↑

Properties

Label 2.1.16.28c
Base 2.1.1.0a1.1
Degree \(16\)
e \(16\)
f \(1\)
c \(28\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{16} + 2 b_{15} x^{15} + 2 a_{13} x^{13} + 2 a_{6} x^{6} + 4 c_{20} x^{4} + 4 b_{19} x^{3} + 4 b_{17} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $16$
Base field: $\Q_{2}$
Ramification index $e$: $16$
Residue field degree $f$: $1$
Discriminant exponent $c$: $28$
Artin slopes: $[\frac{10}{7},\frac{10}{7},\frac{10}{7},\frac{9}{4}]$
Swan slopes: $[\frac{3}{7},\frac{3}{7},\frac{3}{7},\frac{5}{4}]$
Means: $\langle\frac{3}{14},\frac{9}{28},\frac{3}{8},\frac{13}{16}\rangle$
Rams: $(\frac{3}{7},\frac{3}{7},\frac{3}{7},7)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

Indices of inseparability: $[13,6,6,6,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,4,11,27]$

Fields

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

Showing all 16

  displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.1.16.28c1.1 $x^{16} + 2 x^{13} + 2 x^{6} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.2 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.3 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.4 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x^{3} + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.5 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.6 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.7 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{3} + 4 x + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.8 $x^{16} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.9 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.10 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.11 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{3} + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.12 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x^{3} + 2$ $C_2^7.F_8:C_6$ (as 16T1841) $43008$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.13 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.14 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.15 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{3} + 4 x + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
2.1.16.28c1.16 $x^{16} + 2 x^{15} + 2 x^{13} + 2 x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 2$ $C_2^7:F_8:C_3$ (as 16T1800) $21504$ $2$ not computed not computed not computed not computed $[13, 6, 6, 6, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4, 11, 27]$
  displayed columns for results