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Properties

Label 2.1.4.11a1.5-1.4.24b
Base 2.1.4.11a1.5
Degree \(4\)
e \(4\)
f \(1\)
c \(24\)

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Defining polynomial

$x^{4} + \left(b_{35} \pi^{9} + b_{31} \pi^{8} + b_{27} \pi^{7} + b_{23} \pi^{6}\right) x^{3} + \left(b_{10} \pi^{3} + a_{6} \pi^{2}\right) x^{2} + \left(b_{33} \pi^{9} + b_{29} \pi^{8} + b_{25} \pi^{7} + a_{21} \pi^{6}\right) x + c_{36} \pi^{10} + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.5
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Absolute Artin slopes: $[3,\frac{7}{2},4,\frac{21}{4}]$
Swan slopes: $[3,9]$
Means: $\langle\frac{3}{2},\frac{21}{4}\rangle$
Rams: $(3,15)$
Field count: $256$ (complete)
Ambiguity: $4$
Mass: $256$
Absolute Mass: $128$

Diagrams

Varying

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

Galois group: $C_2^6:\SD_{16}$ (show 32), $(C_2^2\times C_4^2):\SD_{16}$ (show 32), $C_2^6:D_8$ (show 32), $(C_2^2\times C_4^2):D_8$ (show 32), $C_2^4.C_2\wr D_4$ (show 128) (incomplete)
Hidden Artin slopes: $[2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ (show 32), $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ (show 16), $[\frac{17}{4},\frac{19}{4},2,\frac{39}{8},\frac{9}{2}]_{2}$ (show 96), not computed (show 112) (incomplete)
Indices of inseparability: $[53,38,28,16,0]$
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 16

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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
2.1.16.68e1.3137 $x^{16} + 8 x^{14} + 4 x^{12} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3145 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3146 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 32 x^{3} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3154 $x^{16} + 16 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3159 $x^{16} + 8 x^{14} + 4 x^{12} + 16 x^{9} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3163 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{9} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3164 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{9} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 32 x^{3} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3166 $x^{16} + 16 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3175 $x^{16} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3176 $x^{16} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 32 x^{4} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3181 $x^{16} + 16 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3182 $x^{16} + 16 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 32 x^{3} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3189 $x^{16} + 8 x^{14} + 4 x^{12} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3190 $x^{16} + 8 x^{14} + 4 x^{12} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 32 x^{4} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3199 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68e1.3200 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 32 x^{3} + 10$ $C_2^4.C_2\wr D_4$ (as 16T1452) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]^{2}$ $[2,2,\frac{17}{4},\frac{9}{2},\frac{19}{4},\frac{39}{8}]^{2}$ $[1,1,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[53, 38, 28, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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