| Label |
$n$ |
Polynomial |
$p$ |
$f$ |
$e$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible Artin slopes |
Visible Swan slopes |
Artin slope content |
Swan slope content |
Hidden Artin slopes |
Hidden Swan slopes |
$\#\Aut(K/\Q_p)$ |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
Resid. Poly |
Jump Set |
| 2.4.2.8a5.1 |
$8$ |
$( x^{4} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 ) + 2$ |
$2$ |
$4$ |
$2$ |
$8$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[2]$ |
$[1]$ |
$[2, 2, 2]^{4}$ |
$[1,1,1]^{4}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 4 t^{2} + 2$ |
$[1, 0]$ |
$[1]$ |
$z + (t^3 + t + 1)$ |
$[1, 3]$ |
| 2.4.2.12a1.6 |
$8$ |
$( x^{4} + x + 1 )^{2} + 4 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[3]$ |
$[2]$ |
$[2, 2, 3]^{4}$ |
$[1,1,2]^{4}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t^{2} + 4\right) x + 8 t^{3} + 2$ |
$[2, 0]$ |
$[1]$ |
$z + 1$ |
$[1, 3]$ |
| 2.2.4.12a6.2 |
$8$ |
$( x^{2} + x + 1 )^{4} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{3} + 2 x ( x^{2} + x + 1 )^{2} + 4 x + 2$ |
$2$ |
$2$ |
$4$ |
$12$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[2, 2]$ |
$[1,1]$ |
$[2, 2, 2]^{4}$ |
$[1,1,1]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{3} + 2 t x^{2} + 4 t + 2$ |
$[3, 2, 0]$ |
$[2]$ |
$z^3 + t z + t$ |
$[1, 3, 8]$ |
| 2.2.4.16b1.6 |
$8$ |
$( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{2} + 4 x + 2$ |
$2$ |
$2$ |
$4$ |
$16$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[2, 3]$ |
$[1,2]$ |
$[2, 2, 3]^{4}$ |
$[1,1,2]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{2} + 4 x + 4 t + 2$ |
$[5, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 8]$ |
| 2.2.4.22a1.23 |
$8$ |
$( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ |
$2$ |
$2$ |
$4$ |
$22$ |
$(C_8:C_2):C_2$ (as 8T16) |
$2$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, 4]^{2}$ |
$[1,1,2,3]^{2}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 8 t + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.2.4.22a1.24 |
$8$ |
$( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 24 x + 2$ |
$2$ |
$2$ |
$4$ |
$22$ |
$(C_8:C_2):C_2$ (as 8T16) |
$2$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, 4]^{2}$ |
$[1,1,2,3]^{2}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 24 t + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.2.4.22a1.63 |
$8$ |
$( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ |
$2$ |
$2$ |
$4$ |
$22$ |
$(C_8:C_2):C_2$ (as 8T16) |
$2$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, 4]^{2}$ |
$[1,1,2,3]^{2}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 8 t x^{3} + 4 x^{2} + 8 x + 8 t + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.2.4.22a1.64 |
$8$ |
$( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 24 x + 2$ |
$2$ |
$2$ |
$4$ |
$22$ |
$(C_8:C_2):C_2$ (as 8T16) |
$2$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, 4]^{2}$ |
$[1,1,2,3]^{2}$ |
$[2,2]$ |
$[1,1]$ |
$2$ |
$t^{2} + t + 1$ |
$x^{4} + 8 t x^{3} + 4 x^{2} + 8 x + 24 t + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.8.31a1.145 |
$8$ |
$x^{8} + 4 x^{4} + 2$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 2$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.146 |
$8$ |
$x^{8} + 4 x^{4} + 34$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 34$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.155 |
$8$ |
$x^{8} + 4 x^{4} + 18$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 18$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.156 |
$8$ |
$x^{8} + 4 x^{4} + 50$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 50$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.171 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 2$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 2$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.172 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 34$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 34$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.183 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 18$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 18$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.184 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 50$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 50$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.221 |
$8$ |
$x^{8} + 4 x^{4} + 10$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 10$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.222 |
$8$ |
$x^{8} + 16 x^{5} + 4 x^{4} + 10$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 16 x^{5} + 4 x^{4} + 10$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.233 |
$8$ |
$x^{8} + 4 x^{4} + 26$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{4} + 26$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.234 |
$8$ |
$x^{8} + 16 x^{5} + 4 x^{4} + 26$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 16 x^{5} + 4 x^{4} + 26$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.249 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 10$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 10$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.250 |
$8$ |
$x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 10$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 10$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.259 |
$8$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 26$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 26$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.31a1.260 |
$8$ |
$x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 26$ |
$2$ |
$1$ |
$8$ |
$31$ |
$(C_8:C_2):C_2$ (as 8T16) |
$1$ |
$1$ |
$[3, 4, 5]$ |
$[2,3,4]$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[2,\frac{7}{2}]$ |
$[1,\frac{5}{2}]$ |
$2$ |
$t + 1$ |
$x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 26$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
