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Properties

Label 2.1.22.26a
Base 2.1.1.0a1.1
Degree \(22\)
e \(22\)
f \(1\)
c \(26\)

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Defining polynomial

$x^{22} + 2 c_{10} x^{10} + 2 b_{9} x^{9} + 2 b_{7} x^{7} + 2 a_{5} x^{5} + 2$

Invariants

Residue field characteristic: $2$
Degree: $22$
Base field: $\Q_{2}$
Ramification index $e$: $22$
Residue field degree $f$: $1$
Discriminant exponent $c$: $26$
Artin slopes: $[\frac{16}{11}]$
Swan slopes: $[\frac{5}{11}]$
Means: $\langle\frac{5}{22}\rangle$
Rams: $(5)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[10,1]$
Jump Set: $[11,27]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
2.1.22.26a1.1 $x^{22} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.2 $x^{22} + 2 x^{10} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.3 $x^{22} + 2 x^{9} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.4 $x^{22} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2$ $C_2^{10}.F_{11}$ (as 22T34) $112640$ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.5 $x^{22} + 2 x^{7} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.6 $x^{22} + 2 x^{10} + 2 x^{7} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.7 $x^{22} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2$ $C_2^{10}.F_{11}$ (as 22T35) $112640$ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
2.1.22.26a1.8 $x^{22} + 2 x^{10} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 0]$ $[10, 1]$ $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1,z + 1$ $[11, 27]$
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