Properties

Label 97.22.20.1
Base \(\Q_{97}\)
Degree \(22\)
e \(11\)
f \(2\)
c \(20\)
Galois group $C_{11}:C_{10}$ (as 22T5)

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Defining polynomial

\(x^{22} + 1056 x^{21} + 506935 x^{20} + 146034240 x^{19} + 28051247455 x^{18} + 3772862308512 x^{17} + 362615550328977 x^{16} + 24910879636304640 x^{15} + 1199348378045912490 x^{14} + 38586738761105071680 x^{13} + 749260822859538763686 x^{12} + 6767016092180686452290 x^{11} + 3746304114297693920862 x^{10} + 964668469027675906010 x^{9} + 149918547269876213730 x^{8} + 15569302485550255860 x^{7} + 1133538005222545989 x^{6} + 93915386222823924 x^{5} + 2398449762565016435 x^{4} + 114958017018624674730 x^{3} + 3676657838275715684435 x^{2} + 70553511155902657916242 x + 615405492102580008082686\) Copy content Toggle raw display

Invariants

Base field: $\Q_{97}$
Degree $d$: $22$
Ramification exponent $e$: $11$
Residue field degree $f$: $2$
Discriminant exponent $c$: $20$
Discriminant root field: $\Q_{97}(\sqrt{5})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 97 }) }$: $2$
This field is not Galois over $\Q_{97}.$
Visible slopes:None

Intermediate fields

$\Q_{97}(\sqrt{5})$, 97.11.10.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{97}(\sqrt{5})$ $\cong \Q_{97}(t)$ where $t$ is a root of \( x^{2} + 96 x + 5 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{11} + 97 \) $\ \in\Q_{97}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{10} + 11z^{9} + 55z^{8} + 68z^{7} + 39z^{6} + 74z^{5} + 74z^{4} + 39z^{3} + 68z^{2} + 55z + 11$
Associated inertia:$5$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{11}:C_{10}$ (as 22T5)
Inertia group:Intransitive group isomorphic to $C_{11}$
Wild inertia group:$C_1$
Unramified degree:$10$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:Not computed