pp-adic field 53.3.5.12a1.1

• Label: 53.3.5.12a1.1
• Base: $\Q_{53}$
• Degree: $15$
• e: $5$
• f: $3$
• c: $12$
• Galois group: $F_5\times C_3$ (as 15T8)

Defining polynomial

$( x^{3} + 3 x + 51 )^{5} + 53$

Invariants

Base field:	$\Q_{53}$
Degree $d$:	$15$
Ramification index $e$:	$5$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$12$
Discriminant root field:	$\Q_{53}(\sqrt{2})$
Root number:	$1$
$\Aut(K/\Q_{53})$:	$C_3$
This field is not Galois over $\Q_{53}.$
Visible Artin slopes:	$[\ ]$
Visible Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Jump set:	undefined
Roots of unity:	$148876 = (53^{ 3 } - 1)$

Intermediate fields

53.3.1.0a1.1, 53.1.5.4a1.1

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

Canonical tower

Unramified subfield:	53.3.1.0a1.1 $\cong \Q_{53}(t)$ where $t$ is a root of $x^{3} + 3 x + 51$
Relative Eisenstein polynomial:	$x^{5} + 53$ $\ \in\Q_{53}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$
Associated inertia:	$4$
Indices of inseparability:	$[0]$

Invariants of the Galois closure

Galois degree:	$60$
Galois group:	$C_3\times F_5$ (as 15T8)
Inertia group:	Intransitive group isomorphic to $C_5$
Wild inertia group:	$C_1$
Galois unramified degree:	$12$
Galois tame degree:	$5$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[\ ]$
Galois mean slope:	$0.8$
Galois splitting model:	not computed