↑

Properties

Label 2.1.4.11a1.5-1.4.18a
Base 2.1.4.11a1.5
Degree \(4\)
e \(4\)
f \(1\)
c \(18\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + \left(b_{19} \pi^{5} + a_{15} \pi^{4}\right) x^{3} + \left(b_{18} \pi^{5} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + b_{17} \pi^{5} x + c_{20} \pi^{6} + \pi$

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 $c$: $18$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{17}{4}]$
Swan slopes: $[5,5]$
Means: $\langle\frac{5}{2},\frac{15}{4}\rangle$
Rams: $(5,5)$
Field count: $24$ (complete)
Ambiguity: $2$
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: $C_2^4.(C_4\times D_4)$ (show 16), $C_2^6:(C_2\times A_4)$ (show 8) (incomplete)
Hidden Artin slopes: $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 16), not computed (show 6), $[2,2,2,\frac{7}{2},\frac{7}{2}]^{3}$ (show 2) (incomplete)
Indices of inseparability: $[47,42,32,16,0]$ (show 8), $[47,46,32,16,0]$ (show 8), $[47,47,32,16,0]$ (show 8)
Associated inertia: $[1,1,2]$ (show 16), $[1,1,3]$ (show 8)
Jump Set: $[1,3,7,15,31]$

Fields

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

Showing all 24

  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
2.1.16.62g1.65 $x^{16} + 8 x^{15} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.66 $x^{16} + 8 x^{15} + 16 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.67 $x^{16} + 8 x^{15} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.68 $x^{16} + 8 x^{15} + 16 x^{4} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.73 $x^{16} + 8 x^{15} + 8 x^{14} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.74 $x^{16} + 8 x^{15} + 8 x^{14} + 16 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.75 $x^{16} + 8 x^{15} + 8 x^{14} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.76 $x^{16} + 8 x^{15} + 8 x^{14} + 16 x^{4} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.81 $x^{16} + 8 x^{15} + 8 x^{12} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.82 $x^{16} + 8 x^{15} + 8 x^{12} + 16 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.83 $x^{16} + 8 x^{15} + 8 x^{12} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.84 $x^{16} + 8 x^{15} + 8 x^{12} + 16 x^{4} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.89 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.90 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.91 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.92 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 16 x^{4} + 16 x^{2} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T824) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.51 $x^{16} + 8 x^{15} + 8 x^{10} + 8 x^{4} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.52 $x^{16} + 8 x^{15} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.55 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{10} + 8 x^{4} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.56 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.59 $x^{16} + 8 x^{15} + 8 x^{12} + 8 x^{10} + 8 x^{4} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.60 $x^{16} + 8 x^{15} + 8 x^{12} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ not computed not computed not computed not computed $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.63 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 8 x^{4} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]^{3}$ $[1,1,1,\frac{5}{2},\frac{5}{2}]^{3}$ $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g2.64 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 26$ $C_2^6:(C_2\times A_4)$ (as 16T1299) $1536$ $1$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]^{3}$ $[1,1,1,\frac{5}{2},\frac{5}{2}]^{3}$ $[47, 42, 32, 16, 0]$ $[1, 1, 3]$ $z^8 + 1,z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15, 31]$
  displayed columns for results