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Properties

Label 7.1.3.2a1.2-1.7.16a
Base 7.1.3.2a1.2
Degree \(7\)
e \(7\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{7} + b_{11} \pi^{2} x^{4} + a_{10} \pi^{2} x^{3} + \pi$

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 $c$: $16$
Absolute Artin slopes: $[\frac{14}{9}]$
Swan slopes: $[\frac{5}{3}]$
Means: $\langle\frac{10}{7}\rangle$
Rams: $(\frac{5}{3})$
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: $[10,0]$
Associated inertia: $[1,1]$ (show 7), $[1,2]$ (show 7)
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
7.1.21.30a1.8 $x^{21} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.9 $x^{21} + 7 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.10 $x^{21} + 14 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.11 $x^{21} + 21 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.12 $x^{21} + 28 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.13 $x^{21} + 35 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a1.14 $x^{21} + 42 x^{11} + 7 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^2 + 2$ undefined
7.1.21.30a2.8 $x^{21} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.9 $x^{21} + 7 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.10 $x^{21} + 14 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.11 $x^{21} + 21 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.12 $x^{21} + 28 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.13 $x^{21} + 35 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
7.1.21.30a2.14 $x^{21} + 42 x^{11} + 21 x^{10} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 0]$ $[1, 2]$ $z^{14} + 3 z^7 + 3,3 z^2 + 6$ undefined
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