Normalized defining polynomial
\( x^{16} - 4 x^{15} + 8 x^{14} + 14 x^{12} - 32 x^{11} + 184 x^{10} - 120 x^{9} + 310 x^{8} - 192 x^{7} + 592 x^{6} - 40 x^{5} + 1196 x^{4} - 240 x^{3} + 1136 x^{2} - 176 x + 484 \)
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: | \(9316515344280166632259584=2^{40}\cdot 3^{14}\cdot 11^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.36$ | 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 a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{22} a^{13} + \frac{3}{22} a^{12} - \frac{1}{11} a^{11} + \frac{3}{22} a^{10} - \frac{1}{11} a^{9} + \frac{2}{11} a^{8} + \frac{5}{11} a^{7} + \frac{1}{11} a^{6} - \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a$, $\frac{1}{3278} a^{14} - \frac{30}{1639} a^{13} + \frac{361}{1639} a^{12} + \frac{767}{3278} a^{11} + \frac{20}{1639} a^{10} - \frac{144}{1639} a^{9} + \frac{29}{298} a^{8} - \frac{468}{1639} a^{7} + \frac{219}{1639} a^{6} + \frac{592}{1639} a^{5} - \frac{536}{1639} a^{4} + \frac{797}{1639} a^{3} + \frac{293}{1639} a^{2} - \frac{203}{1639} a - \frac{42}{149}$, $\frac{1}{295003939857951346} a^{15} + \frac{15822893897430}{147501969928975673} a^{14} - \frac{1547571785947430}{147501969928975673} a^{13} + \frac{16231300145659320}{147501969928975673} a^{12} - \frac{52467709421887827}{295003939857951346} a^{11} + \frac{66952003602756881}{295003939857951346} a^{10} + \frac{59362982228821221}{295003939857951346} a^{9} - \frac{62980897036018673}{295003939857951346} a^{8} + \frac{57728836313734908}{147501969928975673} a^{7} - \frac{35569562076831918}{147501969928975673} a^{6} - \frac{28236054927097880}{147501969928975673} a^{5} + \frac{19212824139615644}{147501969928975673} a^{4} + \frac{33481539188658419}{147501969928975673} a^{3} + \frac{47311389769836617}{147501969928975673} a^{2} - \frac{10129347805417376}{147501969928975673} a - \frac{6505628542746432}{13409269993543243}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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: | \( 19975.482948 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $D_4:D_4$ |
| Character table for $D_4:D_4$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), 4.4.4752.1, 4.4.76032.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.190768545792.5 x2, 8.0.3052296732672.17 x2, 8.8.5780865024.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.8.6.2 | $x^{8} - 781 x^{4} + 290521$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |