Normalized defining polynomial
\( x^{16} - 2 x^{15} - 2 x^{14} + 24 x^{13} - 50 x^{12} - 38 x^{11} + 250 x^{10} - 308 x^{9} - 110 x^{8} + 1014 x^{7} - 1178 x^{6} - 188 x^{5} + 1484 x^{4} - 1626 x^{3} + 862 x^{2} - 220 x + 137 \)
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: | \(2324101713520230400000000=2^{28}\cdot 5^{8}\cdot 53^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.33$ | 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: | $2, 5, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{145} a^{13} + \frac{54}{145} a^{12} + \frac{5}{29} a^{11} + \frac{24}{145} a^{10} - \frac{69}{145} a^{9} - \frac{7}{29} a^{8} - \frac{11}{29} a^{7} - \frac{9}{29} a^{6} + \frac{7}{29} a^{5} - \frac{26}{145} a^{4} + \frac{46}{145} a^{3} + \frac{8}{29} a^{2} - \frac{41}{145} a - \frac{34}{145}$, $\frac{1}{145} a^{14} + \frac{9}{145} a^{12} - \frac{21}{145} a^{11} - \frac{12}{29} a^{10} + \frac{66}{145} a^{9} - \frac{10}{29} a^{8} + \frac{5}{29} a^{7} - \frac{31}{145} a^{5} + \frac{21}{145} a^{3} - \frac{26}{145} a^{2} + \frac{1}{29} a - \frac{49}{145}$, $\frac{1}{895391967285925} a^{15} + \frac{339758935434}{179078393457185} a^{14} - \frac{585674700862}{895391967285925} a^{13} + \frac{70042278373307}{179078393457185} a^{12} - \frac{48218824321536}{179078393457185} a^{11} + \frac{108652141252477}{895391967285925} a^{10} + \frac{285007436524219}{895391967285925} a^{9} - \frac{32280222813118}{179078393457185} a^{8} + \frac{1293059271798}{6175117015765} a^{7} - \frac{7828314228154}{30875585078825} a^{6} + \frac{49053104134594}{179078393457185} a^{5} - \frac{117897182009248}{895391967285925} a^{4} - \frac{218924829520472}{895391967285925} a^{3} + \frac{88282190465958}{179078393457185} a^{2} - \frac{296354753467283}{895391967285925} a - \frac{245813364681346}{895391967285925}$
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{404003535}{14006913841} a^{15} + \frac{319275493}{14006913841} a^{14} + \frac{1125714908}{14006913841} a^{13} - \frac{8251706477}{14006913841} a^{12} + \frac{10421999247}{14006913841} a^{11} + \frac{26497476498}{14006913841} a^{10} - \frac{66888376270}{14006913841} a^{9} + \frac{48205910545}{14006913841} a^{8} + \frac{89176021773}{14006913841} a^{7} - \frac{294140394787}{14006913841} a^{6} + \frac{140293561628}{14006913841} a^{5} + \frac{189306259903}{14006913841} a^{4} - \frac{342555240417}{14006913841} a^{3} + \frac{296174656622}{14006913841} a^{2} - \frac{65930399304}{14006913841} a + \frac{51675917960}{14006913841} \) (order $4$) | 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: | \( 639020.132944 \) (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
| \(\Q(\sqrt{-1}) \), 4.4.21200.1, 8.0.7191040000.3 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.3.2.1 | $x^{3} - 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.3.2.1 | $x^{3} - 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 25 x^{3} + 200$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 53 | Data not computed | ||||||