Properties

Label 2.1.8.22d1.22-1.2.10a
Base 2.1.8.22d1.22
Degree \(2\)
e \(2\)
f \(1\)
c \(10\)

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Defining polynomial

$x^{2} + \left(b_{17} \pi^{9} + b_{15} \pi^{8} + b_{13} \pi^{7} + b_{11} \pi^{6} + a_{9} \pi^{5}\right) x + c_{18} \pi^{10} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.22d1.22
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[2,3,\frac{7}{2},4]$
Swan slopes: $[9]$
Means: $\langle\frac{9}{2}\rangle$
Rams: $(9)$
Field count: $10$ (complete)
Ambiguity: $2$
Mass: $16$
Absolute Mass: $2$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $C_4:C_4$ (show 2), $C_2^2\wr C_2$ (show 1), $C_4:D_4$ (show 1), $C_2^2.D_4$ (show 1), $C_2\wr C_2^2$ (show 1), $C_2^3.D_4$ (show 1), $C_4^2:D_4$ (show 2), $C_4^2:D_4$ (show 1)
Hidden Artin slopes: $[\ ]$ (show 2), $[\ ]^{2}$ (show 3), $[2,\frac{7}{2}]^{2}$ (show 3), $[2]^{2}$ (show 2)
Indices of inseparability: $[39,30,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.1.16.54o1.291 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ $C_4^2:D_4$ (as 16T400) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.292 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_4^2:D_4$ (as 16T402) $128$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.293 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_4^2:D_4$ (as 16T400) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.294 $x^{16} + 4 x^{14} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_4:D_4$ (as 16T43) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.295 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_2^2\wr C_2$ (as 16T39) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.296 $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.297 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ $C_4:C_4$ (as 16T8) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.298 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_4:C_4$ (as 16T8) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.299 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_2^3.D_4$ (as 16T177) $64$ $4$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.300 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$ $C_2\wr C_2^2$ (as 16T150) $64$ $4$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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