↑

Properties

Label 2.1.2.2a1.1-1.4.18a
Base 2.1.2.2a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(18\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + \left(b_{19} \pi^{5} + a_{15} \pi^{4}\right) x^{3} + \left(c_{22} \pi^{6} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-1})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Absolute Artin slopes: $[2,\frac{7}{2},\frac{17}{4}]$
Swan slopes: $[4,\frac{11}{2}]$
Means: $\langle2,\frac{15}{4}\rangle$
Rams: $(4,7)$
Field count: $32$ (complete)
Ambiguity: $4$
Mass: $32$
Absolute Mass: $16$

Diagrams

Varying

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

Galois group: $C_2^3 : C_4 $ (show 16), $C_2 \wr C_2\wr C_2$ (show 16)
Hidden Artin slopes: $[3,4]$ (show 16), $[2,3,4]^{2}$ (show 16)
Indices of inseparability: $[19,12,4,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,5,13,21]$

Fields

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

Showing all 32

  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
2.1.8.26b1.1 $x^{8} + 2 x^{4} + 8 x^{3} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.2 $x^{8} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.3 $x^{8} + 2 x^{4} + 8 x^{3} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.4 $x^{8} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.5 $x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.6 $x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.7 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.8 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.9 $x^{8} + 10 x^{4} + 8 x^{3} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.10 $x^{8} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.11 $x^{8} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.12 $x^{8} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.13 $x^{8} + 8 x^{7} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.14 $x^{8} + 8 x^{7} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.15 $x^{8} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.16 $x^{8} + 8 x^{7} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.17 $x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.18 $x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.19 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.20 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.21 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.22 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.23 $x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.24 $x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.25 $x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.26 $x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.27 $x^{8} + 4 x^{6} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.28 $x^{8} + 4 x^{6} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.29 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.30 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[3,4]$ $[2,3]$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.31 $x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.1.8.26b1.32 $x^{8} + 8 x^{7} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,3,4]^{2}$ $[1,2,3]^{2}$ $[19, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
  displayed columns for results