Properties

Label 2.1.8.25b1.59-1.2.8a
Base 2.1.8.25b1.59
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: 2.1.8.25b1.59
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2,3,4,\frac{17}{4}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $16$ (incomplete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $4$ ($15/4$ currently in the LMFDB)

Diagrams

Varying

The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.

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

Galois group: $C_2\wr C_2^2$ (show 2), $C_2\wr C_2^2$ (show 2), $C_2^4.D_4$ (show 2), $C_2^4.D_4$ (show 2), $C_2^4:Q_8$ (show 2), $C_2^4:Q_8$ (show 2), $C_2^4:D_4$ (show 2), $C_2^4:D_4$ (show 2) (incomplete)
Hidden Artin slopes: $[\frac{7}{2}]^{2}$ (show 4), $[3,\frac{7}{2}]^{2}$ (show 8), $[2,\frac{7}{2}]^{2}$ (show 4) (incomplete)
Indices of inseparability: $[43,34,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 16

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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
2.1.16.58n1.736 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 20 x^{4} + 8 x^{2} + 30$ $C_2\wr C_2^2$ (as 16T128) $64$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[\frac{7}{2}]^{2}$ $[\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.743 $x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 30$ $C_2^4:D_4$ (as 16T350) $128$ $4$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.746 $x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 30$ $C_2^4:D_4$ (as 16T350) $128$ $4$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.749 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 14$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[\frac{7}{2}]^{2}$ $[\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.761 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 20 x^{4} + 8 x^{2} + 30$ $C_2^4:Q_8$ (as 16T347) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.765 $x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$ $C_2^4.D_4$ (as 16T208) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.772 $x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 20 x^{4} + 8 x^{2} + 16 x + 30$ $C_2^4.D_4$ (as 16T208) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.776 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 20 x^{4} + 8 x^{2} + 16 x + 30$ $C_2^4:Q_8$ (as 16T347) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.779 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$ $C_2^4:Q_8$ (as 16T333) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.780 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 8 x^{2} + 16 x + 30$ $C_2^4:Q_8$ (as 16T333) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.786 $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 30$ $C_2^4.D_4$ (as 16T217) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.788 $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 30$ $C_2^4.D_4$ (as 16T217) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[3,\frac{7}{2}]^{2}$ $[2,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.805 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 30$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[\frac{7}{2}]^{2}$ $[\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.812 $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 30$ $C_2^4:D_4$ (as 16T373) $128$ $4$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.813 $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 8 x^{2} + 16 x + 30$ $C_2^4:D_4$ (as 16T373) $128$ $4$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.58n1.822 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 8 x^{2} + 16 x + 14$ $C_2\wr C_2^2$ (as 16T128) $64$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[\frac{7}{2}]^{2}$ $[\frac{5}{2}]^{2}$ $[43, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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