$p$-adic field 2.3.4.30a3.32

• Label: 2.3.4.30a3.32
• Base: \(\Q_{2}\)
• Degree: \(12\)
• e: \(4\)
• f: \(3\)
• c: \(30\)
• Galois group: $C_2^4:A_4$ (as 12T87)

Defining polynomial

$( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 12 x^{2} ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$12$
Ramification index $e$:	$4$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$30$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, \frac{7}{2}]$
Visible Swan slopes:	$[2,\frac{5}{2}]$
Means:	$\langle1, \frac{7}{4}\rangle$
Rams:	$(2, 3)$
Jump set:	$[1, 3, 7]$
Roots of unity:	$14 = (2^{ 3 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.3.1.0a1.1, 2.3.2.9a1.6

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

Canonical tower

Unramified subfield:	2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{4} + \left(4 t^{2} + 4 t + 4\right) x^{3} + 4 t x^{2} + 8 x + 10 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^2 + (t + 1)$,$(t + 1) z + (t + 1)$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[7, 4, 0]$

Invariants of the Galois closure

Galois degree:	$192$
Galois group:	$C_2^4:A_4$ (as 12T87)
Inertia group:	Intransitive group isomorphic to $C_2^3:D_4$
Wild inertia group:	$C_2^3:D_4$
Galois unramified degree:	$3$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}]$
Galois Swan slopes:	$[1,1,1,2,\frac{5}{2},\frac{5}{2}]$
Galois mean slope:	$3.21875$
Galois splitting model:	$x^{12} - 10 x^{10} + 7 x^{8} + 74 x^{6} - 8 x^{4} - 64 x^{2} + 25$