pp-adic field 5.6.2.6a1.2

• Label: 5.6.2.6a1.2
• Base: $\Q_{5}$
• Degree: $12$
• e: $2$
• f: $6$
• c: $6$
• Galois group: $C_6\times C_2$ (as 12T2)

Defining polynomial

$( x^{6} + x^{4} + 4 x^{3} + x^{2} + 2 )^{2} + 5$

Invariants

Base field:	$\Q_{5}$
Degree $d$:	$12$
Ramification index $e$:	$2$
Residue field degree $f$:	$6$
Discriminant exponent $c$:	$6$
Discriminant root field:	$\Q_{5}$
Root number:	$-1$
$\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$:	$C_2\times C_6$
This field is Galois and abelian over $\Q_{5}.$
Visible Artin slopes:	$[\ ]$
Visible Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Jump set:	undefined
Roots of unity:	$15624 = (5^{ 6 } - 1)$

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.1.0a1.1, 5.2.2.2a1.2, 5.6.1.0a1.1, 5.3.2.3a1.2, 5.3.2.3a1.1

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

Canonical tower

Unramified subfield:	5.6.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$
Relative Eisenstein polynomial:	$x^{2} + 5$ $\ \in\Q_{5}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$12$
Galois group:	$C_2\times C_6$ (as 12T2)
Inertia group:	Intransitive group isomorphic to $C_2$
Wild inertia group:	$C_1$
Galois unramified degree:	$6$
Galois tame degree:	$2$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[\ ]$
Galois mean slope:	$0.5$
Galois splitting model:	$x^{12} - x^{11} + 2 x^{10} - 3 x^{9} + 5 x^{8} - 8 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1$