Base \(\Q_{191}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

Related objects

Learn more about

Defining polynomial

\( x^{6} - 382 x^{4} + 36481 x^{2} - 564397551 \)


Base field: $\Q_{191}$
Degree $d$ : $6$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $3$
Discriminant root field: $\Q_{191}(\sqrt{191})$
Root number: $i$
$|\Gal(K/\Q_{ 191 })|$: $6$
This field is Galois and abelian over $\Q_{191}$.

Intermediate fields


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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{191}(t)$ where $t$ is a root of \( x^{3} - x + 9 \)
Relative Eisenstein polynomial:$ x^{2} - 191 t^{2} \in\Q_{191}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed