↑

Properties

Label 2.1.4.11a1.5-1.4.12a
Base 2.1.4.11a1.5
Degree \(4\)
e \(4\)
f \(1\)
c \(12\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + b_{11} \pi^{3} x^{3} + \left(b_{10} \pi^{3} + b_{6} \pi^{2}\right) x^{2} + a_{9} \pi^{3} x + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.5
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{7}{2},4]$
Swan slopes: $[3,3]$
Means: $\langle\frac{3}{2},\frac{9}{4}\rangle$
Rams: $(3,3)$
Field count: $12$ (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: $C_2^4:D_4$ (show 8), $C_2\wr C_6$ (show 4)
Hidden Artin slopes: $[2,2]^{2}$ (show 8), $[2,2,2]^{3}$ (show 4)
Indices of inseparability: $[41,34,28,16,0]$ (show 4), $[41,34,32,16,0]$ (show 8)
Associated inertia: $[1,2,1]$ (show 8), $[1,3,1]$ (show 4)
Jump Set: $[1,3,7,15,31]$

Fields

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

Showing all 12

  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
2.1.16.56k1.50 $x^{16} + 8 x^{13} + 8 x^{9} + 8 x^{8} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.51 $x^{16} + 8 x^{15} + 8 x^{13} + 8 x^{9} + 8 x^{8} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.59 $x^{16} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 8 x^{8} + 8 x^{6} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.60 $x^{16} + 8 x^{15} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 8 x^{8} + 8 x^{6} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.61 $x^{16} + 8 x^{9} + 8 x^{4} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.62 $x^{16} + 8 x^{15} + 8 x^{9} + 8 x^{4} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.77 $x^{16} + 8 x^{11} + 8 x^{9} + 8 x^{6} + 8 x^{4} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k1.78 $x^{16} + 8 x^{15} + 8 x^{11} + 8 x^{9} + 8 x^{6} + 8 x^{4} + 8 x^{2} + 10$ $C_2^4:D_4$ (as 16T392) $128$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[41, 34, 32, 16, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k2.121 $x^{16} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 4 x^{8} + 8 x^{4} + 8 x^{2} + 10$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[41, 34, 28, 16, 0]$ $[1, 3, 1]$ $z^8 + 1,z^6 + z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k2.122 $x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 4 x^{8} + 8 x^{4} + 8 x^{2} + 10$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[41, 34, 28, 16, 0]$ $[1, 3, 1]$ $z^8 + 1,z^6 + z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k2.125 $x^{16} + 4 x^{12} + 8 x^{9} + 4 x^{8} + 8 x^{6} + 8 x^{4} + 8 x^{2} + 10$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[41, 34, 28, 16, 0]$ $[1, 3, 1]$ $z^8 + 1,z^6 + z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.56k2.126 $x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{9} + 4 x^{8} + 8 x^{6} + 8 x^{4} + 8 x^{2} + 10$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[41, 34, 28, 16, 0]$ $[1, 3, 1]$ $z^8 + 1,z^6 + z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
  displayed columns for results