$p$-adic field 2.1.18.28a1.45

• Label: 2.1.18.28a1.45
• Base: \(\Q_{2}\)
• Degree: \(18\)
• e: \(18\)
• f: \(1\)
• c: \(28\)
• Galois group: $C_2^6:C_{18}:C_6$ (as 18T512)

Defining polynomial

\(x^{18} + 2 x^{17} + 2 x^{13} + 2 x^{11} + 4 x + 2\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$18$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$28$
Discriminant root field:	$\Q_{2}(\sqrt{-5})$
Root number:	$-i$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[\frac{20}{9}]$
Visible Swan slopes:	$[\frac{11}{9}]$
Means:	$\langle\frac{11}{18}\rangle$
Rams:	$(11)$
Jump set:	$[9, 27]$
Roots of unity:	$2$

Intermediate fields

2.1.3.2a1.1, 2.1.9.8a1.1

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

Canonical tower

Unramified subfield:	$\Q_{2}$
Relative Eisenstein polynomial:	\( x^{18} + 2 x^{17} + 2 x^{13} + 2 x^{11} + 4 x + 2 \)

Ramification polygon

Residual polynomials:	$z^{16} + z^{14} + 1$,$z + 1$
Associated inertia:	$6$,$1$
Indices of inseparability:	$[11, 0]$

Invariants of the Galois closure

Galois degree:	$6912$
Galois group:	$C_2^6:C_{18}:C_6$ (as 18T512)
Inertia group:	$C_2^6:C_{18}$ (as 18T264)
Wild inertia group:	$C_2^7$
Galois unramified degree:	$6$
Galois tame degree:	$9$
Galois Artin slopes:	$[2, \frac{20}{9}, \frac{20}{9}, \frac{20}{9}, \frac{20}{9}, \frac{20}{9}, \frac{20}{9}]$
Galois Swan slopes:	$[1,\frac{11}{9},\frac{11}{9},\frac{11}{9},\frac{11}{9},\frac{11}{9},\frac{11}{9}]$
Galois mean slope:	$2.2100694444444446$
Galois splitting model:	$x^{18} - 41 x^{16} + 898 x^{14} - 8134 x^{12} + 47824 x^{10} - 187068 x^{8} + 287770 x^{6} - 263742 x^{4} + 106803 x^{2} - 45387$