Normalized defining polynomial
\( x^{16} - 4 x^{15} + 14 x^{14} - 64 x^{13} + 178 x^{12} - 332 x^{11} + 758 x^{10} - 908 x^{9} + 1407 x^{8} - 1344 x^{7} + 1242 x^{6} - 924 x^{5} + 488 x^{4} - 192 x^{3} + 56 x^{2} - 8 x + 1 \)
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: | \(73821320237390233600000000=2^{36}\cdot 5^{8}\cdot 229^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $41.38$ | 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: | $2, 5, 229$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{89} a^{13} + \frac{44}{89} a^{12} + \frac{6}{89} a^{11} + \frac{5}{89} a^{10} + \frac{41}{89} a^{9} + \frac{5}{89} a^{8} - \frac{24}{89} a^{7} - \frac{40}{89} a^{6} + \frac{6}{89} a^{5} - \frac{14}{89} a^{4} + \frac{28}{89} a^{3} - \frac{2}{89} a^{2} - \frac{33}{89} a + \frac{27}{89}$, $\frac{1}{89} a^{14} + \frac{28}{89} a^{12} + \frac{8}{89} a^{11} - \frac{1}{89} a^{10} - \frac{19}{89} a^{9} + \frac{23}{89} a^{8} + \frac{37}{89} a^{7} - \frac{14}{89} a^{6} - \frac{11}{89} a^{5} + \frac{21}{89} a^{4} + \frac{12}{89} a^{3} - \frac{34}{89} a^{2} - \frac{34}{89} a - \frac{31}{89}$, $\frac{1}{1182571569} a^{15} + \frac{590204}{131396841} a^{14} - \frac{36014}{13287321} a^{13} - \frac{85011317}{1182571569} a^{12} + \frac{220030562}{1182571569} a^{11} + \frac{40796677}{131396841} a^{10} + \frac{120677672}{1182571569} a^{9} + \frac{40503460}{131396841} a^{8} + \frac{125164876}{394190523} a^{7} - \frac{21066403}{43798947} a^{6} + \frac{133342}{356089} a^{5} + \frac{89351197}{394190523} a^{4} + \frac{506770754}{1182571569} a^{3} - \frac{374616172}{1182571569} a^{2} - \frac{517863827}{1182571569} a + \frac{124632524}{1182571569}$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (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: | \( 323470.282474 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^2:S_4:C_2$ (as 16T724):
| A solvable group of order 384 |
| The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$ |
| Character table for $C_2\times C_2^2:S_4:C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.2147983360000.8 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 229 | Data not computed | ||||||