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Properties

Label 2.2.4.22a1.39-1.2.4a
Base 2.2.4.22a1.39
Degree \(2\)
e \(2\)
f \(1\)
c \(4\)

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Defining polynomial

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.22a1.39
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Absolute Artin slopes: $[3,\frac{7}{2},4]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $24$ (complete)
Ambiguity: $2$
Mass: $12$
Absolute Mass: $6$

Diagrams

Varying

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

Galois group: $C_2^4.D_4$ (show 8), $C_2^6.D_4$ (show 8), $C_2^6.D_4$ (show 8)
Hidden Artin slopes: $[2,2,\frac{7}{2}]$ (show 8), $[2,2,3,\frac{7}{2}]^{2}$ (show 8), $[2,2,3,3]^{2}$ (show 8)
Indices of inseparability: $[19,14,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 24

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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
2.2.8.52c1.182 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.207 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.363 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.369 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + \left(12 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.387 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.389 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.403 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{5} + \left(12 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c1.405 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T912) $512$ $4$ $[2, 2, 3, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,2,\frac{5}{2},3]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.52c5.102 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.117 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.132 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.139 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.160 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.175 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.178 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.185 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^4.D_4$ (as 16T389) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.491 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.506 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.519 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.526 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.529 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.548 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.565 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.52c5.576 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.D_4$ (as 16T938) $512$ $4$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,3,\frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[19, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
  displayed columns for results