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Properties

Label 5.1.4.3a1.2-1.5.15a
Base 5.1.4.3a1.2
Degree \(5\)
e \(5\)
f \(1\)
c \(15\)

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Defining polynomial

$x^{5} + b_{13} \pi^{3} x^{3} + b_{12} \pi^{3} x^{2} + a_{11} \pi^{3} x + \pi$

Invariants

Residue field characteristic: $5$
Degree: $5$
Base field: 5.1.4.3a1.2
Ramification index $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Absolute Artin slopes: $[\frac{27}{16}]$
Swan slopes: $[\frac{11}{4}]$
Means: $\langle\frac{11}{5}\rangle$
Rams: $(\frac{11}{4})$
Field count: $25$ (complete)
Ambiguity: $1$
Mass: $100$
Absolute Mass: $25$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.

Indices of inseparability: $[11,0]$
Associated inertia: $[1,1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 25

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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
5.1.20.30a1.26 $x^{20} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.27 $x^{20} + 5 x^{13} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.28 $x^{20} + 10 x^{13} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.29 $x^{20} + 15 x^{13} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.30 $x^{20} + 20 x^{13} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.31 $x^{20} + 5 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.32 $x^{20} + 5 x^{13} + 5 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.33 $x^{20} + 10 x^{13} + 5 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.34 $x^{20} + 15 x^{13} + 5 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.35 $x^{20} + 20 x^{13} + 5 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.36 $x^{20} + 10 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.37 $x^{20} + 5 x^{13} + 10 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.38 $x^{20} + 10 x^{13} + 10 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.39 $x^{20} + 15 x^{13} + 10 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.40 $x^{20} + 20 x^{13} + 10 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.41 $x^{20} + 15 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.42 $x^{20} + 5 x^{13} + 15 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.43 $x^{20} + 10 x^{13} + 15 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.44 $x^{20} + 15 x^{13} + 15 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.45 $x^{20} + 20 x^{13} + 15 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.46 $x^{20} + 20 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.47 $x^{20} + 5 x^{13} + 20 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.48 $x^{20} + 10 x^{13} + 20 x^{12} + 5 x^{11} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.49 $x^{20} + 15 x^{13} + 20 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
5.1.20.30a1.50 $x^{20} + 20 x^{13} + 20 x^{12} + 5 x^{11} + 10$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[11, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ undefined
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