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Properties

Label 7.1.3.2a1.2-1.7.9a
Base 7.1.3.2a1.2
Degree \(7\)
e \(7\)
f \(1\)
c \(9\)

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Defining polynomial

$x^{7} + a_{3} \pi x^{3} + \pi$

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 $c$: $9$
Absolute Artin slopes: $[\frac{7}{6}]$
Swan slopes: $[\frac{1}{2}]$
Means: $\langle\frac{3}{7}\rangle$
Rams: $(\frac{1}{2})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $6$
Absolute Mass: $2$

Diagrams

Varying

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

Indices of inseparability: $[3,0]$
Associated inertia: $[1,1]$ (show 2), $[1,3]$ (show 4)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
7.1.21.23a1.3 $x^{21} + 21 x^{3} + 14$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^3 + 6$ undefined
7.1.21.23a1.4 $x^{21} + 28 x^{3} + 14$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^3 + 1$ undefined
7.1.21.23a2.3 $x^{21} + 7 x^{3} + 14$ $C_3\times F_7$ (as 21T9) $126$ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^3 + 2$ undefined
7.1.21.23a2.4 $x^{21} + 42 x^{3} + 14$ $C_3\times F_7$ (as 21T9) $126$ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 3]$ $z^{14} + 3 z^7 + 3,3 z^3 + 5$ undefined
7.1.21.23a3.3 $x^{21} + 14 x^{3} + 14$ $F_7$ (as 21T4) $42$ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^3 + 4$ undefined
7.1.21.23a3.4 $x^{21} + 35 x^{3} + 14$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[3, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z^3 + 3$ undefined
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