Properties

Label 2.10.15.6
Base \(\Q_{2}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(15\)
Galois group $C_2 \times (C_2^4 : C_5)$ (as 10T14)

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Defining polynomial

\(x^{10} - 12 x^{9} + 34 x^{8} + 1040 x^{7} - 168 x^{6} + 19712 x^{5} - 16464 x^{4} + 66752 x^{3} - 45424 x^{2} + 23232 x - 17632\) 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: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

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: \( x^{2} + \left(4 t^{4} + 4 t^{3} + 4 t^{2}\right) x + 4 t^{3} + 4 t + 6 \) $\ \in\Q_{2}(t)[x]$ 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:$C_2\wr C_5$ (as 10T14)
Inertia group:Intransitive group isomorphic to $C_2^5$
Wild inertia group:$C_2^5$
Unramified degree:$5$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2, 3]$
Galois mean slope:$39/16$
Galois splitting model:$x^{10} - 44 x^{6} + 352 x^{2} + 352$