↑

Properties

Label 2.2.2.6a1.4-1.4.10a
Base 2.2.2.6a1.4
Degree \(4\)
e \(4\)
f \(1\)
c \(10\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + a_{7} \pi^{2} x^{3} + b_{6} \pi^{2} x^{2} + b_{9} \pi^{3} x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.2.2.6a1.4
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[3,\frac{19}{6},\frac{19}{6}]$
Swan slopes: $[\frac{7}{3},\frac{7}{3}]$
Means: $\langle\frac{7}{6},\frac{7}{4}\rangle$
Rams: $(\frac{7}{3},\frac{7}{3})$
Field count: $24$ (complete)
Ambiguity: $1$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

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

Galois group: $C_2^6:S_4$ (show 8), $C_2^6.A_4\wr C_2$ (show 16) (incomplete)
Hidden Artin slopes: not computed (show 14), $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3},\frac{8}{3}]_{3}$ (show 4), $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{3}_{3}$ (show 2), $[\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]_{3}$ (show 4) (incomplete)
Indices of inseparability: $[15,14,8,0]$ (show 18), $[15,15,8,0]$ (show 6)
Associated inertia: $[1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing all 24

  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
2.2.8.44c1.6 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3},\frac{8}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.10 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3},\frac{8}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.13 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.19 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]_{3}$ $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.28 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3},\frac{8}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.32 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3},\frac{8}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.33 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c1.35 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]_{3}$ $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.44c2.3 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.11 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.12 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.23 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, \frac{19}{6}, \frac{19}{6}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},\frac{13}{6},\frac{13}{6}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{3}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{3}_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.24 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, \frac{19}{6}, \frac{19}{6}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},\frac{13}{6},\frac{13}{6}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{3}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{3}_{3}$ $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.31 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.32 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.35 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.36 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 15, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.43 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.44 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.55 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.56 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.63 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.44c2.64 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[15, 14, 8, 0]$ $[1, 1]$ $z^4 + 1,z + t$ $[1, 3, 7, 15]$
  displayed columns for results