Defining polynomial
\(x^{2} + 123\) |
Invariants
Base field: | $\Q_{41}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $1$ |
Discriminant root field: | $\Q_{41}(\sqrt{41\cdot 3})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 41 }) }$: | $2$ |
This field is Galois and abelian over $\Q_{41}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 41 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{41}$ |
Relative Eisenstein polynomial: | \( x^{2} + 123 \) |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_2$ (as 2T1) |
Inertia group: | $C_2$ (as 2T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{2} + 246$ |