 3$
3.6.8.4 $x^{6} + 18 x^{5} + 114 x^{4} + 344 x^{3} + 732 x^{2} + 744 x + 296$ $3$ $3$ $2$ $8$ $C_6$ (as 6T1) $[2]$ $[2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 12$
3.6.8.5 $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ $3$ $3$ $2$ $8$ $S_3$ (as 6T2) $[2]$ $[2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 3 x^{2} + 3$
3.6.8.6 $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ $3$ $3$ $2$ $8$ $C_6$ (as 6T1) $[2]$ $[2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 21$
3.6.8.7 $x^{6} + 6 x^{5} + 9 x^{4} + 6 x^{3} + 18 x^{2} + 90$ $3$ $3$ $2$ $8$ $S_3\times C_3$ (as 6T5) $[2]$ $[2]^{6}$ $t^{2} + 2 t + 2$ $x^{3} + 3 x^{2} + 9 t + 12$
3.6.8.8 $x^{6} + 9 x^{4} + 24 x^{3} + 144$ $3$ $3$ $2$ $8$ $S_3\times C_3$ (as 6T5) $[2]$ $[2, 2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(3 t + 3\right) x^{2} + 12$
3.6.8.9 $x^{6} + 36 x^{4} + 6 x^{3} + 9$ $3$ $3$ $2$ $8$ $S_3\times C_3$ (as 6T5) $[2]$ $[2, 2]^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(6 t + 6\right) x^{2} + 3$
3.6.9.1 $x^{6} + 3 x^{4} + 15$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 3 x^{4} + 15$
3.6.9.10 $x^{6} + 6 x^{4} + 3$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 6 x^{4} + 3$
3.6.9.11 $x^{6} + 6 x^{4} + 12$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 6 x^{4} + 12$
3.6.9.12 $x^{6} + 3 x^{4} + 3$ $3$ $6$ $1$ $9$ $D_{6}$ (as 6T3) $[2]$ $[2]_{2}^{2}$ $t + 1$ $x^{6} + 3 x^{4} + 3$
3.6.9.13 $x^{6} + 3 x^{5} + 6 x^{4} + 21$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 3 x^{5} + 6 x^{4} + 21$
3.6.9.14 $x^{6} + 3 x^{5} + 6 x^{4} + 12$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 3 x^{5} + 6 x^{4} + 12$
3.6.9.15 $x^{6} + 6 x^{5} + 6 x^{4} + 3$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 6 x^{5} + 6 x^{4} + 3$
3.6.9.16 $x^{6} + 3 x^{5} + 3 x^{4} + 12$ $3$ $6$ $1$ $9$ $S_3^2$ (as 6T9) $[2]$ $[3/2, 2]_{2}^{2}$ $t + 1$ $x^{6} + 3 x^{5} + 3 x^{4} + 12$
3.6.9.2 $x^{6} + 3 x^{4} + 6$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 3 x^{4} + 6$
3.6.9.3 $x^{6} + 3 x^{4} + 24$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 3 x^{4} + 24$
3.6.9.4 $x^{6} + 6 x^{4} + 6$ $3$ $6$ $1$ $9$ $D_{6}$ (as 6T3) $[2]$ $[2]_{2}^{2}$ $t + 1$ $x^{6} + 6 x^{4} + 6$
3.6.9.5 $x^{6} + 3 x^{5} + 3 x^{4} + 6$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 3 x^{5} + 3 x^{4} + 6$
3.6.9.6 $x^{6} + 6 x^{5} + 3 x^{4} + 24$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 6 x^{5} + 3 x^{4} + 24$
3.6.9.7 $x^{6} + 3 x^{5} + 3 x^{4} + 15$ $3$ $6$ $1$ $9$ $S_3\times C_3$ (as 6T5) $[2]$ $[3/2, 2]_{2}$ $t + 1$ $x^{6} + 3 x^{5} + 3 x^{4} + 15$
3.6.9.8 $x^{6} + 6 x^{5} + 6 x^{4} + 15$ $3$ $6$ $1$ $9$ $S_3^2$ (as 6T9) $[2]$ $[3/2, 2]_{2}^{2}$ $t + 1$ $x^{6} + 6 x^{5} + 6 x^{4} + 15$
3.6.9.9 $x^{6} + 6 x^{4} + 21$ $3$ $6$ $1$ $9$ $C_6$ (as 6T1) $[2]$ $[2]_{2}$ $t + 1$ $x^{6} + 6 x^{4} + 21$
3.6.10.1 $x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$ $3$ $3$ $2$ $10$ $D_{6}$ (as 6T3) $[5/2]$ $[5/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(9 t + 18\right) x + 9 t + 12$
3.6.10.10 $x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 15$ $3$ $6$ $1$ $10$ $C_3^2:D_4$ (as 6T13) $[9/4]$ $[9/4, 9/4]_{4}^{2}$ $t + 1$ $x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 15$
3.6.10.11 $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ $3$ $6$ $1$ $10$ $C_3^2:D_4$ (as 6T13) $[9/4]$ $[9/4, 9/4]_{4}^{2}$ $t + 1$ $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$
3.6.10.12 $x^{6} + 3 x^{5} + 3$ $3$ $6$ $1$ $10$ $C_3^2:D_4$ (as 6T13) $[9/4]$ $[9/4, 9/4]_{4}^{2}$ $t + 1$ $x^{6} + 3 x^{5} + 3$
3.6.10.2 $x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$ $3$ $3$ $2$ $10$ $D_{6}$ (as 6T3) $[5/2]$ $[5/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 18 t x + 18 t + 12$
3.6.10.3 $x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ $3$ $3$ $2$ $10$ $D_{6}$ (as 6T3) $[5/2]$ $[5/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 9 x + 12$
3.6.10.4 $x^{6} + 6 x^{3} + 81 x^{2} + 9$ $3$ $3$ $2$ $10$ $S_3^2$ (as 6T9) $[5/2]$ $[3/2, 5/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + \left(9 t + 9\right) x + 3$
3.6.10.5 $x^{6} - 12 x^{3} + 360$ $3$ $3$ $2$ $10$ $S_3^2$ (as 6T9) $[5/2]$ $[3/2, 5/2]_{2}^{2}$ $t^{2} + 2 t + 2$ $x^{3} + 18 t + 12$
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