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Properties

Label 2.1.2.3a1.3-1.6.12a
Base 2.1.2.3a1.3
Degree \(6\)
e \(6\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{6} + b_{11} \pi^{2} x^{5} + b_{9} \pi^{2} x^{3} + c_{14} \pi^{3} x^{2} + \left(b_{13} \pi^{3} + a_{7} \pi^{2}\right) x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[3,\frac{19}{6}]$
Swan slopes: $[\frac{7}{3}]$
Means: $\langle\frac{7}{6}\rangle$
Rams: $(7)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $8$
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: $C_2^3:S_4$ (show 2), $C_2^3:S_4$ (show 2), $C_2^4:S_4$ (show 4)
Hidden Artin slopes: $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6}]^{2}$ (show 4), $[\frac{4}{3},\frac{4}{3},\frac{19}{6}]^{2}$ (show 4)
Indices of inseparability: $[19,12,0]$
Associated inertia: $[2,1,1]$
Jump Set: $[3,9,21]$

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
2.1.12.30b1.81 $x^{12} + 4 x^{9} + 4 x^{7} + 4 x^{2} + 2$ $C_2^4:S_4$ (as 12T136) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.82 $x^{12} + 4 x^{9} + 4 x^{7} + 12 x^{2} + 2$ $C_2^4:S_4$ (as 12T136) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.93 $x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{2} + 2$ $C_2^3:S_4$ (as 12T109) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.94 $x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 12 x^{2} + 2$ $C_2^3:S_4$ (as 12T108) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.97 $x^{12} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 2$ $C_2^3:S_4$ (as 12T109) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.98 $x^{12} + 4 x^{7} + 4 x^{6} + 12 x^{2} + 2$ $C_2^3:S_4$ (as 12T108) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.109 $x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 2$ $C_2^4:S_4$ (as 12T136) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.30b1.110 $x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 12 x^{2} + 2$ $C_2^4:S_4$ (as 12T136) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6}]^{2}$ $[19, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
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