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Properties

Label 3.1.6.11a1.7-1.3.7a
Base 3.1.6.11a1.7
Degree \(3\)
e \(3\)
f \(1\)
c \(7\)

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Defining polynomial

$x^{3} + a_{5} \pi^{2} x^{2} + b_{7} \pi^{3} x + \pi$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: 3.1.6.11a1.7
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Absolute Artin slopes: $[\frac{9}{4},\frac{5}{2}]$
Swan slopes: $[\frac{5}{2}]$
Means: $\langle\frac{5}{3}\rangle$
Rams: $(\frac{5}{2})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $6$
Absolute Mass: $2$

Diagrams

Varying

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

Indices of inseparability: $[23,15,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,7,25]$

Fields

Galois group Galois Artin Indices
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
3.1.18.40d2.13 $x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.25 $x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.31 $x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.43 $x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.61 $x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 1$ $[1, 7, 25]$
3.1.18.40d2.67 $x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 1$ $[1, 7, 25]$
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