Properties

Label 2.10.15.13
Base \(\Q_{2}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(15\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $10$
Ramification exponent $e$: $2$
Residue field degree $f$: $5$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{2}(\sqrt{-2\cdot 5})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 2 }) }$: $10$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

$\Q_{2}(\sqrt{-2\cdot 5})$, 2.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 12 t^{4} + 4 t^{3} + 12 t^{2} + 12 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$5$
Tame degree:$1$
Wild slopes:$[3]$
Galois mean slope:$3/2$
Galois splitting model:$x^{10} - 22 x^{8} + 176 x^{6} - 616 x^{4} + 880 x^{2} - 352$