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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
2.12.12.18 $x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$ $2$ $2$ $6$ $12$ $D_4 \times C_3$ (as 12T14) $[2]$ $[2, 2]^{6}$ $t^{6} + t^{4} + t^{3} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t^{2} + 2 t\right) x + 4 t^{5} + 4 t^{4} + 4 t^{3} + 2$
2.12.12.24 $x^{12} - 10 x^{11} + 72 x^{10} - 292 x^{9} + 1028 x^{8} - 2144 x^{7} + 5280 x^{6} - 5568 x^{5} + 12400 x^{4} - 12384 x^{3} + 24832 x^{2} - 14784 x + 11200$ $2$ $2$ $6$ $12$ $D_4 \times C_3$ (as 12T14) $[2]$ $[2, 2]^{6}$ $t^{6} + t^{4} + t^{3} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t^{2} + 2 t\right) x + 4 t^{4} + 4 t^{3} + 2$
2.12.18.12 $x^{12} - 20 x^{11} + 316 x^{10} - 3064 x^{9} + 21548 x^{8} - 105376 x^{7} + 379296 x^{6} - 877760 x^{5} + 1334896 x^{4} + 373696 x^{3} - 1816128 x^{2} + 8702592 x + 8239936$ $2$ $2$ $6$ $18$ $D_4 \times C_3$ (as 12T14) $[3]$ $[2, 3]^{6}$ $t^{6} + t^{4} + t^{3} + t + 1$ $x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 8 t^{5} + 8 t^{4} + 8 t^{3} + 8 t^{2} + 8 t + 14$
2.12.18.25 $x^{12} - 20 x^{11} + 292 x^{10} - 2840 x^{9} + 18972 x^{8} - 90272 x^{7} + 343776 x^{6} - 825280 x^{5} + 1529840 x^{4} - 1040448 x^{3} + 101952 x^{2} + 1661056 x + 935488$ $2$ $2$ $6$ $18$ $D_4 \times C_3$ (as 12T14) $[3]$ $[2, 3]^{6}$ $t^{6} + t^{4} + t^{3} + t + 1$ $x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 8 t^{5} + 8 t^{4} + 8 t^{2} + 8 t + 6$
2.12.18.63 $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 28 x^{4} + 56 x^{3} - 24$ $2$ $4$ $3$ $18$ $D_4 \times C_3$ (as 12T14) $[2, 2]$ $[2, 2]^{6}$ $t^{3} + t + 1$ $x^{4} + 2 x^{3} + 4 t + 2$
2.12.18.65 $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 10 x^{8} + 40 x^{7} + 40 x^{6} + 92 x^{4} + 184 x^{3} + 344$ $2$ $4$ $3$ $18$ $D_4 \times C_3$ (as 12T14) $[2, 2]$ $[2, 2]^{6}$ $t^{3} + t + 1$ $x^{4} + 2 x^{3} + 4 t^{2} + 4 t + 6$
2.12.24.313 $x^{12} + 12 x^{11} + 66 x^{10} + 220 x^{9} + 510 x^{8} + 912 x^{7} + 1344 x^{6} + 1632 x^{5} + 1684 x^{4} + 1616 x^{3} + 1320 x^{2} + 688 x + 88$ $2$ $4$ $3$ $24$ $D_4 \times C_3$ (as 12T14) $[2, 3]$ $[2, 3]^{6}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 8 t + 6$
2.12.24.78 $x^{12} + 18 x^{10} + 12 x^{9} + 98 x^{8} + 144 x^{7} + 144 x^{6} + 352 x^{5} - 60 x^{4} - 416 x^{3} - 88 x^{2} + 48 x + 584$ $2$ $4$ $3$ $24$ $D_4 \times C_3$ (as 12T14) $[2, 3]$ $[2, 3]^{6}$ $t^{3} + t + 1$ $x^{4} + 6 x^{2} + 4 x + 8 t^{2} + 2$
2.12.27.11 $x^{12} - 2 x^{10} + 8 x^{9} + 314 x^{8} + 256 x^{7} - 1312 x^{6} + 2688 x^{5} + 5444 x^{4} + 3072 x^{3} - 200 x^{2} + 800 x + 1000$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(4 t^{2} + 12 t + 2\right) x^{2} + \left(8 t^{2} + 8\right) x + 10$
2.12.27.182 $x^{12} - 2 x^{10} + 24 x^{9} + 146 x^{8} - 32 x^{7} + 128 x^{6} + 1088 x^{5} + 516 x^{4} + 1984 x^{3} + 120 x^{2} + 96 x + 584$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(4 t^{2} + 8 t + 2\right) x^{2} + 8 x + 8 t^{2} + 2$
2.12.27.187 $x^{12} + 12 x^{11} + 86 x^{10} + 272 x^{9} + 870 x^{8} + 1344 x^{7} + 1632 x^{6} + 3936 x^{5} + 6276 x^{4} + 1584 x^{3} + 7896 x^{2} + 896 x + 3032$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(4 t + 4\right) x^{3} + \left(4 t^{2} + 10\right) x^{2} + 8 t^{2} x + 12 t + 14$
