LMFDB - p-adic field 2.4.4.44a1.597

Defining polynomial

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

x^16 + 12*x^13 + 16*x^12 + 30*x^10 + 76*x^9 + 42*x^8 + 28*x^7 + 104*x^6 + 116*x^5 + 49*x^4 + 44*x^3 + 74*x^2 + 52*x + 15

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$44$
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:	$[3, 4]$
Visible Swan slopes:	$[2,3]$
Means:	$\langle1, 2\rangle$
Rams:	$(2, 4)$
Jump set:	$[1, 3, 7]$
Roots of unity:	$30 = (2^{ 4 } - 1) \cdot 2$

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

Unramified subfield:	2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of $ x^{4} + x + 1 $ x^4 + x + 1
Relative Eisenstein polynomial:	$ x^{4} + \left(8 t + 8\right) x^{3} + 4 t x^{2} + 2 $ x^4 + (8t + 8)x^3 + 4tx^2 + 2 $\ \in\Q_{2}(t)[x]$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi