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Properties

Label 5.1.15.17a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(17\)

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Defining polynomial

$x^{15} + 5a_{3} x^{3} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
Artin slopes: $[\frac{5}{4}]$
Swan slopes: $[\frac{1}{4}]$
Means: $\langle\frac{1}{5}\rangle$
Rams: $(\frac{3}{4})$
Field count: $4$ (complete)
Ambiguity: $1$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[3,0]$
Associated inertia: $[2,1]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
5.1.15.17a1.1 4 $x^{15} + 5 x^{3} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{5}{4}]_{12}^{2}$ $[\frac{1}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[3, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 2$ undefined
5.1.15.17a1.2 4 $x^{15} + 10 x^{3} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{5}{4}]_{12}^{2}$ $[\frac{1}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[3, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 4$ undefined
5.1.15.17a1.3 4 $x^{15} + 15 x^{3} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{5}{4}]_{12}^{2}$ $[\frac{1}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[3, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 1$ undefined
5.1.15.17a1.4 4 $x^{15} + 20 x^{3} + 5$ $F_5 \times S_3$ (as 15T11) $120$ $1$ $[\frac{5}{4}]_{12}^{2}$ $[\frac{1}{4}]_{12}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[3, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z + 3$ undefined
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