Normalized defining polynomial
\( x^{16} + 4 x^{14} - 2 x^{12} - 20 x^{11} + 32 x^{10} + 20 x^{9} + 46 x^{8} - 184 x^{7} - 268 x^{6} - 184 x^{5} + 398 x^{4} + 376 x^{3} + 304 x^{2} + 144 x + 46 \)
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: | \(2821046020387461136384=2^{40}\cdot 37^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.91$ | 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: | $2, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{63} a^{14} + \frac{5}{63} a^{13} + \frac{1}{7} a^{12} + \frac{8}{63} a^{11} - \frac{16}{63} a^{10} - \frac{8}{63} a^{9} - \frac{1}{21} a^{8} - \frac{8}{21} a^{7} - \frac{2}{9} a^{6} - \frac{26}{63} a^{5} + \frac{8}{63} a^{4} - \frac{2}{63} a^{3} - \frac{8}{21} a^{2} - \frac{19}{63} a - \frac{4}{63}$, $\frac{1}{664871705839650519} a^{15} - \frac{254572339150693}{221623901946550173} a^{14} - \frac{104296899173752858}{664871705839650519} a^{13} - \frac{65426597375116675}{664871705839650519} a^{12} + \frac{6608072142887497}{664871705839650519} a^{11} - \frac{31797570582653312}{221623901946550173} a^{10} + \frac{2951854169340457}{94981672262807217} a^{9} - \frac{33480330082073432}{221623901946550173} a^{8} + \frac{117081182802962170}{664871705839650519} a^{7} + \frac{10173245845366193}{664871705839650519} a^{6} + \frac{30101541272496928}{221623901946550173} a^{5} + \frac{8683306101814807}{73874633982183391} a^{4} - \frac{241974854085744938}{664871705839650519} a^{3} + \frac{247141893674709506}{664871705839650519} a^{2} + \frac{217389618912707917}{664871705839650519} a - \frac{3235023988609505}{28907465471289153}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 36121.9610236 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$Q_8:C_2^2.D_6$ (as 16T754):
| A solvable group of order 384 |
| The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$ |
| Character table for $Q_8:C_2^2.D_6$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.592.1, 8.0.89718784.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| $37$ | 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 37.8.4.1 | $x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |