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Properties

Label 5.1.15.20a
Base 5.1.1.0a1.1
Degree \(15\)
e \(15\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{15} + 5 b_{7} x^{7} + 5 a_{6} x^{6} + 5$

Invariants

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

Diagrams

Varying

Indices of inseparability: $[6,0]$
Associated inertia: $[2,1]$ (show 10), $[2,2]$ (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 10

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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
5.1.15.20a2.1 2 $x^{15} + 10 x^{6} + 5$ $D_5\times S_3$ (as 15T7) $60$ $1$ $[\frac{3}{2}]_{6}^{2}$ $[\frac{1}{2}]_{6}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.20a2.2 8 $x^{15} + 5 x^{7} + 10 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.20a2.3 8 $x^{15} + 10 x^{7} + 10 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.20a2.4 8 $x^{15} + 15 x^{7} + 10 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.20a2.5 8 $x^{15} + 20 x^{7} + 10 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 3$ undefined
5.1.15.20a2.6 2 $x^{15} + 15 x^{6} + 5$ $D_5\times S_3$ (as 15T7) $60$ $1$ $[\frac{3}{2}]_{6}^{2}$ $[\frac{1}{2}]_{6}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
5.1.15.20a2.7 8 $x^{15} + 5 x^{7} + 15 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
5.1.15.20a2.8 8 $x^{15} + 10 x^{7} + 15 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
5.1.15.20a2.9 8 $x^{15} + 15 x^{7} + 15 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
5.1.15.20a2.10 8 $x^{15} + 20 x^{7} + 15 x^{6} + 5$ $C_5^3:D_6$ (as 15T40) $1500$ $1$ $[\frac{7}{6}, \frac{7}{6}, \frac{3}{2}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6},\frac{1}{2}]_{6}^{2}$ $[\frac{7}{6},\frac{7}{6}]^{2}_{2}$ $[\frac{1}{6},\frac{1}{6}]^{2}_{2}$ $[6, 0]$ $[2, 1]$ $z^{10} + 3 z^5 + 3,3 z^2 + 2$ undefined
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