$p$-adic family 2.1.10.10a

• Label: 2.1.10.10a
• Base: 2.1.1.0a1.1
• Degree: \(10\)
• e: \(10\)
• f: \(1\)
• c: \(10\)

Defining polynomial

$x^{10} + 2 c_{2} x^{2} + 2 a_{1} x + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$10$
Base field:	$\Q_{2}$
Ramification index $e$:	$10$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$10$
Artin slopes:	$[\frac{6}{5}]$
Swan slopes:	$[\frac{1}{5}]$
Means:	$\langle\frac{1}{10}\rangle$
Rams:	$(1)$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1$

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[4,1]$
Jump Set:	$[5,11]$

Galois groups and Hidden Artin slopes

		Galois groups
		$C_2^4:F_5$ (as 10T24)	$C_2^4:F_5$ (as 10T25)	Total
hidden slopes	$[\frac{6}{5},\frac{6}{5},\frac{6}{5}]^{4}$	1	1	2

Fields

Showing all 2

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Label	Packet size	Polynomial	Galois group	Galois degree	$\#\Aut(K/\Q_p)$	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
2.1.10.10a1.1	1	$x^{10} + 2 x + 2$	$(C_2^4 : C_5):C_4$ (as 10T24)	$320$	$2$	$[\frac{6}{5}, \frac{6}{5}, \frac{6}{5}, \frac{6}{5}]_{5}^{4}$	$[\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5}]_{5}^{4}$	$[\frac{6}{5},\frac{6}{5},\frac{6}{5}]^{4}$	$[\frac{1}{5},\frac{1}{5},\frac{1}{5}]^{4}$	$[1, 0]$	$[4, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 11]$
2.1.10.10a1.2	1	$x^{10} + 2 x^{2} + 2 x + 2$	$(C_2^4 : C_5):C_4$ (as 10T25)	$320$	$2$	$[\frac{6}{5}, \frac{6}{5}, \frac{6}{5}, \frac{6}{5}]_{5}^{4}$	$[\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5}]_{5}^{4}$	$[\frac{6}{5},\frac{6}{5},\frac{6}{5}]^{4}$	$[\frac{1}{5},\frac{1}{5},\frac{1}{5}]^{4}$	$[1, 0]$	$[4, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 11]$

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