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Properties

Label 3.1.21.24a
Base 3.1.1.0a1.1
Degree \(21\)
e \(21\)
f \(1\)
c \(24\)

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Defining polynomial

$x^{21} + 3 c_{6} x^{6} + 3 b_{5} x^{5} + 3 a_{4} x^{4} + 3$

Invariants

Residue field characteristic: $3$
Degree: $21$
Base field: $\Q_{3}$
Ramification index $e$: $21$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Artin slopes: $[\frac{9}{7}]$
Swan slopes: $[\frac{2}{7}]$
Means: $\langle\frac{4}{21}\rangle$
Rams: $(2)$
Field count: $12$ (complete)
Ambiguity: $3$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[4,0]$
Associated inertia: $[6,1]$ (show 9), $[6,2]$ (show 3)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
3.1.21.24a1.1 $x^{21} + 6 x^{4} + 3$ $C_3^6:F_7$ (as 21T78) $30618$ $1$ not computed not computed not computed not computed $[4, 0]$ $[6, 2]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$ undefined
3.1.21.24a1.2 $x^{21} + 3 x^{5} + 6 x^{4} + 3$ $C_3^6:F_7$ (as 21T78) $30618$ $1$ not computed not computed not computed not computed $[4, 0]$ $[6, 2]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$ undefined
3.1.21.24a1.3 $x^{21} + 6 x^{5} + 6 x^{4} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 0]$ $[6, 2]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$ undefined
3.1.21.24a2.1 $x^{21} + 3 x^{4} + 3$ $C_3^6:F_7$ (as 21T81) $30618$ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.2 $x^{21} + 3 x^{6} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.3 $x^{21} + 6 x^{6} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.4 $x^{21} + 3 x^{5} + 3 x^{4} + 3$ $C_3^6:F_7$ (as 21T81) $30618$ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.5 $x^{21} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.6 $x^{21} + 6 x^{6} + 3 x^{5} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.7 $x^{21} + 6 x^{5} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.8 $x^{21} + 3 x^{6} + 6 x^{5} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
3.1.21.24a2.9 $x^{21} + 6 x^{6} + 6 x^{5} + 3 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[6, 1]$ $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 2$ undefined
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