Normalized defining polynomial
\( x^{16} - 2 x^{15} + 5 x^{14} + 10 x^{13} + 5 x^{12} + 114 x^{11} - 83 x^{10} + 145 x^{9} + 150 x^{8} - 1095 x^{7} + 642 x^{6} - 614 x^{5} - 1580 x^{4} + 1160 x^{3} + 1405 x^{2} - 1383 x - 419 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(243680579501983642578125=5^{15}\cdot 41^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.95$ | 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: | $5, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{33} a^{14} - \frac{2}{33} a^{13} - \frac{3}{11} a^{12} + \frac{5}{33} a^{11} - \frac{1}{33} a^{10} + \frac{1}{3} a^{9} - \frac{1}{11} a^{8} - \frac{3}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{16}{33} a^{3} + \frac{4}{33} a^{2} + \frac{4}{11} a - \frac{4}{33}$, $\frac{1}{1111624703810051721316297623} a^{15} - \frac{3276836535367758218299768}{1111624703810051721316297623} a^{14} - \frac{264370020977566382800528445}{1111624703810051721316297623} a^{13} + \frac{454820289169160597065525328}{1111624703810051721316297623} a^{12} + \frac{78274291505916676537099684}{1111624703810051721316297623} a^{11} + \frac{312542193376257877677263281}{1111624703810051721316297623} a^{10} - \frac{386611218373404957099181501}{1111624703810051721316297623} a^{9} - \frac{6998169269334579191757197}{33685597085153082464130231} a^{8} - \frac{182079112680487442279200664}{370541567936683907105432541} a^{7} - \frac{20045919183649503756360439}{370541567936683907105432541} a^{6} - \frac{49483682787587008758725946}{123513855978894635701810847} a^{5} + \frac{141054500082026435043989041}{1111624703810051721316297623} a^{4} + \frac{23060540064176541891011710}{101056791255459247392390693} a^{3} - \frac{27304371816656855297690614}{101056791255459247392390693} a^{2} + \frac{472277121984978823005541778}{1111624703810051721316297623} a - \frac{162081962780875415026964698}{1111624703810051721316297623}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 333394.754185 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4:D_4.D_4$ (as 16T681):
| A solvable group of order 256 |
| The 19 conjugacy class representatives for $C_4:D_4.D_4$ |
| Character table for $C_4:D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.5125.1, 8.4.131328125.1 |
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 | $16$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | $16$ | R | $16$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | Data not computed | ||||||
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.8.6.2 | $x^{8} + 943 x^{4} + 242064$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |