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Properties

Label 2.1.8.31a1.87-1.2.10a
Base 2.1.8.31a1.87
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: 2.1.8.31a1.87
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[3,4,5,\frac{41}{8}]$
Swan slopes: $[9]$
Means: $\langle\frac{9}{2}\rangle$
Rams: $(9)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $16$
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: $C_2^4.Q_{16}$ (show 4), $C_2^6.D_4$ (show 4)
Hidden Artin slopes: $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]$ (show 4), $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]^{2}$ (show 4)
Indices of inseparability: $[57,48,32,16,0]$
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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Showing all 8

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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
2.1.16.72e1.1279 $x^{16} + 16 x^{15} + 16 x^{11} + 16 x^{9} + 16 x^{2} + 26$ $C_2^4.Q_{16}$ (as 16T698) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1280 $x^{16} + 16 x^{15} + 16 x^{11} + 16 x^{9} + 48 x^{2} + 26$ $C_2^4.Q_{16}$ (as 16T698) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1281 $x^{16} + 16 x^{14} + 16 x^{11} + 16 x^{9} + 16 x^{2} + 26$ $C_2^4.Q_{16}$ (as 16T698) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1282 $x^{16} + 16 x^{14} + 16 x^{11} + 16 x^{9} + 48 x^{2} + 26$ $C_2^4.Q_{16}$ (as 16T698) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1293 $x^{16} + 16 x^{9} + 16 x^{6} + 16 x^{2} + 26$ $C_2^6.D_4$ (as 16T900) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]^{2}$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]^{2}$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1296 $x^{16} + 16 x^{15} + 16 x^{14} + 16 x^{9} + 16 x^{6} + 16 x^{2} + 26$ $C_2^6.D_4$ (as 16T900) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]^{2}$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]^{2}$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1313 $x^{16} + 16 x^{13} + 16 x^{10} + 16 x^{9} + 16 x^{6} + 16 x^{2} + 26$ $C_2^6.D_4$ (as 16T900) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]^{2}$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]^{2}$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.72e1.1316 $x^{16} + 16 x^{15} + 16 x^{14} + 16 x^{13} + 16 x^{10} + 16 x^{9} + 16 x^{6} + 16 x^{2} + 26$ $C_2^6.D_4$ (as 16T900) $512$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]^{2}$ $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]^{2}$ $[57, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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