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Properties

Label 2.1.8.28b1.10-1.2.10a
Base 2.1.8.28b1.10
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: 2.1.8.28b1.10
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{19}{4}]$
Swan slopes: $[9]$
Means: $\langle\frac{9}{2}\rangle$
Rams: $(9)$
Field count: $20$ (complete)
Ambiguity: $2$
Mass: $16$
Absolute Mass: $8$

Diagrams

Varying

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

Galois group: $C_4^2:C_4$ (show 2), $C_2^4.D_4$ (show 2), $C_2^4.(C_4\times D_4)$ (show 4), $C_2^5.(C_2\times D_4)$ (show 4), $C_2^6.D_4$ (show 8)
Hidden Artin slopes: $[2,3,\frac{7}{2},4]^{2}$ (show 12), $[2,\frac{7}{2}]$ (show 2), $[2,\frac{7}{2}]^{2}$ (show 2), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 4)
Indices of inseparability: $[51,42,32,16,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,3,7,15,31]$

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
2.1.16.66j1.221 $x^{16} + 8 x^{14} + 8 x^{10} + 16 x^{7} + 24 x^{4} + 16 x^{3} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.222 $x^{16} + 8 x^{14} + 16 x^{12} + 8 x^{10} + 16 x^{7} + 24 x^{4} + 16 x^{3} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.223 $x^{16} + 8 x^{14} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 24 x^{4} + 16 x^{3} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.224 $x^{16} + 8 x^{14} + 16 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 24 x^{4} + 16 x^{3} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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