pp-adic field 5.12.1.0a1.1

• Label: 5.12.1.0a1.1
• Base: $\Q_{5}$
• Degree: $12$
• e: $1$
• f: $12$
• c: $0$
• Galois group: $C_{12}$ (as 12T1)

Defining polynomial

$x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$

Invariants

Base field:	$\Q_{5}$
Degree $d$:	$12$
Ramification index $e$:	$1$
Residue field degree $f$:	$12$
Discriminant exponent $c$:	$0$
Discriminant root field:	$\Q_{5}(\sqrt{2})$
Root number:	$1$
$\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$:	$C_{12}$
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:	$244140624 = (5^{ 12 } - 1)$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.1.0a1.1, 5.4.1.0a1.1, 5.6.1.0a1.1

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

Canonical tower

Unramified subfield:	5.12.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$
Relative Eisenstein polynomial:	$x - 5$ $\ \in\Q_{5}(t)[x]$

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois degree:	$12$
Galois group:	$C_{12}$ (as 12T1)
Inertia group:	trivial
Wild inertia group:	$C_1$
Galois unramified degree:	$12$
Galois tame degree:	$1$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[]$
Galois mean slope:	$0.0$
Galois splitting model:	$x^{12} - 20 x^{10} + 122 x^{8} - 280 x^{6} + 264 x^{4} - 96 x^{2} + 8$