$p$-adic field 2.1.16.66j1.1122

• Label: 2.1.16.66j1.1122
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(66\)
• Galois group: $C_2^5:C_4$ (as 16T259)

Defining polynomial

\(x^{16} + 16 x^{12} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$66$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, 4, \frac{17}{4}, \frac{19}{4}]$
Visible Swan slopes:	$[2,3,\frac{13}{4},\frac{15}{4}]$
Means:	$\langle1, 2, \frac{21}{8}, \frac{51}{16}\rangle$
Rams:	$(2, 4, 5, 9)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.11a1.15, 2.1.8.28b1.57, 2.1.8.30a1.105, 2.1.8.30a1.108

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:	\( x^{16} + 16 x^{12} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 10 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[51, 42, 32, 16, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_2^5:C_4$ (as 16T259)
Inertia group:	$C_2\wr C_4$ (as 16T159)
Wild inertia group:	$C_2\wr C_4$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
Galois Swan slopes:	$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
Galois mean slope:	$4.28125$
Galois splitting model:	$x^{16} - 848 x^{14} - 10080 x^{13} + 210344 x^{12} + 6137600 x^{11} + 30105888 x^{10} - 836841600 x^{9} - 15212283544 x^{8} - 81774201600 x^{7} + 412337155520 x^{6} + 8288797372800 x^{5} + 40129986371680 x^{4} - 41831897676800 x^{3} - 1255866750115200 x^{2} - 5075968133376000 x - 6968043823676400$