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Properties

Label 7.1.21.25a
Base 7.1.1.0a1.1
Degree \(21\)
e \(21\)
f \(1\)
c \(25\)

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Defining polynomial

$x^{21} + 7 a_{5} x^{5} + 7 d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $21$
Base field: $\Q_{7}$
Ramification index $e$: $21$
Residue field degree $f$: $1$
Discriminant exponent $c$: $25$
Artin slopes: $[\frac{23}{18}]$
Swan slopes: $[\frac{5}{18}]$
Means: $\langle\frac{5}{21}\rangle$
Rams: $(\frac{5}{6})$
Field count: $6$ (complete)
Ambiguity: $3$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[1,1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
7.1.21.25a1.1 $x^{21} + 7 x^{5} + 7$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.25a1.2 $x^{21} + 21 x^{5} + 7$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.25a1.3 $x^{21} + 7 x^{5} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 1$ undefined
7.1.21.25a1.4 $x^{21} + 21 x^{5} + 14$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.25a1.5 $x^{21} + 7 x^{5} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.25a1.6 $x^{21} + 21 x^{5} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
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