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Properties

Label 2.3.2.9a1.6-1.2.5a
Base 2.3.2.9a1.6
Degree \(2\)
e \(2\)
f \(1\)
c \(5\)

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Defining polynomial

$x^{2} + \left(b_{7} \pi^{4} + b_{5} \pi^{3}\right) x + c_{8} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.3.2.9a1.6
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $28$ (complete)
Ambiguity: $2$
Mass: $64$
Absolute Mass: $32/3$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $C_{12}$ (show 4), $D_4 \times C_3$ (show 2), $C_4\times A_4$ (show 4), $D_4\times A_4$ (show 2), $C_2^4:C_{12}$ (show 8), $C_2\wr C_6$ (show 8)
Hidden Artin slopes: $[2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2,2,2]$ (show 2), $[2,2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[\ ]$ (show 4), $[2,2]$ (show 4), $[2]$ (show 2)
Indices of inseparability: $[8,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 28

  displayed columns for results
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
2.3.4.33a1.205 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 10$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.206 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 26$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.207 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 10$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.208 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 26$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.209 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 10$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.210 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 26$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.211 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 10$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.212 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 26$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.213 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.214 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.215 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.216 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.217 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.218 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.219 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.220 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.221 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.222 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 26$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.223 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.224 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 26$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.225 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.226 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.227 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.228 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.229 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.230 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.231 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.232 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
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