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Properties

Label 2.1.2.3a1.3-1.6.10a
Base 2.1.2.3a1.3
Degree \(6\)
e \(6\)
f \(1\)
c \(10\)

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Defining polynomial

$x^{6} + a_{5} \pi x^{5} + c_{10} \pi^{2} x^{4} + b_{9} \pi^{2} x^{3} + b_{7} \pi^{2} x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[\frac{8}{3},3]$
Swan slopes: $[\frac{5}{3}]$
Means: $\langle\frac{5}{6}\rangle$
Rams: $(5)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $4$
Absolute Mass: $2$

Diagrams

Varying

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

Galois group: $C_2 \times S_4$ (show 2), $C_2 \times S_4$ (show 2), $C_2^2\times S_4$ (show 4)
Hidden Artin slopes: $[\frac{8}{3}]^{2}$ (show 4), $[2,\frac{8}{3}]^{2}$ (show 4)
Indices of inseparability: $[17,10,0]$
Associated inertia: $[2,1,1]$
Jump Set: $[3,9,21]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
2.1.12.28c1.7 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 2$ $C_2 \times S_4$ (as 12T24) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.13 $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{8} + 4 x^{5} + 2$ $C_2 \times S_4$ (as 12T23) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.27 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 2$ $C_2 \times S_4$ (as 12T24) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.33 $x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 2$ $C_2 \times S_4$ (as 12T23) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.63 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 2$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.69 $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 2$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.83 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.89 $x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2$ $C_2^2\times S_4$ (as 12T48) $96$ $4$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
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