$p$-adic field 2.1.8.21a1.15

• Label: 2.1.8.21a1.15
• Base: \(\Q_{2}\)
• Degree: \(8\)
• e: \(8\)
• f: \(1\)
• c: \(21\)
• Galois group: $C_2 \wr C_2\wr C_2$ (as 8T35)

Defining polynomial

\(x^{8} + 4 x^{7} + 2 x^{6} + 8 x^{5} + 8 x^{3} + 2\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$8$
Ramification index $e$:	$8$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$21$
Discriminant root field:	$\Q_{2}(\sqrt{2\cdot 5})$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 2, \frac{15}{4}]$
Visible Swan slopes:	$[1,1,\frac{11}{4}]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{7}{4}\rangle$
Rams:	$(1, 1, 8)$
Jump set:	$[1, 3, 11, 19]$
Roots of unity:	$4 = 2^{ 2 }$

Intermediate fields

$\Q_{2}(\sqrt{-1})$, 2.1.4.6a1.1

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:	\( x^{8} + 4 x^{7} + 2 x^{6} + 8 x^{5} + 8 x^{3} + 2 \)

Ramification polygon

Residual polynomials:	$z^6 + 1$,$z + 1$
Associated inertia:	$2$,$1$
Indices of inseparability:	$[14, 6, 6, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_2\wr D_4$ (as 8T35)
Inertia group:	$C_2\wr C_2^2$ (as 8T31)
Wild inertia group:	$C_2\wr C_2^2$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]$
Galois mean slope:	$3.46875$
Galois splitting model:	$x^{8} - 2 x^{6} + 4 x^{4} + 10$