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Properties

Label 2.1.8.28b1.7-1.2.14a
Base 2.1.8.28b1.7
Degree \(2\)
e \(2\)
f \(1\)
c \(14\)

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Defining polynomial

$x^{2} + \left(b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + b_{15} \pi^{8} + a_{13} \pi^{7}\right) x + c_{26} \pi^{14} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.28b1.7
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{21}{4}]$
Swan slopes: $[13]$
Means: $\langle\frac{13}{2}\rangle$
Rams: $(13)$
Field count: $80$ (complete)
Ambiguity: $2$
Mass: $64$
Absolute Mass: $32$

Diagrams

Varying

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

Galois group: $C_4^2.(C_2\times D_4)$ (show 8), $(C_4\times C_8).D_4$ (show 4), $D_4^2:C_2^2$ (show 4), $C_4^3:D_4$ (show 4), $C_4^3.D_4$ (show 8), $C_4^3:D_4$ (show 8), $C_4^3.D_4$ (show 4), $(C_4\times \OD_{16}).D_4$ (show 4), $(C_4\times \OD_{16}):D_4$ (show 4), $D_4^2:Q_8$ (show 8), $C_4^3.D_4$ (show 8), $Q_8^2:D_4$ (show 4), $Q_8^2:Q_8$ (show 8), $D_4^2:D_4$ (show 4)
Hidden Artin slopes: $[2,\frac{7}{2},\frac{19}{4}]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{19}{4}]^{2}$ (show 16), $[2,3,\frac{7}{2},\frac{19}{4}]^{2}$ (show 48)
Indices of inseparability: $[55,42,32,16,0]$
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 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
2.1.16.70e1.481 $x^{16} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{7} + 2$ $D_4^2:C_2^2$ (as 16T696) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.482 $x^{16} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{7} + 32 x^{4} + 2$ $D_4^2:C_2^2$ (as 16T696) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.501 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.502 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 32 x^{4} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.503 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 32 x + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.504 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 32 x^{4} + 32 x + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.515 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 2$ $(C_4\times C_8).D_4$ (as 16T677) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.516 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 32 x^{4} + 2$ $(C_4\times C_8).D_4$ (as 16T677) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.517 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 32 x + 2$ $(C_4\times C_8).D_4$ (as 16T677) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.518 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 32 x^{4} + 32 x + 2$ $(C_4\times C_8).D_4$ (as 16T677) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.527 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.528 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 32 x^{4} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.607 $x^{16} + 8 x^{14} + 16 x^{13} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 16 x^{2} + 2$ $D_4^2:C_2^2$ (as 16T696) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.608 $x^{16} + 8 x^{14} + 16 x^{13} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{7} + 32 x^{4} + 16 x^{2} + 2$ $D_4^2:C_2^2$ (as 16T696) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.613 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 16 x^{2} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70e1.614 $x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 32 x^{4} + 16 x^{2} + 2$ $C_4^2.(C_2\times D_4)$ (as 16T662) $256$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},\frac{17}{4}]^{2}$ $[2,\frac{7}{2},\frac{19}{4}]^{2}$ $[1,\frac{5}{2},\frac{15}{4}]^{2}$ $[55, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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