$p$-adic field 2.1.16.74a1.41

• Label: 2.1.16.74a1.41
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(74\)
• Galois group: $C_2^7.C_2\wr C_4$ (as 16T1704)

Defining polynomial

\(x^{16} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 32 x^{5} + 2\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$74$
Discriminant root field:	$\Q_{2}(\sqrt{-1})$
Root number:	$-i$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, \frac{7}{2}, \frac{19}{4}, \frac{45}{8}]$
Visible Swan slopes:	$[2,\frac{5}{2},\frac{15}{4},\frac{37}{8}]$
Means:	$\langle1, \frac{7}{4}, \frac{11}{4}, \frac{59}{16}\rangle$
Rams:	$(2, 3, 8, 15)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{2})$, 2.1.4.10a1.1, 2.1.8.29a1.3

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:	\( x^{16} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 32 x^{5} + 2 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[59, 44, 28, 16, 0]$

Invariants of the Galois closure

Galois degree:	$8192$
Galois group:	$C_2^7.C_2\wr C_4$ (as 16T1704)
Inertia group:	$C_2^6.C_2\wr C_4$ (as 16T1572)
Wild inertia group:	not computed
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	not computed
Galois Swan slopes:	not computed
Galois mean slope:	not computed
Galois splitting model:	$x^{16} + 24 x^{14} + 224 x^{12} + 960 x^{10} - 5350 x^{8} - 45000 x^{6} - 100000 x^{4} - 100000 x^{2} - 15625$