Properties

Label 5.1.4.3a1.1-1.5.12a
Base 5.1.4.3a1.1
Degree \(5\)
e \(5\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{5} + b_{9} \pi^{2} x^{4} + a_{8} \pi^{2} x^{3} + c_{10} \pi^{3} + \pi$

Invariants

Residue field characteristic: $5$
Degree: $5$
Base field: $\Q_{5}(\sqrt[4]{-5})$
Ramification index $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[2]$
Means: $\langle\frac{8}{5}\rangle$
Rams: $(2)$
Field count: $14$ (complete)
Ambiguity: $5$
Mass: $20$
Absolute Mass: $5$

Diagrams

Varying

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

Galois group: $C_5:C_4$ (show 1), $C_4\times D_5$ (show 1), $C_4\times F_5$ (show 2), $C_5:C_{20}$ (show 2), $C_5:F_5$ (show 1), $D_5\times F_5$ (show 1), $F_5^2$ (show 2), $C_5^2:C_{20}$ (show 4) (incomplete)
Hidden Artin slopes: not computed (show 13), $[\ ]$ (show 1) (incomplete)
Indices of inseparability: $[8,0]$
Associated inertia: $[1,1]$ (show 8), $[1,2]$ (show 2), $[1,4]$ (show 4)
Jump Set: $[1,13]$

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
5.1.20.27a1.1 $x^{20} + 15 x^{8} + 5$ $C_5:C_4$ (as 20T2) $20$ $20$ $[\frac{3}{2}]_{4}$ $[\frac{1}{2}]_{4}$ $[\ ]$ $[\ ]$ $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.2 $x^{20} + 5 x^{10} + 15 x^{8} + 5$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.3 $x^{20} + 10 x^{10} + 15 x^{8} + 5$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.4 $x^{20} + 5 x^{9} + 15 x^{8} + 5$ $C_5:F_5$ (as 20T26) $100$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.5 $x^{20} + 5 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.6 $x^{20} + 10 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.7 $x^{20} + 15 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.8 $x^{20} + 20 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a2.1 $x^{20} + 5 x^{8} + 5$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ $[1, 13]$
5.1.20.27a2.2 $x^{20} + 5 x^{9} + 5 x^{8} + 5$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ $[1, 13]$
5.1.20.27a3.1 $x^{20} + 20 x^{8} + 5$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ $[1, 13]$
5.1.20.27a3.2 $x^{20} + 5 x^{9} + 20 x^{8} + 5$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ $[1, 13]$
5.1.20.27a4.1 $x^{20} + 10 x^{8} + 5$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\frac{3}{2}]_{4}^{2}$ $[\frac{1}{2}]_{4}^{2}$ not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ $[1, 13]$
5.1.20.27a4.2 $x^{20} + 5 x^{9} + 10 x^{8} + 5$ $D_5\times F_5$ (as 20T51) $200$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ $[1, 13]$
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