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Properties

Label 2.1.8.20c
Base 2.1.1.0a1.1
Degree \(8\)
e \(8\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{8} + 4 b_{15} x^{7} + 4 a_{13} x^{5} + 4 b_{12} x^{4} + 4 a_{10} x^{2} + 8 c_{16} + 2$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Artin slopes: $[\frac{8}{3},\frac{8}{3},3]$
Swan slopes: $[\frac{5}{3},\frac{5}{3},2]$
Means: $\langle\frac{5}{6},\frac{5}{4},\frac{13}{8}\rangle$
Rams: $(\frac{5}{3},\frac{5}{3},3)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[13,10,8,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7,15]$

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 8

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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
2.1.8.20c1.1 8 $x^{8} + 4 x^{5} + 4 x^{2} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.2 8 $x^{8} + 4 x^{5} + 4 x^{2} + 10$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.3 8 $x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.4 8 $x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 10$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.5 8 $x^{8} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.6 8 $x^{8} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 10$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.7 8 $x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 2$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.20c1.8 8 $x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 10$ $S_4\times C_2$ (as 8T24) $48$ $2$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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