Normalized defining polynomial
\( x^{16} - 4 x^{15} - 2 x^{14} + 24 x^{13} + 48 x^{12} - 328 x^{11} + 222 x^{10} + 972 x^{9} - 1718 x^{8} + 188 x^{7} + 1654 x^{6} - 1400 x^{5} + 120 x^{4} + 168 x^{3} - 34 x^{2} - 4 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(64698023224167823835136=2^{36}\cdot 3^{12}\cdot 11^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.65$ | 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: | $2, 3, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{124} a^{14} - \frac{1}{62} a^{13} + \frac{5}{62} a^{12} + \frac{3}{31} a^{11} + \frac{15}{124} a^{10} + \frac{9}{62} a^{9} + \frac{1}{62} a^{8} + \frac{6}{31} a^{7} + \frac{5}{124} a^{6} - \frac{19}{62} a^{5} + \frac{23}{62} a^{4} - \frac{14}{31} a^{3} + \frac{1}{4} a^{2} - \frac{23}{62} a - \frac{1}{62}$, $\frac{1}{5481545628988} a^{15} - \frac{629827547}{1370386407247} a^{14} + \frac{254387140439}{5481545628988} a^{13} + \frac{8481078140}{1370386407247} a^{12} - \frac{912303728049}{5481545628988} a^{11} - \frac{25574791100}{1370386407247} a^{10} - \frac{719788584731}{5481545628988} a^{9} + \frac{6655153001}{29157157601} a^{8} + \frac{730882961685}{5481545628988} a^{7} - \frac{25462091038}{105414339019} a^{6} - \frac{81206307931}{498322329908} a^{5} - \frac{635271550813}{1370386407247} a^{4} + \frac{2308357518495}{5481545628988} a^{3} - \frac{27604372183}{124580582477} a^{2} + \frac{1109330800885}{5481545628988} a + \frac{448153252531}{1370386407247}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 312626.292657 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T657):
| A solvable group of order 256 |
| The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.4.76032.1, 4.4.4752.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.5780865024.1 |
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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| 3 | Data not computed | ||||||
| $11$ | 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 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.8.6.1 | $x^{8} + 143 x^{4} + 5929$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |