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Properties

Label 2.1.8.20a1.2-1.2.10a
Base 2.1.8.20a1.2
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: 2.1.8.20a1.2
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[\frac{20}{7},\frac{20}{7},\frac{20}{7},\frac{15}{4}]$
Swan slopes: $[9]$
Means: $\langle\frac{9}{2}\rangle$
Rams: $(9)$
Field count: $32$ (complete)
Ambiguity: $2$
Mass: $16$
Absolute Mass: $16$

Diagrams

Varying

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

Indices of inseparability: $[35,26,26,16,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 32

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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
2.1.16.50b1.33 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.34 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.35 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.36 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.37 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.38 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.39 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.40 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.41 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.42 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.43 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.44 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.45 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.46 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.47 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.48 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.49 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.50 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.51 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.52 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.53 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.54 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.55 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.56 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.57 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.58 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.59 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.60 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.61 $x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.62 $x^{16} + 12 x^{12} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.63 $x^{16} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.50b1.64 $x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[35, 26, 26, 16, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
  displayed columns for results