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Properties

Label 5.1.15.23a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(23\)

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Defining polynomial

$x^{15} + 5 b_{11} x^{11} + 5 a_{9} x^{9} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $23$
Artin slopes: $[\frac{7}{4}]$
Swan slopes: $[\frac{3}{4}]$
Means: $\langle\frac{3}{5}\rangle$
Rams: $(\frac{9}{4})$
Field count: $20$ (complete)
Ambiguity: $1$
Mass: $20$
Absolute Mass: $20$

Diagrams

Varying

Indices of inseparability: $[9,0]$
Associated inertia: $[2,1]$
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 20

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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
5.1.15.23a1.1 4 $x^{15} + 5 x^{9} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{7}{4}]_{12}^{2}$ $[\frac{3}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.23a1.2 16 $x^{15} + 5 x^{11} + 5 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.23a1.3 16 $x^{15} + 10 x^{11} + 5 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.23a1.4 16 $x^{15} + 15 x^{11} + 5 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.23a1.5 16 $x^{15} + 20 x^{11} + 5 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.23a1.6 4 $x^{15} + 10 x^{9} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{7}{4}]_{12}^{2}$ $[\frac{3}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.23a1.7 16 $x^{15} + 5 x^{11} + 10 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.23a1.8 16 $x^{15} + 10 x^{11} + 10 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.23a1.9 16 $x^{15} + 15 x^{11} + 10 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.23a1.10 16 $x^{15} + 20 x^{11} + 10 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.23a1.11 4 $x^{15} + 15 x^{9} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{7}{4}]_{12}^{2}$ $[\frac{3}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
5.1.15.23a1.12 16 $x^{15} + 5 x^{11} + 15 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
5.1.15.23a1.13 16 $x^{15} + 10 x^{11} + 15 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
5.1.15.23a1.14 16 $x^{15} + 15 x^{11} + 15 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
5.1.15.23a1.15 16 $x^{15} + 20 x^{11} + 15 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
5.1.15.23a1.16 4 $x^{15} + 20 x^{9} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{7}{4}]_{12}^{2}$ $[\frac{3}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
5.1.15.23a1.17 16 $x^{15} + 5 x^{11} + 20 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
5.1.15.23a1.18 16 $x^{15} + 10 x^{11} + 20 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
5.1.15.23a1.19 16 $x^{15} + 15 x^{11} + 20 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
5.1.15.23a1.20 16 $x^{15} + 20 x^{11} + 20 x^{9} + 5$ $C_5^3:(C_4\times S_3)$ (as 15T49) $3000$ $1$ $[\frac{13}{12}, \frac{13}{12}, \frac{7}{4}]_{12}^{2}$ $[\frac{1}{12},\frac{1}{12},\frac{3}{4}]_{12}^{2}$ $[\frac{13}{12},\frac{13}{12}]^{2}_{4}$ $[\frac{1}{12},\frac{1}{12}]^{2}_{4}$ $[9, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
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