Normalized defining polynomial
\( x^{16} - 4 x^{15} - 2 x^{14} + 23 x^{13} - 21 x^{12} + 84 x^{11} - 77 x^{10} - 498 x^{9} + 629 x^{8} + 919 x^{7} - 636 x^{6} - 1695 x^{5} + 212 x^{4} + 1689 x^{3} + 583 x^{2} - 1867 x + 661 \)
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: | \(8334464799024100000000=2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.45$ | 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: | $2, 5, 11, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{62} a^{14} - \frac{29}{62} a^{13} + \frac{15}{31} a^{12} + \frac{13}{31} a^{11} - \frac{9}{62} a^{10} + \frac{23}{62} a^{9} + \frac{5}{62} a^{8} - \frac{4}{31} a^{7} + \frac{15}{31} a^{6} + \frac{9}{62} a^{5} - \frac{13}{62} a^{4} + \frac{19}{62} a^{3} + \frac{2}{31} a^{2} + \frac{8}{31} a + \frac{15}{62}$, $\frac{1}{3733151729975051480878} a^{15} + \frac{11247418228674181080}{1866575864987525740439} a^{14} - \frac{500276824584042069037}{3733151729975051480878} a^{13} + \frac{832766034863917591693}{1866575864987525740439} a^{12} - \frac{606715208132906711079}{3733151729975051480878} a^{11} - \frac{883446753608388832143}{1866575864987525740439} a^{10} + \frac{126049556215783586829}{1866575864987525740439} a^{9} + \frac{569960799887808778643}{3733151729975051480878} a^{8} + \frac{240817236898345952125}{1866575864987525740439} a^{7} - \frac{639306528421478814463}{3733151729975051480878} a^{6} + \frac{199946349285218447754}{1866575864987525740439} a^{5} + \frac{588696037133617834994}{1866575864987525740439} a^{4} - \frac{279784213448651001139}{3733151729975051480878} a^{3} - \frac{916856881391235541305}{1866575864987525740439} a^{2} + \frac{249204150601764393513}{3733151729975051480878} a + \frac{718595833203167754209}{3733151729975051480878}$
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: | \( 38869.6846855 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T511):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$ |
| Character table for $C_2^2.C_2^5.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.5225.1, 4.2.475.1, 4.2.275.1, 8.4.91293290000.2, 8.4.91293290000.1, 8.4.27300625.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.8.6.2 | $x^{8} - 781 x^{4} + 290521$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.3.1 | $x^{4} + 76$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 19.4.3.1 | $x^{4} + 76$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |