Normalized defining polynomial
\( x^{16} - x^{15} + 12 x^{14} + 12 x^{13} + 74 x^{12} + 96 x^{11} + 460 x^{10} + 670 x^{9} + 1236 x^{8} + 1529 x^{7} + 2882 x^{6} + 3853 x^{5} + 4475 x^{4} + 3423 x^{3} + 1909 x^{2} + 627 x + 121 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(29099519213052891015625=5^{8}\cdot 19^{4}\cdot 29^{6}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.35$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 19, 29, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{156597318115918627998061} a^{15} - \frac{75186115757425880761258}{156597318115918627998061} a^{14} - \frac{32744263974481325311088}{156597318115918627998061} a^{13} - \frac{28177030435251292776708}{156597318115918627998061} a^{12} - \frac{55186996053714288172544}{156597318115918627998061} a^{11} - \frac{24474549469535083099503}{156597318115918627998061} a^{10} + \frac{40253592834605658986014}{156597318115918627998061} a^{9} - \frac{66785225931143451795350}{156597318115918627998061} a^{8} - \frac{46430191670216573311660}{156597318115918627998061} a^{7} - \frac{26759241241308063136}{14236119828719875272551} a^{6} + \frac{4236663076383451625875}{14236119828719875272551} a^{5} - \frac{9722273742314974839897}{156597318115918627998061} a^{4} - \frac{43672811024263559357727}{156597318115918627998061} a^{3} + \frac{50218344443990844313568}{156597318115918627998061} a^{2} + \frac{26349310772427333005552}{156597318115918627998061} a + \frac{1199015851361667121963}{14236119828719875272551}$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3033.18899555 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 65 conjugacy class representatives for t16n876 are not computed |
| Character table for t16n876 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.5502768125.2, 8.8.309593125.1, 8.0.8978200625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | R | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 31 | Data not computed | ||||||