Defining polynomial
\(x^{12} + 5550 x^{11} + 12834597 x^{10} + 15830089320 x^{9} + 10983311908793 x^{8} + 4064880121459160 x^{7} + 627211360763929855 x^{6} + 150784964321118570 x^{5} + 157305214180175641 x^{4} + 21952692460017249320 x^{3} + 5259154456776668411 x^{2} + 19105722471444557170 x + 2009397505586138476\) |
Invariants
Base field: | $\Q_{37}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{37}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 37 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{37}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{37}(\sqrt{2})$, $\Q_{37}(\sqrt{37})$, $\Q_{37}(\sqrt{37\cdot 2})$, 37.3.0.1, 37.4.2.1, 37.6.0.1, 37.6.3.1, 37.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 37.6.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of \( x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 925 x + 37 \) $\ \in\Q_{37}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_6$ (as 12T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |