Properties

Label 107.13.0.1
Base \(\Q_{107}\)
Degree \(13\)
e \(1\)
f \(13\)
c \(0\)
Galois group $C_{13}$ (as 13T1)

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Defining polynomial

\(x^{13} + 4 x + 105\) Copy content Toggle raw display

Invariants

Base field: $\Q_{107}$
Degree $d$: $13$
Ramification exponent $e$: $1$
Residue field degree $f$: $13$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{107}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 107 }) }$: $13$
This field is Galois and abelian over $\Q_{107}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 107 }$.

Unramified/totally ramified tower

Unramified subfield:107.13.0.1 $\cong \Q_{107}(t)$ where $t$ is a root of \( x^{13} + 4 x + 105 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 107 \) $\ \in\Q_{107}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{13}$ (as 13T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$13$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{13} - x^{12} - 36 x^{11} + 77 x^{10} + 365 x^{9} - 1193 x^{8} - 617 x^{7} + 5541 x^{6} - 4414 x^{5} - 4575 x^{4} + 6321 x^{3} + 411 x^{2} - 2196 x + 293$