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Properties

Label 7.1.3.2a1.2-1.7.19a
Base 7.1.3.2a1.2
Degree \(7\)
e \(7\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{7} + a_{13} \pi^{2} x^{6} + b_{15} \pi^{3} x + \pi$

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 $c$: $19$
Absolute Artin slopes: $[\frac{31}{18}]$
Swan slopes: $[\frac{13}{6}]$
Means: $\langle\frac{13}{7}\rangle$
Rams: $(\frac{13}{6})$
Field count: $14$ (complete)
Ambiguity: $1$
Mass: $42$
Absolute Mass: $14$

Diagrams

Varying

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

Indices of inseparability: $[13,0]$
Associated inertia: $[1,1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 14

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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
7.1.21.33a1.15 $x^{21} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.16 $x^{21} + 7 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.17 $x^{21} + 14 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.18 $x^{21} + 21 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.19 $x^{21} + 28 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.20 $x^{21} + 35 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.21 $x^{21} + 42 x^{15} + 7 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 4$ undefined
7.1.21.33a1.22 $x^{21} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.23 $x^{21} + 7 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.24 $x^{21} + 14 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.25 $x^{21} + 21 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.26 $x^{21} + 28 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.27 $x^{21} + 35 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
7.1.21.33a1.28 $x^{21} + 42 x^{15} + 21 x^{13} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[13, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 5$ undefined
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