↑

Properties

Label 2.1.8.20d1.3-1.2.8a
Base 2.1.8.20d1.3
Degree \(2\)
e \(2\)
f \(1\)
c \(8\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.20d1.3
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2,3,3,\frac{7}{2}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $10$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $4$

Diagrams

Varying

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

Galois group: $C_2^3.D_4$ (show 2), $C_2^4.D_4$ (show 4), $C_2^5.(C_2\times D_4)$ (show 4)
Hidden Artin slopes: $[2,2,3]^{4}$ (show 4), $[2]^{4}$ (show 4), $[\ ]^{4}$ (show 2)
Indices of inseparability: $[33,26,24,8,0]$
Associated inertia: $[1,2,1]$
Jump Set: $[1,5,15,31,47]$

Fields

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

Showing all 10

  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
2.1.16.48o1.13 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T874) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.14 $x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T874) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.15 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T874) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.16 $x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T874) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.17 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{3} + 8 x + 2$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.18 $x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{3} + 8 x + 2$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.19 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.20 $x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.21 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$ $C_2^3.D_4$ (as 16T148) $64$ $2$ $[2, 3, 3, \frac{7}{2}]^{4}$ $[1,2,2,\frac{5}{2}]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
2.1.16.48o1.22 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 2$ $C_2^3.D_4$ (as 16T148) $64$ $2$ $[2, 3, 3, \frac{7}{2}]^{4}$ $[1,2,2,\frac{5}{2}]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 15, 31, 47]$
  displayed columns for results