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Properties

Label 2.1.6.11a1.15-1.2.4a
Base 2.1.6.11a1.15
Degree \(2\)
e \(2\)
f \(1\)
c \(4\)

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Defining polynomial

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.6.11a1.15
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Absolute Artin slopes: $[2,3]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $4$ (complete)
Ambiguity: $2$
Mass: $2$
Absolute Mass: $1$

Diagrams

Varying

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

Galois group: $C_2^2\times S_4$ (show 2), $C_2^2\wr S_3$ (show 2)
Hidden Artin slopes: $[\frac{4}{3},\frac{4}{3},\frac{8}{3},\frac{8}{3}]^{2}$ (show 2), $[\frac{8}{3},\frac{8}{3}]^{2}$ (show 2)
Indices of inseparability: $[15,6,0]$
Associated inertia: $[2,1,1]$
Jump Set: $[3,6,24]$ (show 1), $[3,11,23]$ (show 2), $[3,17,29]$ (show 1)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 4

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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
2.1.12.26b1.17 $x^{12} + 4 x^{9} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3},\frac{8}{3}]^{2}$ $[\frac{5}{3},\frac{5}{3}]^{2}$ $[15, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 17, 29]$
2.1.12.26b1.37 $x^{12} + 4 x^{11} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3},\frac{8}{3}]^{2}$ $[\frac{5}{3},\frac{5}{3}]^{2}$ $[15, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 6, 24]$
2.1.12.26b1.43 $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{6} + 4 x^{3} + 2$ $C_2^2\wr S_3$ (as 12T139) $384$ $4$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{8}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]^{2}$ $[15, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 11, 23]$
2.1.12.26b1.63 $x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{6} + 4 x^{3} + 6$ $C_2^2\wr S_3$ (as 12T139) $384$ $4$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{8}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]^{2}$ $[15, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 11, 23]$
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