↑

Properties

Label 2.1.2.2a1.2-3.2.12a
Base 2.1.2.2a1.2
Degree \(6\)
e \(2\)
f \(3\)
c \(12\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{-5})$
Ramification index $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[2,3]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $22$ (complete)
Ambiguity: $6$
Mass: $56$
Absolute Mass: $28/3$

Diagrams

Varying

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

Galois group: $C_6\times C_2$ (show 2), $D_4 \times C_3$ (show 1), $C_2^2 \times A_4$ (show 2), $D_4\times A_4$ (show 1), $C_2^3:A_4$ (show 2), $C_2^4:A_4$ (show 6), $C_2\wr C_6$ (show 8)
Hidden Artin slopes: $[2,2,2,3]^{2}$ (show 4), $[2,2,3]$ (show 2), $[2,2]^{2}$ (show 1), $[2,2,3,3]$ (show 4), $[2,2,3]^{2}$ (show 2), $[2,2,3,3]^{2}$ (show 4), $[2,2]$ (show 2), $[\ ]$ (show 2), $[\ ]^{2}$ (show 1)
Indices of inseparability: $[5,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,8]$

Fields

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

Showing all 22

  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
2.3.4.24b1.7 $( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 6$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3]^{6}$ $[1,2]^{6}$ $[\ ]^{2}$ $[\ ]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b1.8 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 6$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3]^{6}$ $[1,1,1,2]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b1.9 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 6$ $C_6\times C_2$ (as 12T2) $12$ $12$ $[2, 3]^{3}$ $[1,2]^{3}$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b1.10 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 14$ $C_6\times C_2$ (as 12T2) $12$ $12$ $[2, 3]^{3}$ $[1,2]^{3}$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b1.11 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 6$ $C_2^2 \times A_4$ (as 12T25) $48$ $4$ $[2, 2, 2, 3]^{3}$ $[1,1,1,2]^{3}$ $[2,2]$ $[1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b1.12 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 14$ $C_2^2 \times A_4$ (as 12T25) $48$ $4$ $[2, 2, 2, 3]^{3}$ $[1,1,1,2]^{3}$ $[2,2]$ $[1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + t^2$ $[1, 2, 8]$
2.3.4.24b2.9 $( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, 3, 3]^{6}$ $[1,1,1,2,2,2]^{6}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.10 $( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 14$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, 3, 3]^{6}$ $[1,1,1,2,2,2]^{6}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.11 $( x^{3} + x + 1 )^{4} + 4 x^{2} ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3, 3]^{3}$ $[1,1,1,2,2,2]^{3}$ $[2,2,3,3]$ $[1,1,2,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.12 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3, 3]^{3}$ $[1,1,1,2,2,2]^{3}$ $[2,2,3,3]$ $[1,1,2,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.13 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3, 3]^{3}$ $[1,1,1,2,2,2]^{3}$ $[2,2,3,3]$ $[1,1,2,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.14 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, 3, 3]^{6}$ $[1,1,1,2,2,2]^{6}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.15 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 14$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, 3, 3]^{6}$ $[1,1,1,2,2,2]^{6}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b2.16 $( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 4 x ( x^{3} + x + 1 ) + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3, 3]^{3}$ $[1,1,1,2,2,2]^{3}$ $[2,2,3,3]$ $[1,1,2,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t + 1)$ $[1, 2, 8]$
2.3.4.24b3.9 $( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,2,3]^{2}$ $[1,1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.10 $( x^{3} + x + 1 )^{4} + 4 x^{2} ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2^3:A_4$ (as 12T58) $96$ $2$ $[2, 2, 2, 3, 3]^{3}$ $[1,1,1,2,2]^{3}$ $[2,2,3]$ $[1,1,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.11 $( x^{3} + x + 1 )^{4} + 4 x^{2} ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 8 x^{2} + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3]^{6}$ $[1,1,1,2,2]^{6}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.12 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,2,3]^{2}$ $[1,1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.13 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,2,3]^{2}$ $[1,1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.14 $( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2^3:A_4$ (as 12T58) $96$ $2$ $[2, 2, 2, 3, 3]^{3}$ $[1,1,1,2,2]^{3}$ $[2,2,3]$ $[1,1,2]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.15 $( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 8 x^{2} + 6$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, 3]^{6}$ $[1,1,1,2,2]^{6}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
2.3.4.24b3.16 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 6$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,2,3]^{2}$ $[1,1,1,2]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t + 1),(t^2 + t + 1) z + (t^2 + t + 1)$ $[1, 2, 8]$
  displayed columns for results