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Properties

Label 5.1.15.16a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{15} + 5a_{2} x^{2} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Artin slopes: $[\frac{7}{6}]$
Swan slopes: $[\frac{1}{6}]$
Means: $\langle\frac{2}{15}\rangle$
Rams: $(\frac{1}{2})$
Field count: $4$ (complete)
Ambiguity: $1$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

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

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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
5.1.15.16a1.1 2 $x^{15} + 10 x^{2} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6}]^{2}_{2}$ $[2, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^2 + 1$ undefined
5.1.15.16a1.2 2 $x^{15} + 15 x^{2} + 5$ $(C_5^2 : C_3):C_4$ (as 15T17) $300$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6}]^{2}_{2}$ $[2, 0]$ $[2, 2]$ $z^{10} + 3 z^5 + 3,3 z^2 + 4$ undefined
5.1.15.16a2.1 2 $x^{15} + 5 x^{2} + 5$ $((C_5^2 : C_3):C_2):C_2$ (as 15T18) $300$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6}]^{2}_{2}$ $[2, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.16a2.2 2 $x^{15} + 20 x^{2} + 5$ $((C_5^2 : C_3):C_2):C_2$ (as 15T18) $300$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6}]^{2}_{2}$ $[2, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
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