$p$-adic field 2.7.2.14a8.1

• Label: 2.7.2.14a8.1
• Base: \(\Q_{2}\)
• Degree: \(14\)
• e: \(2\)
• f: \(7\)
• c: \(14\)
• Galois group: $C_2 \wr C_7$ (as 14T29)

Defining polynomial

$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2}\right) ( x^{7} + x + 1 ) + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$14$
Ramification index $e$:	$2$
Residue field degree $f$:	$7$
Discriminant exponent $c$:	$14$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2]$
Visible Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Jump set:	$[1, 3]$
Roots of unity:	$254 = (2^{ 7 } - 1) \cdot 2$

Intermediate fields

2.7.1.0a1.1

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

Canonical tower

Unramified subfield:	2.7.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{2} + \left(2 t^{6} + 2 t^{5} + 2 t^{3} + 2 t^{2}\right) x + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z + (t^2 + t)$
Associated inertia:	$1$
Indices of inseparability:	$[1, 0]$

Invariants of the Galois closure

Galois degree:	$896$
Galois group:	$C_2\wr C_7$ (as 14T29)
Inertia group:	Intransitive group isomorphic to $C_2^6$
Wild inertia group:	$C_2^6$
Galois unramified degree:	$14$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 2, 2]$
Galois Swan slopes:	$[1,1,1,1,1,1]$
Galois mean slope:	$1.96875$
Galois splitting model:	$x^{14} - 21 x^{12} - 126 x^{10} + 1890 x^{8} + 3969 x^{6} - 34020 x^{4} + 15309 x^{2} + 2187$