Normalized defining polynomial
\( x^{16} - x^{15} - 37 x^{14} + 24 x^{13} + 442 x^{12} - 140 x^{11} - 1518 x^{10} - 378 x^{9} - 4525 x^{8} + 8032 x^{7} + 31797 x^{6} - 55905 x^{5} - 70683 x^{4} + 157319 x^{3} + 136913 x^{2} - 82011 x - 57769 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(121003215064784995361328125=5^{12}\cdot 29^{6}\cdot 941^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.68$ | 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: | $5, 29, 941$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{211} a^{14} - \frac{24}{211} a^{13} + \frac{89}{211} a^{12} - \frac{28}{211} a^{11} + \frac{97}{211} a^{10} + \frac{62}{211} a^{9} + \frac{44}{211} a^{8} + \frac{50}{211} a^{7} + \frac{57}{211} a^{6} - \frac{20}{211} a^{5} - \frac{43}{211} a^{4} + \frac{24}{211} a^{3} + \frac{44}{211} a^{2} + \frac{71}{211} a + \frac{64}{211}$, $\frac{1}{72529407507719957517532875110762221} a^{15} - \frac{571973179242182392457576085093}{343741267809099324727643957870911} a^{14} - \frac{35817641175575760020286238356609472}{72529407507719957517532875110762221} a^{13} - \frac{34311889299165969761868729178782771}{72529407507719957517532875110762221} a^{12} + \frac{6632184811199894873850082635258507}{72529407507719957517532875110762221} a^{11} + \frac{29757345460889295627092833452428540}{72529407507719957517532875110762221} a^{10} - \frac{28964499228516945598053539439589108}{72529407507719957517532875110762221} a^{9} + \frac{11793499908848567113897675424817623}{72529407507719957517532875110762221} a^{8} - \frac{26970816703417233829538870158529178}{72529407507719957517532875110762221} a^{7} + \frac{1017566136312657810271560887604726}{3817337237248418816712256584776959} a^{6} + \frac{22525731716580011567845727637991787}{72529407507719957517532875110762221} a^{5} + \frac{176297560254136744630695108831879}{2339658306700643790888157261637491} a^{4} + \frac{608349711371017956605228782718489}{1229311991656270466398862290012919} a^{3} + \frac{18429091698792998162229923849295102}{72529407507719957517532875110762221} a^{2} - \frac{15944683569686070786127634947715304}{72529407507719957517532875110762221} a + \frac{3672322067566189602564474637941766}{72529407507719957517532875110762221}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 26541057.2176 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 74 conjugacy class representatives for t16n1477 are not computed |
| Character table for t16n1477 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.494613125.1 |
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 | $16$ | $16$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | $16$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 941 | Data not computed | ||||||