| 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 |
| 2.4.4.44a1.44 |
$16$ |
$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, \frac{7}{2}, 4]^{4}$ |
$[1,1,2,\frac{5}{2},3]^{4}$ |
$[2,2,\frac{7}{2}]$ |
$[1,1,\frac{5}{2}]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + 8 t^{3} x^{3} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.52 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, \frac{7}{2}, 4]^{4}$ |
$[1,1,2,\frac{5}{2},3]^{4}$ |
$[2,2,\frac{7}{2}]$ |
$[1,1,\frac{5}{2}]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8\right) x^{3} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.55 |
$16$ |
$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + 8 t^{3} x^{3} + 8 x + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.56 |
$16$ |
$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + 8 t^{3} x^{3} + 8 x + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.63 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + 8 x + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.64 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8\right) x^{3} + 8 x + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1149 |
$16$ |
$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1150 |
$16$ |
$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1157 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1158 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 2, 3, 4]^{4}$ |
$[1,1,1,2,3]^{4}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 16 t^{3} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1197 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, \frac{7}{2}, 4]^{4}$ |
$[1,1,2,\frac{5}{2},3]^{4}$ |
$[2,2,\frac{7}{2}]$ |
$[1,1,\frac{5}{2}]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 4 x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.4.4.44a1.1201 |
$16$ |
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ |
$2$ |
$4$ |
$4$ |
$44$ |
$(C_2^3\times C_4):C_4$ (as 16T292) |
$4$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[2, 2, 3, \frac{7}{2}, 4]^{4}$ |
$[1,1,2,\frac{5}{2},3]^{4}$ |
$[2,2,\frac{7}{2}]$ |
$[1,1,\frac{5}{2}]$ |
$4$ |
$t^{4} + t + 1$ |
$x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + 4 x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
