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Properties

Label 7.1.3.2a1.2-1.7.24a
Base 7.1.3.2a1.2
Degree \(7\)
e \(7\)
f \(1\)
c \(24\)

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Defining polynomial

$x^{7} + b_{20} \pi^{3} x^{6} + b_{19} \pi^{3} x^{5} + a_{18} \pi^{3} x^{4} + c_{21} \pi^{4} + \pi$

Invariants

Residue field characteristic: $7$
Degree: $7$
Base field: 7.1.3.2a1.2
Ramification index $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Absolute Artin slopes: $[2]$
Swan slopes: $[3]$
Means: $\langle\frac{18}{7}\rangle$
Rams: $(3)$
Field count: $204$ (complete)
Ambiguity: $7$
Mass: $294$
Absolute Mass: $98$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.

Indices of inseparability: $[18,0]$
Associated inertia: $[1,1]$ (show 119), $[1,2]$ (show 17), $[1,3]$ (show 34), $[1,6]$ (show 34)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 34

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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
7.1.21.38a1.18 $x^{21} + 21 x^{18} + 14$ $C_{21}:C_3$ (as 21T7) $63$ $3$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.19 $x^{21} + 7 x^{20} + 21 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.20 $x^{21} + 21 x^{20} + 21 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.21 $x^{21} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.22 $x^{21} + 7 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.23 $x^{21} + 14 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.24 $x^{21} + 21 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.25 $x^{21} + 28 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.26 $x^{21} + 35 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.27 $x^{21} + 42 x^{20} + 7 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.28 $x^{21} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.29 $x^{21} + 7 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.30 $x^{21} + 14 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.31 $x^{21} + 21 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.32 $x^{21} + 28 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.33 $x^{21} + 35 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a1.34 $x^{21} + 42 x^{20} + 21 x^{19} + 21 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 1$ undefined
7.1.21.38a2.18 $x^{21} + 42 x^{18} + 14$ $C_{21}:C_3$ (as 21T7) $63$ $3$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.19 $x^{21} + 7 x^{20} + 42 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.20 $x^{21} + 21 x^{20} + 42 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.21 $x^{21} + 7 x^{19} + 42 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.22 $x^{21} + 7 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.23 $x^{21} + 14 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.24 $x^{21} + 21 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.25 $x^{21} + 28 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.26 $x^{21} + 35 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.27 $x^{21} + 42 x^{20} + 7 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.28 $x^{21} + 21 x^{19} + 42 x^{18} + 14$ $C_7^2:C_3^2$ (as 21T21) $441$ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.29 $x^{21} + 7 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.30 $x^{21} + 14 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.31 $x^{21} + 21 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.32 $x^{21} + 28 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.33 $x^{21} + 35 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
7.1.21.38a2.34 $x^{21} + 42 x^{20} + 21 x^{19} + 42 x^{18} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[18, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^6 + 2$ undefined
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