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Properties

Label 2.1.14.20a
Base 2.1.1.0a1.1
Degree \(14\)
e \(14\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{14} + 2 b_{13} x^{13} + 2 b_{11} x^{11} + 2 b_{9} x^{9} + 2 a_{7} x^{7} + 4 c_{14} + 2$

Invariants

Residue field characteristic: $2$
Degree: $14$
Base field: $\Q_{2}$
Ramification index $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{1}{2}\rangle$
Rams: $(7)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

Indices of inseparability: $[7,0]$
Associated inertia: $[3,1]$
Jump Set: $[7,14]$ (show 1), $[7,23]$ (show 8), $[7,25]$ (show 4), $[7,27]$ (show 2), $[7,28]$ (show 1)

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 16

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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
2.1.14.20a1.1 2 $x^{14} + 2 x^{7} + 2$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[2]_{7}^{3}$ $[1]_{7}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 14]$
2.1.14.20a1.2 2 $x^{14} + 2 x^{7} + 6$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[2]_{7}^{3}$ $[1]_{7}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 28]$
2.1.14.20a1.3 2 $x^{14} + 2 x^{13} + 2 x^{7} + 2$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 27]$
2.1.14.20a1.4 2 $x^{14} + 2 x^{13} + 2 x^{7} + 6$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 27]$
2.1.14.20a1.5 2 $x^{14} + 2 x^{11} + 2 x^{7} + 2$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$ $[\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$ $[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ $[\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 25]$
2.1.14.20a1.6 2 $x^{14} + 2 x^{11} + 2 x^{7} + 6$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$ $[\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$ $[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ $[\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 25]$
2.1.14.20a1.7 2 $x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 25]$
2.1.14.20a1.8 2 $x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 6$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 25]$
2.1.14.20a1.9 4 $x^{14} + 2 x^{9} + 2 x^{7} + 2$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.10 4 $x^{14} + 2 x^{9} + 2 x^{7} + 6$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.11 4 $x^{14} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 2$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.12 4 $x^{14} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 6$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.13 4 $x^{14} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.14 4 $x^{14} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 6$ $C_2\wr C_7:C_3$ (as 14T44) $2688$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.15 4 $x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
2.1.14.20a1.16 4 $x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 6$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$ $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 23]$
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