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Properties

Label 2.1.22.42a
Base 2.1.1.0a1.1
Degree \(22\)
e \(22\)
f \(1\)
c \(42\)

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Defining polynomial

$x^{22} + 2 a_{21} x^{21} + 4 c_{42} x^{20} + 4 b_{41} x^{19} + 4 b_{39} x^{17} + 4 b_{37} x^{15} + 4 b_{35} x^{13} + 4 b_{33} x^{11} + 4 b_{31} x^{9} + 4 b_{29} x^{7} + 4 b_{27} x^{5} + 4 b_{25} x^{3} + 4 b_{23} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $22$
Base field: $\Q_{2}$
Ramification index $e$: $22$
Residue field degree $f$: $1$
Discriminant exponent $c$: $42$
Artin slopes: $[\frac{32}{11}]$
Swan slopes: $[\frac{21}{11}]$
Means: $\langle\frac{21}{22}\rangle$
Rams: $(21)$
Field count: $2048$ (complete)
Ambiguity: $2$
Mass: $1024$
Absolute Mass: $1024$

Diagrams

Varying

Indices of inseparability: $[21,0]$
Associated inertia: $[10,1]$
Jump Set: $[11,33]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of at least 1000

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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
2.1.22.42a1.1 $x^{22} + 2 x^{21} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.2 $x^{22} + 2 x^{21} + 4 x^{20} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.3 $x^{22} + 2 x^{21} + 4 x^{19} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.4 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.5 $x^{22} + 2 x^{21} + 4 x^{17} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.6 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.7 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.8 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.9 $x^{22} + 2 x^{21} + 4 x^{15} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.10 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{15} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.11 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{15} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.12 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{15} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.13 $x^{22} + 2 x^{21} + 4 x^{17} + 4 x^{15} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.14 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 4 x^{15} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.15 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.16 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.17 $x^{22} + 2 x^{21} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.18 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.19 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.20 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.21 $x^{22} + 2 x^{21} + 4 x^{17} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.22 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.23 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.24 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.25 $x^{22} + 2 x^{21} + 4 x^{15} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.26 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{15} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.27 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{15} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.28 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{15} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.29 $x^{22} + 2 x^{21} + 4 x^{17} + 4 x^{15} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.30 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 4 x^{15} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.31 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{13} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.32 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.33 $x^{22} + 2 x^{21} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.34 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.35 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.36 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.37 $x^{22} + 2 x^{21} + 4 x^{17} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.38 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.39 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.40 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.41 $x^{22} + 2 x^{21} + 4 x^{15} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.42 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.43 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.44 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.45 $x^{22} + 2 x^{21} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.46 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.47 $x^{22} + 2 x^{21} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.48 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.49 $x^{22} + 2 x^{21} + 4 x^{13} + 4 x^{11} + 2$ $C_2\times C_2^{10}.F_{11}$ (as 22T37) $225280$ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
2.1.22.42a1.50 $x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{13} + 4 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 33]$
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