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Properties

Label 43.1.3.2a1.2-1.7.6a
Base 43.1.3.2a1.2
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{7} + d_{0} \pi$

Invariants

Residue field characteristic: $43$
Degree: $7$
Base field: 43.1.3.2a1.2
Ramification index $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $7$ (complete)
Ambiguity: $7$
Mass: $1$
Absolute Mass: $1/3$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 43 }$ within this relative family, not the relative extension.

Galois group: $C_{21}$
Hidden Artin slopes: $[\ ]$
Indices of inseparability: $[0]$
Associated inertia: $[1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 7

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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
43.1.21.20a1.2 $x^{21} + 129$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.5 $x^{21} + 430$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.6 $x^{21} + 516$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.8 $x^{21} + 817$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.10 $x^{21} + 989$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.19 $x^{21} + 1591$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
43.1.21.20a1.20 $x^{21} + 1634$ $C_{21}$ (as 21T1) $21$ $21$ $[\ ]_{21}$ $[\ ]_{21}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{20} + 21 z^{19} + 38 z^{18} + 40 z^{17} + 8 z^{16} + 10 z^{15} + 41 z^{14} + 8 z^{13} + 14 z^{12} + 25 z^{11} + 30 z^{10} + 30 z^9 + 25 z^8 + 14 z^7 + 8 z^6 + 41 z^5 + 10 z^4 + 8 z^3 + 40 z^2 + 38 z + 21$ undefined
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