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Properties

Label 2.1.8.12b
Base 2.1.1.0a1.1
Degree \(8\)
e \(8\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{8} + 2 b_{7} x^{7} + 2 a_{5} x^{5} + 2 a_{2} x^{2} + 4 c_{8} + 2$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Artin slopes: $[\frac{4}{3},\frac{4}{3},2]$
Swan slopes: $[\frac{1}{3},\frac{1}{3},1]$
Means: $\langle\frac{1}{6},\frac{1}{4},\frac{5}{8}\rangle$
Rams: $(\frac{1}{3},\frac{1}{3},3)$
Field count: $4$ (complete)
Ambiguity: $2$
Mass: $2$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[5,2,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,5,10]$ (show 1), $[1,2,5,15]$ (show 2), $[1,2,5,16]$ (show 1)

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.1.8.12b1.1 2 $x^{8} + 2 x^{5} + 2 x^{2} + 2$ $\textrm{GL(2,3)}$ (as 8T23) $48$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 15]$
2.1.8.12b1.2 2 $x^{8} + 2 x^{5} + 2 x^{2} + 6$ $\textrm{GL(2,3)}$ (as 8T23) $48$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 15]$
2.1.8.12b1.3 2 $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 10]$
2.1.8.12b1.4 2 $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 6$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 16]$
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