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Properties

Label 5.1.2.1a1.2-1.10.17a
Base 5.1.2.1a1.2
Degree \(10\)
e \(10\)
f \(1\)
c \(17\)

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Defining polynomial

$x^{10} + b_{9} \pi x^{9} + a_{8} \pi x^{8} + c_{10} \pi^{2} + d_{0} \pi$

Invariants

Residue field characteristic: $5$
Degree: $10$
Base field: $\Q_{5}(\sqrt{5\cdot 2})$
Ramification index $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
Absolute Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[1]$
Means: $\langle\frac{4}{5}\rangle$
Rams: $(2)$
Field count: $28$ (complete)
Ambiguity: $10$
Mass: $20$
Absolute Mass: $10$

Diagrams

Varying

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

Indices of inseparability: $[8,0]$
Associated inertia: $[1,1]$ (show 16), $[1,2]$ (show 4), $[1,4]$ (show 8)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 28

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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
5.1.20.27a1.9 $x^{20} + 5 x^{8} + 10$ $C_5:C_4$ (as 20T2) $20$ $20$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.10 $x^{20} + 5 x^{10} + 5 x^{8} + 10$ not computed $ $not computed$ $ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.11 $x^{20} + 10 x^{10} + 5 x^{8} + 10$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.12 $x^{20} + 5 x^{9} + 5 x^{8} + 10$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.13 $x^{20} + 5 x^{10} + 5 x^{9} + 5 x^{8} + 10$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.14 $x^{20} + 10 x^{10} + 5 x^{9} + 5 x^{8} + 10$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.15 $x^{20} + 15 x^{10} + 5 x^{9} + 5 x^{8} + 10$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.16 $x^{20} + 20 x^{10} + 5 x^{9} + 5 x^{8} + 10$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.17 $x^{20} + 20 x^{8} + 15$ not computed $ $not computed$ $ $20$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.18 $x^{20} + 5 x^{10} + 20 x^{8} + 15$ not computed $ $not computed$ $ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.19 $x^{20} + 10 x^{10} + 20 x^{8} + 15$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.20 $x^{20} + 5 x^{9} + 20 x^{8} + 15$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.21 $x^{20} + 5 x^{10} + 5 x^{9} + 20 x^{8} + 15$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.22 $x^{20} + 10 x^{10} + 5 x^{9} + 20 x^{8} + 15$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.23 $x^{20} + 15 x^{10} + 5 x^{9} + 20 x^{8} + 15$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a1.24 $x^{20} + 20 x^{10} + 5 x^{9} + 20 x^{8} + 15$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ undefined
5.1.20.27a2.3 $x^{20} + 10 x^{8} + 10$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ undefined
5.1.20.27a2.4 $x^{20} + 5 x^{9} + 10 x^{8} + 10$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ undefined
5.1.20.27a2.5 $x^{20} + 15 x^{8} + 15$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ undefined
5.1.20.27a2.6 $x^{20} + 5 x^{9} + 15 x^{8} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ undefined
5.1.20.27a3.3 $x^{20} + 15 x^{8} + 10$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ undefined
5.1.20.27a3.4 $x^{20} + 5 x^{9} + 15 x^{8} + 10$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ undefined
5.1.20.27a3.5 $x^{20} + 10 x^{8} + 15$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ undefined
5.1.20.27a3.6 $x^{20} + 5 x^{9} + 10 x^{8} + 15$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ undefined
5.1.20.27a4.3 $x^{20} + 20 x^{8} + 10$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ undefined
5.1.20.27a4.4 $x^{20} + 5 x^{9} + 20 x^{8} + 10$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ undefined
5.1.20.27a4.5 $x^{20} + 5 x^{8} + 15$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ undefined
5.1.20.27a4.6 $x^{20} + 5 x^{9} + 5 x^{8} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ undefined
  displayed columns for results