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Properties

Label 2.1.8.18c
Base 2.1.1.0a1.1
Degree \(8\)
e \(8\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{8} + 4 b_{13} x^{5} + 2 a_{4} x^{4} + 4 a_{11} x^{3} + 4 b_{10} x^{2} + 4 c_{8} + 2$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Artin slopes: $[2,\frac{8}{3},\frac{8}{3}]$
Swan slopes: $[1,\frac{5}{3},\frac{5}{3}]$
Means: $\langle\frac{1}{2},\frac{13}{12},\frac{11}{8}\rangle$
Rams: $(1,\frac{7}{3},\frac{7}{3})$
Field count: $6$ (complete)
Ambiguity: $2$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[11,10,4,0]$ (show 2), $[11,11,4,0]$ (show 4)
Associated inertia: $[1,1]$
Jump Set: $[1,2,4,16]$ (show 3), $[1,2,11,19]$ (show 1), $[1,5,11,19]$ (show 2)

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 6

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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
2.1.8.18c1.1 4 $x^{8} + 2 x^{4} + 4 x^{3} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[11, 11, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 5, 11, 19]$
2.1.8.18c1.2 4 $x^{8} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[11, 11, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 5, 11, 19]$
2.1.8.18c1.3 2 $x^{8} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$ $V_4^2:(S_3\times C_2)$ (as 8T41) $192$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3}]^{2}_{3}$ $[11, 10, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 2, 11, 19]$
2.1.8.18c1.4 4 $x^{8} + 2 x^{4} + 4 x^{3} + 6$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[11, 11, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.18c1.5 4 $x^{8} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 6$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[11, 11, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.18c1.6 2 $x^{8} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 6$ $V_4^2:(S_3\times C_2)$ (as 8T41) $192$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3}]^{2}_{3}$ $[11, 10, 4, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 2, 4, 16]$
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