$p$-adic field 2.2.8.50b1.278

• Label: 2.2.8.50b1.278
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(50\)
• Galois group: $C_4^2:D_4$ (as 16T400)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$50$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_4$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 3, \frac{17}{4}]$
Visible Swan slopes:	$[1,2,\frac{13}{4}]$
Means:	$\langle\frac{1}{2}, \frac{5}{4}, \frac{9}{4}\rangle$
Rams:	$(1, 3, 8)$
Jump set:	$[1, 7, 15, 23]$
Roots of unity:	$12 = (2^{ 2 } - 1) \cdot 2^{ 2 }$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.2.2.4a1.1, 2.2.2.6a1.3 x2, 2.1.4.8b1.3 x2, 2.2.4.16b1.4

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

Canonical tower

Unramified subfield:	$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{8} + 4 x^{6} + 2 x^{4} + \left(16 t + 4\right) x^{2} + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[18, 10, 4, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_4^2:D_4$ (as 16T400)
Inertia group:	Intransitive group isomorphic to $C_2\wr C_2^2$
Wild inertia group:	$C_2\wr C_2^2$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]$
Galois mean slope:	$3.84375$
Galois splitting model:	$x^{16} + 60 x^{12} + 1000 x^{8} + 1000 x^{4} + 2500$