2.12.27.193 $x^{12} - 8 x^{11} + 22 x^{10} + 710 x^{8} + 1920 x^{7} + 2224 x^{6} + 1280 x^{5} + 3588 x^{4} + 416 x^{3} + 3128 x^{2} + 408$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 t^{2} x^{3} + \left(12 t^{2} + 10\right) x^{2} + 4 t^{2} + 8 t + 6$
2.12.27.292 $x^{12} - 8 x^{11} + 14 x^{10} + 64 x^{9} + 230 x^{8} + 256 x^{7} + 304 x^{6} + 384 x^{5} + 452 x^{4} + 416 x^{3} + 792 x^{2} + 408$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 t^{2} x^{3} + \left(4 t^{2} + 2\right) x^{2} + 4 t^{2} + 8 t + 6$
2.12.27.301 $x^{12} + 12 x^{11} + 70 x^{10} + 288 x^{9} + 974 x^{8} + 1952 x^{7} + 4864 x^{6} + 4928 x^{5} + 2692 x^{4} + 1072 x^{3} + 1176 x^{2} - 2304 x - 648$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(4 t + 4\right) x^{3} + \left(12 t^{2} + 4 t + 10\right) x^{2} + 8 t x + 12 t + 6$
2.12.27.303 $x^{12} + 6 x^{10} + 8 x^{9} + 570 x^{8} + 256 x^{7} + 4192 x^{6} + 8832 x^{5} + 9540 x^{4} + 3072 x^{3} + 600 x^{2} + 800 x + 1000$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(12 t^{2} + 12 t + 10\right) x^{2} + \left(8 t^{2} + 8\right) x + 10$
2.12.27.304 $x^{12} - 2 x^{10} + 74 x^{8} + 288 x^{6} + 1092 x^{4} + 2232 x^{2} + 2088$ $2$ $4$ $3$ $27$ $D_4 \times C_3$ (as 12T14) $[2, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(4 t^{2} + 4 t + 2\right) x^{2} + 8 t^{2} + 8 t + 10$
2.12.30.173 $x^{12} + 12 x^{11} + 48 x^{10} + 88 x^{9} + 210 x^{8} + 528 x^{7} + 544 x^{6} + 1440 x^{5} + 1836 x^{4} + 1200 x^{3} + 1024 x^{2} + 608 x + 88$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + \left(8 t + 8\right) x + 8 t + 6$
2.12.30.176 $x^{12} + 12 x^{11} + 60 x^{10} + 184 x^{9} + 754 x^{8} + 2128 x^{7} + 6496 x^{6} + 7712 x^{5} + 17276 x^{4} + 10736 x^{3} + 18000 x^{2} + 5088 x + 6392$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + \left(12 t^{2} + 8 t + 12\right) x^{2} + 8 x + 12 t^{2} + 12 t + 14$
2.12.30.225 $x^{12} + 12 x^{11} + 48 x^{10} + 64 x^{9} + 82 x^{8} + 528 x^{7} + 480 x^{6} - 640 x^{5} - 2004 x^{4} - 2128 x^{3} - 1920 x^{2} - 768 x + 88$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + 8 t x^{2} + 8 t x + 8 t + 6$
2.12.30.256 $x^{12} + 12 x^{11} + 88 x^{10} + 396 x^{9} + 1308 x^{8} + 3036 x^{7} + 6126 x^{6} + 8928 x^{5} + 9281 x^{4} - 1260 x^{3} - 2242 x^{2} - 6960 x + 3767$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + 12 x^{2} + 14$
2.12.30.290 $x^{12} + 12 x^{11} + 52 x^{10} + 108 x^{9} + 228 x^{8} + 732 x^{7} + 990 x^{6} - 792 x^{5} - 2683 x^{4} - 1500 x^{3} - 1006 x^{2} - 1872 x + 6467$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + 14$
2.12.30.293 $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 322 x^{8} + 720 x^{7} + 944 x^{6} + 448 x^{5} - 948 x^{4} - 1616 x^{3} - 1232 x^{2} - 768 x + 88$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + \left(8 t + 4\right) x^{2} + 8 t x + 8 t + 6$
2.12.30.304 $x^{12} + 12 x^{11} + 84 x^{10} + 352 x^{9} + 1026 x^{8} + 1872 x^{7} + 2512 x^{6} + 2112 x^{5} + 4044 x^{4} + 1712 x^{3} + 912 x^{2} - 768 x + 88$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + 12 x^{2} + 8 t x + 8 t + 6$
2.12.30.68 $x^{12} + 12 x^{11} + 52 x^{10} + 80 x^{9} - 42 x^{8} - 48 x^{7} + 960 x^{6} + 2304 x^{5} + 2076 x^{4} + 2224 x^{3} + 3376 x^{2} + 1216 x + 936$ $2$ $4$ $3$ $30$ $D_4 \times C_3$ (as 12T14) $[3, 7/2]$ $[2, 3, 7/2]^{3}$ $t^{3} + t + 1$ $x^{4} + 4 x^{3} + \left(4 t^{2} + 4\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 4 t^{2} + 4 t + 10$
