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Properties

Label 5.1.15.22a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(22\)

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Defining polynomial

$x^{15} + 5 c_{10} x^{10} + 5 b_{9} x^{9} + 5 a_{8} x^{8} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $22$
Artin slopes: $[\frac{5}{3}]$
Swan slopes: $[\frac{2}{3}]$
Means: $\langle\frac{8}{15}\rangle$
Rams: $(2)$
Field count: $40$ (complete)
Ambiguity: $5$
Mass: $20$
Absolute Mass: $20$

Diagrams

Varying

Indices of inseparability: $[8,0]$
Associated inertia: $[2,1]$ (show 25), $[2,2]$ (show 5), $[2,4]$ (show 10)
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 40

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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
5.1.15.22a1.1 5 $x^{15} + 15 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.22a1.2 5 $x^{15} + 5 x^{9} + 15 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.22a1.3 5 $x^{15} + 10 x^{9} + 15 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.22a1.4 5 $x^{15} + 15 x^{9} + 15 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.22a1.5 5 $x^{15} + 20 x^{9} + 15 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.22a2.1 5 $x^{15} + 5 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.2 20 $x^{15} + 5 x^{10} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.3 20 $x^{15} + 10 x^{10} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.4 20 $x^{15} + 15 x^{10} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.5 20 $x^{15} + 20 x^{10} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.6 5 $x^{15} + 5 x^{9} + 5 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.7 20 $x^{15} + 5 x^{10} + 5 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.8 20 $x^{15} + 10 x^{10} + 5 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.9 20 $x^{15} + 15 x^{10} + 5 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.10 20 $x^{15} + 20 x^{10} + 5 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.11 5 $x^{15} + 10 x^{9} + 5 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.12 20 $x^{15} + 5 x^{10} + 10 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.13 20 $x^{15} + 10 x^{10} + 10 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.14 20 $x^{15} + 15 x^{10} + 10 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.15 20 $x^{15} + 20 x^{10} + 10 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.16 5 $x^{15} + 15 x^{9} + 5 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.17 20 $x^{15} + 5 x^{10} + 15 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.18 20 $x^{15} + 10 x^{10} + 15 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.19 20 $x^{15} + 15 x^{10} + 15 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.20 20 $x^{15} + 20 x^{10} + 15 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.21 5 $x^{15} + 20 x^{9} + 5 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.22 20 $x^{15} + 5 x^{10} + 20 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.23 20 $x^{15} + 10 x^{10} + 20 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.24 20 $x^{15} + 15 x^{10} + 20 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a2.25 20 $x^{15} + 20 x^{10} + 20 x^{9} + 5 x^{8} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{10}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{10}$ $[\frac{5}{3}]^{10}$ $[\frac{2}{3}]^{10}$ $[8, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.22a3.1 5 $x^{15} + 20 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.22a3.2 5 $x^{15} + 5 x^{9} + 20 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.22a3.3 5 $x^{15} + 10 x^{9} + 20 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.22a3.4 5 $x^{15} + 15 x^{9} + 20 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.22a3.5 5 $x^{15} + 20 x^{9} + 20 x^{8} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]^{2}$ $[\frac{2}{3}]^{2}$ $[8, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.22a4.1 5 $x^{15} + 10 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
5.1.15.22a4.2 5 $x^{15} + 5 x^{9} + 10 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
5.1.15.22a4.3 5 $x^{15} + 10 x^{9} + 10 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
5.1.15.22a4.4 5 $x^{15} + 15 x^{9} + 10 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
5.1.15.22a4.5 5 $x^{15} + 20 x^{9} + 10 x^{8} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{4}$ $[\frac{2}{3}]^{4}$ $[8, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
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