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Properties

Label 11.1.11.16a
Base 11.1.1.0a1.1
Degree \(11\)
e \(11\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{11} + 11a_{6} x^{6} + 11$

Invariants

Residue field characteristic: $11$
Degree: $11$
Base field: $\Q_{11}$
Ramification index $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Artin slopes: $[\frac{8}{5}]$
Swan slopes: $[\frac{3}{5}]$
Means: $\langle\frac{6}{11}\rangle$
Rams: $(\frac{3}{5})$
Field count: $10$ (complete)
Ambiguity: $1$
Mass: $10$
Absolute Mass: $10$

Diagrams

Varying

Indices of inseparability: $[6,0]$
Associated inertia: $[1]$ (show 5), $[2]$ (show 5)
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 10

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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
11.1.11.16a1.1 5 $x^{11} + 11 x^{6} + 11$ $F_{11}$ (as 11T4) $110$ $1$ $[\frac{8}{5}]_{5}^{2}$ $[\frac{3}{5}]_{5}^{2}$ $[\ ]^{2}_{5}$ $[\ ]^{2}_{5}$ $[6, 0]$ $[2]$ $z^2 + 5$ undefined
11.1.11.16a1.2 5 $x^{11} + 33 x^{6} + 11$ $F_{11}$ (as 11T4) $110$ $1$ $[\frac{8}{5}]_{5}^{2}$ $[\frac{3}{5}]_{5}^{2}$ $[\ ]^{2}_{5}$ $[\ ]^{2}_{5}$ $[6, 0]$ $[2]$ $z^2 + 4$ undefined
11.1.11.16a1.3 5 $x^{11} + 44 x^{6} + 11$ $F_{11}$ (as 11T4) $110$ $1$ $[\frac{8}{5}]_{5}^{2}$ $[\frac{3}{5}]_{5}^{2}$ $[\ ]^{2}_{5}$ $[\ ]^{2}_{5}$ $[6, 0]$ $[2]$ $z^2 + 9$ undefined
11.1.11.16a1.4 5 $x^{11} + 55 x^{6} + 11$ $F_{11}$ (as 11T4) $110$ $1$ $[\frac{8}{5}]_{5}^{2}$ $[\frac{3}{5}]_{5}^{2}$ $[\ ]^{2}_{5}$ $[\ ]^{2}_{5}$ $[6, 0]$ $[2]$ $z^2 + 3$ undefined
11.1.11.16a1.5 5 $x^{11} + 99 x^{6} + 11$ $F_{11}$ (as 11T4) $110$ $1$ $[\frac{8}{5}]_{5}^{2}$ $[\frac{3}{5}]_{5}^{2}$ $[\ ]^{2}_{5}$ $[\ ]^{2}_{5}$ $[6, 0]$ $[2]$ $z^2 + 1$ undefined
11.1.11.16a2.1 5 $x^{11} + 22 x^{6} + 11$ $C_{11}:C_5$ (as 11T3) $55$ $1$ $[\frac{8}{5}]_{5}$ $[\frac{3}{5}]_{5}$ $[\ ]_{5}$ $[\ ]_{5}$ $[6, 0]$ $[1]$ $z^2 + 10$ undefined
11.1.11.16a2.2 5 $x^{11} + 66 x^{6} + 11$ $C_{11}:C_5$ (as 11T3) $55$ $1$ $[\frac{8}{5}]_{5}$ $[\frac{3}{5}]_{5}$ $[\ ]_{5}$ $[\ ]_{5}$ $[6, 0]$ $[1]$ $z^2 + 8$ undefined
11.1.11.16a2.3 5 $x^{11} + 77 x^{6} + 11$ $C_{11}:C_5$ (as 11T3) $55$ $1$ $[\frac{8}{5}]_{5}$ $[\frac{3}{5}]_{5}$ $[\ ]_{5}$ $[\ ]_{5}$ $[6, 0]$ $[1]$ $z^2 + 2$ undefined
11.1.11.16a2.4 5 $x^{11} + 88 x^{6} + 11$ $C_{11}:C_5$ (as 11T3) $55$ $1$ $[\frac{8}{5}]_{5}$ $[\frac{3}{5}]_{5}$ $[\ ]_{5}$ $[\ ]_{5}$ $[6, 0]$ $[1]$ $z^2 + 7$ undefined
11.1.11.16a2.5 5 $x^{11} + 110 x^{6} + 11$ $C_{11}:C_5$ (as 11T3) $55$ $1$ $[\frac{8}{5}]_{5}$ $[\frac{3}{5}]_{5}$ $[\ ]_{5}$ $[\ ]_{5}$ $[6, 0]$ $[1]$ $z^2 + 6$ undefined
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