$p$-adic family 3.1.6.11a1.7-1.2.1a

• Label: 3.1.6.11a1.7-1.2.1a
• Base: 3.1.6.11a1.7
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(1\)

Defining polynomial

$x^{2} + d_{0} \pi$

Invariants

Residue field characteristic:	$3$
Degree:	$2$
Base field:	3.1.6.11a1.7
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$1$
Absolute Artin slopes:	$[\frac{5}{2}]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1/3$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Galois group:	$C_6\wr C_2$
Hidden Artin slopes:	$[2]^{2}$
Indices of inseparability:	$[12,0]$
Associated inertia:	$[2,1]$
Jump Set:	$[2,14]$

Fields

Showing all 2

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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
3.1.12.23a2.19	$x^{12} + 9 x^{2} + 3$	$C_6\wr C_2$ (as 12T42)	$72$	$6$	$[2, \frac{5}{2}]_{4}^{2}$	$[1,\frac{3}{2}]_{4}^{2}$	$[2]^{2}$	$[1]^{2}$	$[12, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.23a2.37	$x^{12} + 18 x^{2} + 3$	$C_6\wr C_2$ (as 12T42)	$72$	$6$	$[2, \frac{5}{2}]_{4}^{2}$	$[1,\frac{3}{2}]_{4}^{2}$	$[2]^{2}$	$[1]^{2}$	$[12, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$

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