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Properties

Label 5.1.20.21a
Base 5.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(21\)

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Defining polynomial

$x^{20} + 5 a_{2} x^{2} + 5 d_{0}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $21$
Artin slopes: $[\frac{9}{8}]$
Swan slopes: $[\frac{1}{8}]$
Means: $\langle\frac{1}{10}\rangle$
Rams: $(\frac{1}{2})$
Field count: $8$ (complete)
Ambiguity: $4$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1,1]$ (show 4), $[1,2]$ (show 4)
Jump Set: undefined (show 6), $[1,7]$ (show 2)

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
5.1.20.21a1.1 $x^{20} + 10 x^{2} + 5$ $C_5^2:\OD_{16}$ (as 20T109) $400$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 1$ $[1, 7]$
5.1.20.21a1.2 $x^{20} + 5 x^{2} + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 4$ undefined
5.1.20.21a1.3 $x^{20} + 5 x^{2} + 15$ $C_5^2:\OD_{16}$ (as 20T109) $400$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 1$ undefined
5.1.20.21a1.4 $x^{20} + 10 x^{2} + 20$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 4$ undefined
5.1.20.21a2.1 $x^{20} + 5 x^{2} + 5$ $C_5^2:\OD_{16}$ (as 20T104) $400$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 3$ $[1, 7]$
5.1.20.21a2.2 $x^{20} + 10 x^{2} + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 3$ undefined
5.1.20.21a2.3 $x^{20} + 10 x^{2} + 15$ $C_5^2:\OD_{16}$ (as 20T104) $400$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 2$ undefined
5.1.20.21a2.4 $x^{20} + 5 x^{2} + 20$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 2$ undefined
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