↑

Learn more

Refine search

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
Sort order Select

Results (30 matches)

  displayed columns for results
Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
2.8.22.3 $x^{8} + 8 x^{7} + 60 x^{6} + 136 x^{5} + 256 x^{4} + 240 x^{3} + 104 x^{2} + 112 x + 76$ $2$ $4$ $2$ $22$ $Q_8$ (as 8T5) $[3, 4]$ $[3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + \left(12 t + 4\right) x^{2} + \left(8 t + 8\right) x + 4 t + 10$
2.8.22.4 $x^{8} + 16 x^{7} + 68 x^{6} + 24 x^{5} + 48 x^{4} + 128 x^{3} + 568 x^{2} + 336 x + 588$ $2$ $4$ $2$ $22$ $Q_8$ (as 8T5) $[3, 4]$ $[3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + 8 x^{3} + \left(12 t + 8\right) x^{2} + 8 t x + 28 t + 14$
2.8.24.12 $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 14$ $2$ $8$ $1$ $24$ $Q_8$ (as 8T5) $[2, 3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 14$
2.8.24.16 $x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $2$ $8$ $1$ $24$ $Q_8$ (as 8T5) $[2, 3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 6$
2.8.24.4 $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30$ $2$ $8$ $1$ $24$ $Q_8$ (as 8T5) $[2, 3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30$
2.8.24.8 $x^{8} + 8 x^{7} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 6$ $2$ $8$ $1$ $24$ $Q_8$ (as 8T5) $[2, 3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{8} + 8 x^{7} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 6$
3.8.6.1 $x^{8} + 9$ $3$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 2 t + 2$ $x^{4} + 3 t + 3$
7.8.6.1 $x^{8} + 14 x^{4} - 245$ $7$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 6 t + 3$ $x^{4} + 7 t + 28$
11.8.6.1 $x^{8} - 110 x^{4} - 16819$ $11$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 7 t + 2$ $x^{4} + 44 t + 99$
19.8.6.1 $x^{8} + 304 x^{4} - 5415$ $19$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 18 t + 2$ $x^{4} + 19 t + 323$
23.8.6.1 $x^{8} - 138 x^{4} - 217948$ $23$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 21 t + 5$ $x^{4} + 46 t + 414$
31.8.6.2 $x^{8} - 62 x^{4} - 795708$ $31$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 29 t + 3$ $x^{4} + 62 t + 868$
43.8.6.2 $x^{8} + 1634 x^{4} - 142373$ $43$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 42 t + 3$ $x^{4} + 43 t + 1720$
47.8.6.2 $x^{8} - 282 x^{4} - 4409164$ $47$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 45 t + 5$ $x^{4} + 94 t + 1974$
59.8.6.2 $x^{8} + 3304 x^{4} - 191455$ $59$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 58 t + 2$ $x^{4} + 59 t + 3363$
67.8.6.2 $x^{8} - 8174 x^{4} - 54420147$ $67$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 63 t + 2$ $x^{4} + 268 t + 4355$
71.8.6.2 $x^{8} - 710 x^{4} - 23733028$ $71$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 69 t + 7$ $x^{4} + 142 t + 4544$
79.8.6.2 $x^{8} + 5846 x^{4} - 929909$ $79$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 78 t + 3$ $x^{4} + 79 t + 6004$
83.8.6.2 $x^{8} + 6640 x^{4} - 544231$ $83$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 82 t + 2$ $x^{4} + 83 t + 6723$
103.8.6.2 $x^{8} + 9682 x^{4} - 4105683$ $103$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 102 t + 5$ $x^{4} + 103 t + 10094$
107.8.6.2 $x^{8} - 21614 x^{4} - 368692147$ $107$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 103 t + 2$ $x^{4} + 428 t + 11235$
127.8.6.2 $x^{8} + 15494 x^{4} - 3951605$ $127$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 126 t + 3$ $x^{4} + 127 t + 15748$
131.8.6.2 $x^{8} - 32750 x^{4} - 838469299$ $131$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 127 t + 2$ $x^{4} + 524 t + 16899$
139.8.6.2 $x^{8} + 18904 x^{4} - 2608335$ $139$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 138 t + 2$ $x^{4} + 139 t + 19043$
151.8.6.2 $x^{8} - 1208 x^{4} - 505292961$ $151$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 149 t + 6$ $x^{4} + 302 t + 21895$
163.8.6.2 $x^{8} - 51182 x^{4} - 2031014067$ $163$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 159 t + 2$ $x^{4} + 652 t + 26243$
167.8.6.2 $x^{8} + 26386 x^{4} - 17932627$ $167$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 166 t + 5$ $x^{4} + 167 t + 27054$
179.8.6.2 $x^{8} - 152150 x^{4} - 5821240921$ $179$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 172 t + 2$ $x^{4} + 1253 t + 31683$
191.8.6.2 $x^{8} + 29414 x^{4} - 112252037$ $191$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 190 t + 19$ $x^{4} + 191 t + 32852$
199.8.6.2 $x^{8} - 152434 x^{4} - 7462570844$ $199$ $4$ $2$ $6$ $Q_8$ (as 8T5) $[\ ]$ $[\ ]_{4}^{2}$ $t^{2} + 193 t + 3$ $x^{4} + 1194 t + 39004$
  displayed columns for results