$p$-adic family 2.1.8.20a1.2-1.2.16a

• Label: 2.1.8.20a1.2-1.2.16a
• Base: 2.1.8.20a1.2
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(16\)

Defining polynomial

$x^{2} + \left(b_{29} \pi^{15} + b_{27} \pi^{14} + b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + a_{15} \pi^{8}\right) x + c_{30} \pi^{16} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.8.20a1.2
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$16$
Absolute Artin slopes:	$[\frac{20}{7},\frac{20}{7},\frac{20}{7},\frac{9}{2}]$
Swan slopes:	$[15]$
Means:	$\langle\frac{15}{2}\rangle$
Rams:	$(15)$
Field count:	$256$ (complete)
Ambiguity:	$2$
Mass:	$128$
Absolute Mass:	$128$

Diagrams

Varying

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

Indices of inseparability:	$[41,26,26,16,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,3,7,15,31]$

Fields

Showing 1-50 of 256

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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
2.1.16.56a1.257	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.258	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.259	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.260	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.261	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.262	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.263	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.264	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.265	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.266	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.267	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.268	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.269	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.270	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.271	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.272	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.273	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.274	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.275	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.276	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.277	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.278	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.279	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.280	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.281	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.282	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.283	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.284	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.285	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.286	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.287	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.288	$x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.289	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.290	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.291	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.292	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.293	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.294	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.295	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.296	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.297	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.298	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.299	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.300	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.301	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.302	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.303	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.304	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.305	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.56a1.306	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 16 x^{8} + 16 x + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[41, 26, 26, 16, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$

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