$p$-adic field 2.9.2.18a7.2

• Label: 2.9.2.18a7.2
• Base: \(\Q_{2}\)
• Degree: \(18\)
• e: \(2\)
• f: \(9\)
• c: \(18\)
• Galois group: $C_2^6:C_{18}$ (as 18T264)

Defining polynomial

$( x^{9} + x^{4} + 1 )^{2} + 2 x^{5} ( x^{9} + x^{4} + 1 ) + 4 x + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$18$
Ramification index $e$:	$2$
Residue field degree $f$:	$9$
Discriminant exponent $c$:	$18$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2]$
Visible Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Jump set:	$[1, 3]$
Roots of unity:	$1022 = (2^{ 9 } - 1) \cdot 2$

Intermediate fields

2.3.1.0a1.1, 2.9.1.0a1.1

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

Canonical tower

Unramified subfield:	2.9.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{9} + x^{4} + 1 \)
Relative Eisenstein polynomial:	\( x^{2} + \left(2 t^{5} + 2 t^{4} + 2 t^{2} + 2 t\right) x + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z + (t^8 + t^4)$
Associated inertia:	$1$
Indices of inseparability:	$[1, 0]$

Invariants of the Galois closure

Galois degree:	$1152$
Galois group:	$C_2^6:C_{18}$ (as 18T264)
Inertia group:	Intransitive group isomorphic to $C_2^6$
Wild inertia group:	$C_2^6$
Galois unramified degree:	$18$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 2, 2]$
Galois Swan slopes:	$[1,1,1,1,1,1]$
Galois mean slope:	$1.96875$
Galois splitting model:	$x^{18} + 15 x^{16} - 90 x^{14} - 1053 x^{12} + 2754 x^{10} + 18954 x^{8} - 32805 x^{6} - 59049 x^{4} + 6561 x^{2} + 19683$