Normalized defining polynomial
\( x^{16} - 8 x^{15} + 20 x^{14} - 34 x^{12} - 160 x^{11} + 988 x^{10} - 2520 x^{9} + 6291 x^{8} - 13520 x^{7} + 25876 x^{6} - 37312 x^{5} + 55602 x^{4} - 61552 x^{3} + 73496 x^{2} - 47168 x + 55036 \)
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: | \(39615410820716953600000000=2^{44}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.80$ | 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: | $2, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(560=2^{4}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(69,·)$, $\chi_{560}(321,·)$, $\chi_{560}(141,·)$, $\chi_{560}(461,·)$, $\chi_{560}(209,·)$, $\chi_{560}(281,·)$, $\chi_{560}(169,·)$, $\chi_{560}(349,·)$, $\chi_{560}(421,·)$, $\chi_{560}(489,·)$, $\chi_{560}(29,·)$, $\chi_{560}(449,·)$, $\chi_{560}(181,·)$, $\chi_{560}(41,·)$, $\chi_{560}(309,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{15213630121762} a^{14} - \frac{7}{15213630121762} a^{13} + \frac{399050155412}{7606815060881} a^{12} + \frac{1409106598014}{7606815060881} a^{11} + \frac{3454518763187}{15213630121762} a^{10} + \frac{3802478095741}{15213630121762} a^{9} + \frac{777420658819}{15213630121762} a^{8} + \frac{2537896973436}{7606815060881} a^{7} + \frac{2993650546753}{7606815060881} a^{6} - \frac{2177773688846}{7606815060881} a^{5} - \frac{3811350157817}{15213630121762} a^{4} + \frac{403633356699}{7606815060881} a^{3} - \frac{1747672498190}{7606815060881} a^{2} + \frac{1677389937641}{7606815060881} a + \frac{2774098182768}{7606815060881}$, $\frac{1}{823985421024751682} a^{15} + \frac{27073}{823985421024751682} a^{14} - \frac{71336312590881386}{411992710512375841} a^{13} + \frac{124999598122743619}{823985421024751682} a^{12} - \frac{71206649604215864}{411992710512375841} a^{11} + \frac{56324417677248087}{823985421024751682} a^{10} + \frac{40459078083005041}{823985421024751682} a^{9} - \frac{81520273861214893}{823985421024751682} a^{8} + \frac{392172689695987551}{823985421024751682} a^{7} + \frac{134914522848358993}{411992710512375841} a^{6} + \frac{32769399278238957}{823985421024751682} a^{5} - \frac{183438989323614092}{411992710512375841} a^{4} + \frac{40732145035442488}{411992710512375841} a^{3} - \frac{48139223072561826}{411992710512375841} a^{2} - \frac{195542193889087270}{411992710512375841} a - \frac{75165266368468477}{411992710512375841}$
Class group and class number
$C_{2}\times C_{8}\times C_{8}$, which has order $128$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 12198.9512748 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_4$ (as 16T2):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.22.2 | $x^{8} + 10 x^{4} + 16 x + 4$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.22.2 | $x^{8} + 10 x^{4} + 16 x + 4$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |