Defining polynomial
\(x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413\) |
Invariants
Base field: | $\Q_{37}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{37}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 37 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{37}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{37}(\sqrt{2})$, $\Q_{37}(\sqrt{37})$, $\Q_{37}(\sqrt{37\cdot 2})$, 37.4.0.1, 37.4.2.1, 37.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 37.4.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of \( x^{4} + 6 x^{2} + 24 x + 2 \) |
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: | $C_2\times C_4$ (as 8T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |