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Properties

Label 2.1.4.11a1.12-1.4.20d
Base 2.1.4.11a1.12
Degree \(4\)
e \(4\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{4} + \left(b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + \left(b_{18} \pi^{5} + b_{14} \pi^{4} + a_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + a_{17} \pi^{5}\right) x + c_{24} \pi^{7} + c_{20} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.12
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{9}{2}]$
Swan slopes: $[5,6]$
Means: $\langle\frac{5}{2},\frac{17}{4}\rangle$
Rams: $(5,7)$
Field count: $16$ (complete)
Ambiguity: $4$
Mass: $32$
Absolute Mass: $8$

Diagrams

Varying

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

Galois group: $C_2^6.(C_2\times C_4)$ (show 8), $C_2^3\wr C_2:C_4$ (show 8)
Hidden Artin slopes: $[2,3,\frac{7}{2},4]^{2}$
Indices of inseparability: $[49,42,32,16,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
2.1.16.64m1.345 $x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.346 $x^{16} + 8 x^{14} + 8 x^{10} + 20 x^{8} + 8 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.347 $x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 16 x^{5} + 8 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.348 $x^{16} + 8 x^{14} + 8 x^{10} + 20 x^{8} + 16 x^{5} + 8 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.349 $x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 24 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.350 $x^{16} + 8 x^{14} + 8 x^{10} + 20 x^{8} + 24 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.351 $x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 16 x^{5} + 24 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.352 $x^{16} + 8 x^{14} + 8 x^{10} + 20 x^{8} + 16 x^{5} + 24 x^{4} + 16 x + 18$ $C_2^6.(C_2\times C_4)$ (as 16T910) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.361 $x^{16} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.362 $x^{16} + 8 x^{12} + 8 x^{10} + 20 x^{8} + 8 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.363 $x^{16} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 16 x^{5} + 8 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.364 $x^{16} + 8 x^{12} + 8 x^{10} + 20 x^{8} + 16 x^{5} + 8 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.365 $x^{16} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 24 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.366 $x^{16} + 8 x^{12} + 8 x^{10} + 20 x^{8} + 24 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.367 $x^{16} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 16 x^{5} + 24 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.64m1.368 $x^{16} + 8 x^{12} + 8 x^{10} + 20 x^{8} + 16 x^{5} + 24 x^{4} + 16 x + 18$ $C_2^3\wr C_2:C_4$ (as 16T939) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2},4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[49, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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