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Properties

Label 2.1.8.31a1.168-1.2.12a
Base 2.1.8.31a1.168
Degree \(2\)
e \(2\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{2} + \left(b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + b_{15} \pi^{8} + b_{13} \pi^{7} + a_{11} \pi^{6}\right) x + c_{22} \pi^{12} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.31a1.168
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[3,4,5,\frac{43}{8}]$
Swan slopes: $[11]$
Means: $\langle\frac{11}{2}\rangle$
Rams: $(11)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $32$
Absolute Mass: $4$

Diagrams

Varying

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

Galois group: $C_2^6:C_8$
Hidden Artin slopes: $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$
Indices of inseparability: $[59,48,32,16,0]$
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
2.1.16.74c1.4697 $x^{16} + 16 x^{13} + 8 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4698 $x^{16} + 16 x^{13} + 8 x^{12} + 16 x^{11} + 4 x^{8} + 48 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4708 $x^{16} + 16 x^{15} + 16 x^{14} + 16 x^{13} + 8 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4713 $x^{16} + 8 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4714 $x^{16} + 8 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 48 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4715 $x^{16} + 8 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{6} + 32 x^{5} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4716 $x^{16} + 8 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 48 x^{6} + 32 x^{5} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.74c1.4726 $x^{16} + 16 x^{15} + 16 x^{14} + 8 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{6} + 34$ $C_2^6:C_8$ (as 16T817) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}, \frac{43}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8},\frac{35}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4},\frac{33}{8}]$ $[59, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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