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Properties

Label 3.1.12.16a
Base 3.1.1.0a1.1
Degree \(12\)
e \(12\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{12} + 3 b_{7} x^{7} + 3 a_{5} x^{5} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $12$
Base field: $\Q_{3}$
Ramification index $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Artin slopes: $[\frac{13}{8}]$
Swan slopes: $[\frac{5}{8}]$
Means: $\langle\frac{5}{12}\rangle$
Rams: $(\frac{5}{2})$
Field count: $6$ (complete)
Ambiguity: $2$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[2,1]$
Jump Set: undefined (show 3), $[2,11]$ (show 3)

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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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
3.1.12.16a1.1 2 $x^{12} + 3 x^{5} + 3$ $F_9:C_2$ (as 12T84) $144$ $1$ $[\frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{13}{8}]^{2}_{2}$ $[\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 1$ $[2, 11]$
3.1.12.16a1.2 4 $x^{12} + 3 x^{7} + 3 x^{5} + 3$ $C_3^4:\SD_{16}$ (as 12T212) $1296$ $1$ $[\frac{9}{8}, \frac{9}{8}, \frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{9}{8},\frac{9}{8},\frac{13}{8}]^{2}_{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 1$ $[2, 11]$
3.1.12.16a1.3 4 $x^{12} + 6 x^{7} + 3 x^{5} + 3$ $C_3^4:\SD_{16}$ (as 12T212) $1296$ $1$ $[\frac{9}{8}, \frac{9}{8}, \frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{9}{8},\frac{9}{8},\frac{13}{8}]^{2}_{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 1$ $[2, 11]$
3.1.12.16a1.4 2 $x^{12} + 3 x^{5} + 6$ $F_9:C_2$ (as 12T84) $144$ $1$ $[\frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{13}{8}]^{2}_{2}$ $[\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 2$ undefined
3.1.12.16a1.5 4 $x^{12} + 3 x^{7} + 3 x^{5} + 6$ $C_3^4:\SD_{16}$ (as 12T212) $1296$ $1$ $[\frac{9}{8}, \frac{9}{8}, \frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{9}{8},\frac{9}{8},\frac{13}{8}]^{2}_{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 2$ undefined
3.1.12.16a1.6 4 $x^{12} + 6 x^{7} + 3 x^{5} + 6$ $C_3^4:\SD_{16}$ (as 12T212) $1296$ $1$ $[\frac{9}{8}, \frac{9}{8}, \frac{13}{8}, \frac{13}{8}]_{8}^{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8},\frac{5}{8}]_{8}^{2}$ $[\frac{9}{8},\frac{9}{8},\frac{13}{8}]^{2}_{2}$ $[\frac{1}{8},\frac{1}{8},\frac{5}{8}]^{2}_{2}$ $[5, 0]$ $[2, 1]$ $z^9 + z^6 + 1,z + 2$ undefined
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