Normalized defining polynomial
\( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 74 x^{12} - 60 x^{11} - 195 x^{10} + 788 x^{9} - 1383 x^{8} + 1732 x^{7} - 915 x^{6} - 2452 x^{5} + 6664 x^{4} - 8502 x^{3} + 6319 x^{2} - 2810 x + 631 \)
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: | \(114989547959792521183232=2^{24}\cdot 41^{6}\cdot 113^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.62$ | 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: | $2, 41, 113$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{1235321930476707006924335} a^{15} - \frac{11319006688789527480907}{1235321930476707006924335} a^{14} - \frac{357338499005747927508756}{1235321930476707006924335} a^{13} + \frac{480020524260294749601377}{1235321930476707006924335} a^{12} + \frac{269076888803797187580958}{1235321930476707006924335} a^{11} - \frac{102598513317376453541994}{1235321930476707006924335} a^{10} - \frac{70707714325222239067696}{247064386095341401384867} a^{9} + \frac{12012778370953102197959}{1235321930476707006924335} a^{8} + \frac{408372646306014588280546}{1235321930476707006924335} a^{7} - \frac{85139964293378983458983}{1235321930476707006924335} a^{6} + \frac{6281104856758857251429}{1235321930476707006924335} a^{5} + \frac{611333882640579333991233}{1235321930476707006924335} a^{4} + \frac{16990604858970742144222}{247064386095341401384867} a^{3} - \frac{554619280695857773655601}{1235321930476707006924335} a^{2} + \frac{457777578970624667369268}{1235321930476707006924335} a - \frac{208823543439017150634192}{1235321930476707006924335}$
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: | \( 121242.079438 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 8192 |
| The 104 conjugacy class representatives for t16n1722 are not computed |
| Character table for t16n1722 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1 |
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 | R | $16$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | $16$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | $16$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $41$ | 41.4.3.4 | $x^{4} + 8856$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.3.4 | $x^{4} + 8856$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 113 | Data not computed | ||||||