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Properties

Label 3.1.4.3a1.1-1.3.10a
Base 3.1.4.3a1.1
Degree \(3\)
e \(3\)
f \(1\)
c \(10\)

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Defining polynomial

$x^{3} + \left(b_{11} \pi^{4} + a_{8} \pi^{3}\right) x^{2} + b_{10} \pi^{4} x + c_{12} \pi^{5} + \pi$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: 3.1.4.3a1.1
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[2]$
Swan slopes: $[4]$
Means: $\langle\frac{8}{3}\rangle$
Rams: $(4)$
Field count: $24$ (complete)
Ambiguity: $3$
Mass: $18$
Absolute Mass: $9$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Galois group: $D_{12}$ (show 1), $D_4 \times C_3$ (show 3), $C_3:D_{12}$ (show 2), $C_6\wr C_2$ (show 6), $C_3^2:D_{12}$ (show 1), $S_3^2:C_6$ (show 3), $C_3\wr D_4$ (show 6), $C_3^3:D_{12}$ (show 2)
Hidden Artin slopes: $[\ ]^{2}$ (show 4), $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ (show 8), $[\frac{3}{2}]^{2}$ (show 8), $[\frac{5}{4},\frac{5}{4}]^{2}$ (show 4)
Indices of inseparability: $[8,0]$
Associated inertia: $[2,1]$ (show 18), $[2,2]$ (show 6)
Jump Set: $[2,6]$ (show 1), $[2,14]$ (show 6), $[2,16]$ (show 12), $[2,17]$ (show 3), $[2,18]$ (show 2)

Fields

Galois group Galois Artin Indices Hidden content
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
3.1.12.19a1.4 $x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{8} + 3$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a1.6 $x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{8} + 3$ $C_3^3:D_{12}$ (as 12T169) $648$ $1$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 2]$ $z^9 + z^6 + 1,z^2 + 1$ $[2, 14]$
3.1.12.19a2.10 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 3$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.11 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 12$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.12 $x^{12} + 3 x^{11} + 3 x^{10} + 6 x^{8} + 21$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.16 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 3$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.17 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 12$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
3.1.12.19a2.18 $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{8} + 21$ $C_3\wr D_4$ (as 12T167) $648$ $3$ $[\frac{5}{4}, \frac{5}{4}, \frac{3}{2}, 2]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2},1]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$ $[\frac{1}{4},\frac{1}{4},\frac{1}{2}]^{2}$ $[8, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z^2 + 2$ $[2, 16]$
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