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Properties

Label 2.1.4.11a1.15-1.4.22c
Base 2.1.4.11a1.15
Degree \(4\)
e \(4\)
f \(1\)
c \(22\)

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Defining polynomial

$x^{4} + \left(b_{27} \pi^{7} + b_{23} \pi^{6} + a_{19} \pi^{5}\right) x^{3} + \left(b_{18} \pi^{5} + b_{14} \pi^{4} + a_{10} \pi^{3}\right) x^{2} + \left(b_{25} \pi^{7} + b_{21} \pi^{6}\right) x + c_{28} \pi^{8} + c_{20} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.15
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $22$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{19}{4}]$
Swan slopes: $[5,7]$
Means: $\langle\frac{5}{2},\frac{19}{4}\rangle$
Rams: $(5,9)$
Field count: $40$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $16$

Diagrams

Varying

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

Galois group: $C_2\wr C_4$ (show 4), $C_2\wr C_4$ (show 4), $C_2^5:C_4$ (show 8), $C_2^6:(C_2\times C_4)$ (show 16), $C_2^6.(C_2\times C_4)$ (show 8)
Hidden Artin slopes: $[2,\frac{7}{2}]^{2}$ (show 8), $[2,\frac{7}{2}]$ (show 8), $[2,3,\frac{7}{2},4]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 8)
Indices of inseparability: $[51,42,32,16,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices Hidden content
Assoc. Inertia Jump Set
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Showing all 8

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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
2.1.16.66j1.1121 $x^{16} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1122 $x^{16} + 16 x^{12} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1123 $x^{16} + 16 x^{11} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1124 $x^{16} + 16 x^{12} + 16 x^{11} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1125 $x^{16} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1126 $x^{16} + 16 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1127 $x^{16} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.1128 $x^{16} + 16 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10$ $C_2^5:C_4$ (as 16T259) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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