Normalized defining polynomial
\( x^{16} + 4 x^{14} - 30 x^{12} - 64 x^{11} - 16 x^{10} + 384 x^{9} + 154 x^{8} - 768 x^{7} - 128 x^{6} + 552 x^{5} + 100 x^{4} - 160 x^{3} - 48 x^{2} + 16 x + 4 \)
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: | \(2817928042905600000000=2^{40}\cdot 3^{8}\cdot 5^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.91$ | 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: | $2, 3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{6} a^{9} + \frac{1}{18} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{270} a^{14} + \frac{2}{135} a^{13} + \frac{1}{270} a^{12} + \frac{1}{15} a^{11} - \frac{11}{135} a^{10} + \frac{23}{135} a^{9} + \frac{1}{270} a^{8} + \frac{1}{5} a^{7} + \frac{2}{15} a^{6} + \frac{2}{9} a^{5} - \frac{61}{135} a^{4} + \frac{47}{135} a^{3} + \frac{2}{135} a^{2} - \frac{13}{27} a - \frac{52}{135}$, $\frac{1}{1388233890} a^{15} + \frac{110189}{1388233890} a^{14} + \frac{11424013}{694116945} a^{13} - \frac{2155457}{1388233890} a^{12} - \frac{26407661}{694116945} a^{11} + \frac{3157877}{462744630} a^{10} + \frac{10023461}{1388233890} a^{9} + \frac{139525007}{694116945} a^{8} - \frac{12414703}{77124105} a^{7} - \frac{6018472}{46274463} a^{6} + \frac{49293284}{694116945} a^{5} + \frac{112576327}{694116945} a^{4} + \frac{42130867}{694116945} a^{3} - \frac{4722601}{46274463} a^{2} + \frac{61374991}{231372315} a - \frac{41772812}{138823389}$
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: | \( 85695.3684811 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$Q_8:C_2^2$ (as 16T23):
| A solvable group of order 32 |
| The 17 conjugacy class representatives for $Q_8 : C_2^2$ |
| Character table for $Q_8 : C_2^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 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 | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |