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Properties

Label 2.1.8.30a1.97-1.2.8a
Base 2.1.8.30a1.97
Degree \(2\)
e \(2\)
f \(1\)
c \(8\)

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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: 2.1.8.30a1.97
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[3,4,\frac{19}{4},\frac{19}{4}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $8$ (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: $C_2\wr (C_2\times C_4)$
Hidden Artin slopes: $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$
Indices of inseparability: $[53,48,32,16,0]$
Associated inertia: $[1,1,2]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 8

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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
2.1.16.68b1.835 $x^{16} + 8 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{5} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.837 $x^{16} + 8 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{5} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.844 $x^{16} + 24 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{6} + 16 x^{5} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.846 $x^{16} + 24 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{6} + 16 x^{5} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.852 $x^{16} + 24 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{5} + 16 x^{2} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.854 $x^{16} + 24 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{5} + 16 x^{2} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.860 $x^{16} + 24 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{2} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.68b1.862 $x^{16} + 24 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{2} + 10$ $C_2\wr (C_2\times C_4)$ (as 16T1385) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4}]^{2}$ $[53, 48, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
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