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Properties

Label 2.1.2.3a1.3-3.2.15a
Base 2.1.2.3a1.3
Degree \(6\)
e \(2\)
f \(3\)
c \(15\)

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Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $28$ (complete)
Ambiguity: $6$
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: $[2,2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2,2]$ (show 4), $[2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2,2,2]$ (show 2), $[2]$ (show 2), $[\ ]$ (show 4)
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
2.3.4.33a1.177 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 2$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.178 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 18$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.179 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 2$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.180 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 18$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.181 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 2$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.182 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 18$ $C_{12}$ (as 12T1) $12$ $12$ $[3, 4]^{3}$ $[2,3]^{3}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.183 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 2$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.184 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 18$ $C_4\times A_4$ (as 12T29) $48$ $4$ $[2, 2, 3, 4]^{3}$ $[1,1,2,3]^{3}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.185 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.186 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.187 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.188 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.189 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.190 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.191 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.192 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 2$ $C_2^4:C_{12}$ (as 12T105) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.193 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 2$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.194 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 18$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.195 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 2$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.196 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 18$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.197 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.198 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.199 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.200 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.201 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.202 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.203 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.204 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 2$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
  displayed columns for results