Normalized defining polynomial
\( x^{16} - 2 x^{14} - 4 x^{13} + 15 x^{12} + 40 x^{11} - 16 x^{10} - 152 x^{9} + 311 x^{8} - 660 x^{7} + 680 x^{6} - 489 x^{5} + 545 x^{4} - 355 x^{3} + 182 x^{2} - 66 x + 45 \)
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: | \(30468981617476954930037=11^{10}\cdot 53^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.42$ | 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: | $11, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{161} a^{14} - \frac{1}{161} a^{13} + \frac{62}{161} a^{12} - \frac{37}{161} a^{11} - \frac{3}{23} a^{10} - \frac{39}{161} a^{9} + \frac{11}{161} a^{8} + \frac{71}{161} a^{7} + \frac{13}{161} a^{6} + \frac{51}{161} a^{5} + \frac{22}{161} a^{4} - \frac{12}{161} a^{3} - \frac{12}{161} a^{2} + \frac{5}{161} a - \frac{27}{161}$, $\frac{1}{1780077077076527775} a^{15} - \frac{782893692861974}{593359025692175925} a^{14} + \frac{44950689129376}{254296725296646825} a^{13} + \frac{227268254468425292}{1780077077076527775} a^{12} + \frac{35235045722103922}{593359025692175925} a^{11} - \frac{520424664277213562}{1780077077076527775} a^{10} - \frac{255551269387650052}{1780077077076527775} a^{9} + \frac{17438485455096934}{136929005928963675} a^{8} + \frac{231839208628081937}{1780077077076527775} a^{7} - \frac{8550443728934746}{25798218508355475} a^{6} + \frac{353522601403007108}{1780077077076527775} a^{5} + \frac{31553945997776204}{118671805138435185} a^{4} + \frac{4178655653739650}{10171869011865873} a^{3} - \frac{88942020438442046}{356015415415305555} a^{2} + \frac{249213048624397817}{1780077077076527775} a + \frac{7116063578297864}{118671805138435185}$
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: | \( 323277.839532 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$(C_2^3\times C_4).D_4$ (as 16T675):
| A solvable group of order 256 |
| The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$ |
| Character table for $(C_2^3\times C_4).D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 4.0.6413.1, 8.0.2179708157.1 |
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 | $16$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | $16$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.8.6.1 | $x^{8} + 143 x^{4} + 5929$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $53$ | 53.4.3.4 | $x^{4} + 424$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 53.4.0.1 | $x^{4} - x + 18$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 53.8.4.1 | $x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |