pp-adic field 3.3.3.9a8.1

• Label: 3.3.3.9a8.1
• Base: $\Q_{3}$
• Degree: $9$
• e: $3$
• f: $3$
• c: $9$
• Galois group: $C_3^2 : S_3$ (as 9T13)

Defining polynomial

$( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 3\right) ( x^{3} + 2 x + 1 ) + 3$

Invariants

Base field:	$\Q_{3}$
Degree $d$:	$9$
Ramification index $e$:	$3$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$9$
Discriminant root field:	$\Q_{3}(\sqrt{3})$
Root number:	$i$
$\Aut(K/\Q_{3})$:	$C_1$
This field is not Galois over $\Q_{3}.$
Visible Artin slopes:	$[\frac{3}{2}]$
Visible Swan slopes:	$[\frac{1}{2}]$
Means:	$\langle\frac{1}{3}\rangle$
Rams:	$(\frac{1}{2})$
Jump set:	undefined
Roots of unity:	$26 = (3^{ 3 } - 1)$

Intermediate fields

3.3.1.0a1.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:	3.3.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of $x^{3} + 2 x + 1$
Relative Eisenstein polynomial:	$x^{3} + \left(3 t^{2} + 3\right) x + 3$ $\ \in\Q_{3}(t)[x]$

Ramification polygon

Residual polynomials:	$z + (t^{2} + 1)$
Associated inertia:	$1$
Indices of inseparability:	$[1, 0]$

Invariants of the Galois closure

Galois degree:	$54$
Galois group:	$C_3^2:C_6$ (as 9T13)
Inertia group:	Intransitive group isomorphic to $C_3:S_3$
Wild inertia group:	$C_3^2$
Galois unramified degree:	$3$
Galois tame degree:	$2$
Galois Artin slopes:	$[\frac{3}{2}, \frac{3}{2}]$
Galois Swan slopes:	$[\frac{1}{2},\frac{1}{2}]$
Galois mean slope:	$1.3888888888888888$
Galois splitting model:	$x^{9} + 6 x^{7} - x^{6} - 9 x^{5} + 45 x^{4} - 57 x^{3} + 108 x^{2} - 123 x + 43$