2.12.33.160 $x^{12} + 68 x^{10} - 264 x^{9} - 926 x^{8} - 768 x^{7} - 2688 x^{6} - 64 x^{5} + 6012 x^{4} + 10496 x^{3} + 26352 x^{2} + 27680 x + 39224$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 8 t + 4\right) x^{2} + \left(8 t + 8\right) x + 12 t^{2} + 28 t + 30$
2.12.33.193 $x^{12} + 24 x^{10} + 8 x^{9} + 366 x^{8} + 608 x^{7} + 1680 x^{6} + 3392 x^{5} + 6716 x^{4} + 12864 x^{3} + 18432 x^{2} + 18144 x + 11016$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[3, 4]^{6}$ $t^{3} + t + 1$ $x^{4} + \left(12 t + 8\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 12 t^{2} + 24 t + 18$
2.12.33.209 $x^{12} + 8 x^{11} + 536 x^{9} + 174 x^{8} + 736 x^{7} + 6336 x^{6} + 1248 x^{5} + 8236 x^{4} + 30112 x^{3} + 2944 x^{2} + 18784 x + 31656$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[3, 4]^{6}$ $t^{3} + t + 1$ $x^{4} + \left(8 t^{2} + 8\right) x^{3} + 8 x + 16 t^{2} + 24 t + 26$
2.12.33.243 $x^{12} + 8 x^{10} + 294 x^{8} + 2848 x^{6} + 9804 x^{4} + 14880 x^{2} + 7688$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t + 8\right) x^{2} + 8 t^{2} + 24 t + 18$
2.12.33.307 $x^{12} + 8 x^{10} + 214 x^{8} + 2720 x^{6} + 10892 x^{4} + 11552 x^{2} - 8248$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t + 8\right) x^{2} + 24 t^{2} + 24 t + 2$
2.12.33.320 $x^{12} + 12 x^{10} + 24 x^{9} + 86 x^{8} + 192 x^{7} + 560 x^{6} + 480 x^{5} + 812 x^{4} + 4992 x^{3} - 8528 x^{2} + 14944 x - 12024$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(8 t + 4\right) x^{2} + 8 x + 16 t^{2} + 24 t + 2$
2.12.33.332 $x^{12} + 68 x^{10} - 264 x^{9} - 1006 x^{8} - 384 x^{7} - 3584 x^{6} - 4032 x^{5} + 5500 x^{4} - 3072 x^{3} + 15344 x^{2} + 14112 x + 24696$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 8 t + 4\right) x^{2} + \left(8 t + 8\right) x + 28 t^{2} + 28 t + 14$
2.12.33.350 $x^{12} + 24 x^{11} + 196 x^{10} + 596 x^{9} + 696 x^{8} + 2892 x^{7} + 4866 x^{6} + 804 x^{5} - 4735 x^{4} - 10536 x^{3} + 2642 x^{2} - 6140 x + 11903$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + 8 x^{3} + 14$
2.12.33.356 $x^{12} + 24 x^{11} + 188 x^{10} + 456 x^{9} + 326 x^{8} + 5056 x^{7} + 7872 x^{6} + 9024 x^{5} + 23132 x^{4} + 30368 x^{3} + 34192 x^{2} + 19168 x + 8936$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + 8 x^{3} + \left(8 t^{2} + 12 t + 4\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 20 t^{2} + 26$
2.12.33.360 $x^{12} + 64 x^{10} - 360 x^{9} - 1442 x^{8} - 640 x^{7} - 2880 x^{6} - 6944 x^{5} - 20 x^{4} - 6400 x^{3} - 8064 x^{2} + 7008 x - 7064$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[3, 4]^{6}$ $t^{3} + t + 1$ $x^{4} + 8 t x^{3} + 8 t x^{2} + 8 x + 24 t + 10$
2.12.33.365 $x^{12} - 16 x^{11} + 64 x^{10} + 808 x^{9} - 578 x^{8} + 2528 x^{7} + 1520 x^{6} - 1216 x^{5} + 796 x^{4} - 1024 x^{3} - 4704 x^{2} + 2400 x - 2808$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[3, 4]^{6}$ $t^{3} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 12 t x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 12 t^{2} + 16 t + 2$
2.12.33.48 $x^{12} + 8 x^{11} + 276 x^{10} + 1960 x^{9} + 6182 x^{8} + 10848 x^{7} + 21376 x^{6} + 23296 x^{5} + 28892 x^{4} + 21024 x^{3} + 19632 x^{2} + 6624 x + 12648$ $2$ $4$ $3$ $33$ $D_4 \times C_3$ (as 12T14) $[3, 4]$ $[2, 3, 4]^{3}$ $t^{3} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t + 8\right) x^{3} + \left(8 t^{2} + 4 t + 12\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 4 t^{2} + 26$
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