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Properties

Label 3.1.18.25b
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(25\)

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Defining polynomial

$x^{18} + 3 b_{10} x^{10} + 3 a_{8} x^{8} + 3 a_{3} x^{3} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $25$
Artin slopes: $[\frac{5}{4},\frac{19}{12}]$
Swan slopes: $[\frac{1}{4},\frac{7}{12}]$
Means: $\langle\frac{1}{6},\frac{4}{9}\rangle$
Rams: $(\frac{1}{2},\frac{5}{2})$
Field count: $12$ (complete)
Ambiguity: $2$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[8,3,0]$
Associated inertia: $[1,1,1]$
Jump Set: undefined (show 6), $[1,4,17]$ (show 6)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 12

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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
3.1.18.25b1.1 $x^{18} + 3 x^{8} + 3 x^{3} + 3$ $C_3^4.\SOPlus(4,2)$ (as 18T505) $5832$ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 1$ $[1, 4, 17]$
3.1.18.25b1.2 $x^{18} + 3 x^{10} + 3 x^{8} + 3 x^{3} + 3$ $C_3^4.\SOPlus(4,2)$ (as 18T505) $5832$ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 1$ $[1, 4, 17]$
3.1.18.25b1.3 $x^{18} + 6 x^{10} + 3 x^{8} + 3 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 1$ $[1, 4, 17]$
3.1.18.25b1.4 $x^{18} + 6 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 1$ undefined
3.1.18.25b1.5 $x^{18} + 3 x^{10} + 6 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 1$ undefined
3.1.18.25b1.6 $x^{18} + 6 x^{10} + 6 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 1$ undefined
3.1.18.25b2.1 $x^{18} + 6 x^{8} + 3 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 2$ $[1, 4, 17]$
3.1.18.25b2.2 $x^{18} + 3 x^{10} + 6 x^{8} + 3 x^{3} + 3$ $C_3^4.\SOPlus(4,2)$ (as 18T491) $5832$ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 2$ $[1, 4, 17]$
3.1.18.25b2.3 $x^{18} + 6 x^{10} + 6 x^{8} + 3 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 2,2 z + 2$ $[1, 4, 17]$
3.1.18.25b2.4 $x^{18} + 3 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 2$ undefined
3.1.18.25b2.5 $x^{18} + 3 x^{10} + 3 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 2$ undefined
3.1.18.25b2.6 $x^{18} + 6 x^{10} + 3 x^{8} + 3 x^{3} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 3, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z + 2$ undefined
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