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Properties

Label 5.1.15.18a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{15} + 5 c_{5} x^{5} + 5 a_{4} x^{4} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Artin slopes: $[\frac{4}{3}]$
Swan slopes: $[\frac{1}{3}]$
Means: $\langle\frac{4}{15}\rangle$
Rams: $(1)$
Field count: $8$ (complete)
Ambiguity: $5$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[4,0]$
Associated inertia: $[2,1]$ (show 5), $[2,2]$ (show 1), $[2,4]$ (show 2)
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 8

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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
5.1.15.18a1.1 1 $x^{15} + 5 x^{4} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{4}$ $[\frac{4}{3}]^{4}$ $[\frac{1}{3}]^{4}$ $[4, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 1$ undefined
5.1.15.18a2.1 1 $x^{15} + 10 x^{4} + 5$ $(C_5^2 : C_3):C_2$ (as 15T13) $150$ $5$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $[4, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.18a2.2 4 $x^{15} + 5 x^{5} + 10 x^{4} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{10}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{10}$ $[\frac{4}{3}]^{10}$ $[\frac{1}{3}]^{10}$ $[4, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.18a2.3 4 $x^{15} + 10 x^{5} + 10 x^{4} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{10}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{10}$ $[\frac{4}{3}]^{10}$ $[\frac{1}{3}]^{10}$ $[4, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.18a2.4 4 $x^{15} + 15 x^{5} + 10 x^{4} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{10}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{10}$ $[\frac{4}{3}]^{10}$ $[\frac{1}{3}]^{10}$ $[4, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.18a2.5 4 $x^{15} + 20 x^{5} + 10 x^{4} + 5$ $C_5\wr S_3$ (as 15T32) $750$ $5$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{10}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{10}$ $[\frac{4}{3}]^{10}$ $[\frac{1}{3}]^{10}$ $[4, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^4 + 2$ undefined
5.1.15.18a3.1 1 $x^{15} + 15 x^{4} + 5$ $(C_5^2 : C_3):C_2$ (as 15T14) $150$ $1$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $[4, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^4 + 3$ undefined
5.1.15.18a4.1 1 $x^{15} + 20 x^{4} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{4}$ $[\frac{4}{3}]^{4}$ $[\frac{1}{3}]^{4}$ $[4, 0]$ $[2, 4]$ $z^{10} + 3 z^5 + 3,3 z^4 + 4$ undefined
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