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Properties

Label 3.2.3.10a1.1-1.3.7a
Base 3.2.3.10a1.1
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: 3.2.3.10a1.1
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Absolute Artin slopes: $[\frac{5}{2},\frac{17}{6}]$
Swan slopes: $[\frac{5}{2}]$
Means: $\langle\frac{5}{3}\rangle$
Rams: $(\frac{5}{2})$
Field count: $39$ (complete)
Ambiguity: $1$
Mass: $72$
Absolute Mass: $36$

Diagrams

Varying

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

Indices of inseparability: $[14,9,0]$
Associated inertia: $[1,1]$
Jump Set: undefined

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
3.2.9.44b1.1 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:D_6$ (as 18T41) $108$ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b1.2 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{7} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:S_3^2$ (as 18T121) $324$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b1.3 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{7} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:D_6$ (as 18T41) $108$ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b1.4 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b1.5 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{7} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:D_6$ (as 18T41) $108$ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b1.6 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 18 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:S_3^2$ (as 18T121) $324$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2$ undefined
3.2.9.44b2.1 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b2.2 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{7} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b2.3 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{7} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b2.4 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b2.5 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{7} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b2.6 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 9 ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^2:S_3^2$ (as 18T139) $324$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 1$ undefined
3.2.9.44b3.1 $( x^{2} + 2 x + 2 )^{9} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.2 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.3 $( x^{2} + 2 x + 2 )^{9} + 18 x ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.4 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.5 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.6 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.7 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.8 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b3.9 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 18 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + 2 t$ undefined
3.2.9.44b4.1 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:D_6$ (as 18T236) $972$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.2 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.3 $( x^{2} + 2 x + 2 )^{9} + 18 x ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:D_6$ (as 18T236) $972$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.4 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.5 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:D_6$ (as 18T236) $972$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.6 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.7 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.8 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:D_6$ (as 18T236) $972$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b4.9 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + (2 t + 2)$ undefined
3.2.9.44b5.1 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.2 $( x^{2} + 2 x + 2 )^{9} + 9 x ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.3 $( x^{2} + 2 x + 2 )^{9} + 18 x ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.4 $( x^{2} + 2 x + 2 )^{9} + 9 ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.5 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.6 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.7 $( x^{2} + 2 x + 2 )^{9} + 18 ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.8 $( x^{2} + 2 x + 2 )^{9} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ $C_3^4:(C_4\times S_3)$ (as 18T344) $1944$ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
3.2.9.44b5.9 $( x^{2} + 2 x + 2 )^{9} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{7} + 9 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 9, 0]$ $[1, 1]$ $z^3 + (t + 1),(t + 1) z + t$ undefined
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