pp-adic field 5.3.2.3a1.1

• Label: 5.3.2.3a1.1
• Base: $\Q_{5}$
• Degree: $6$
• e: $2$
• f: $3$
• c: $3$
• Galois group: $C_6$ (as 6T1)

Defining polynomial

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

Invariants

Base field:	$\Q_{5}$
Degree $d$:	$6$
Ramification index $e$:	$2$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$3$
Discriminant root field:	$\Q_{5}(\sqrt{5\cdot 2})$
Root number:	$-1$
$\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$:	$C_6$
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:	$124 = (5^{ 3 } - 1)$

Intermediate fields

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

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

Canonical tower

Unramified subfield:	5.3.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of $x^{3} + 3 x + 3$
Relative Eisenstein polynomial:	$x^{2} + 5 t$ $\ \in\Q_{5}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$6$
Galois group:	$C_6$ (as 6T1)
Inertia group:	Intransitive group isomorphic to $C_2$
Wild inertia group:	$C_1$
Galois unramified degree:	$3$
Galois tame degree:	$2$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[\ ]$
Galois mean slope:	$0.5$
Galois splitting model:	$x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$