Normalized defining polynomial
\( x^{16} + 4 x^{12} - 48 x^{10} + 246 x^{8} + 768 x^{6} + 1924 x^{4} + 2352 x^{2} + 2401 \)
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: | \(1495873430980877352960000=2^{52}\cdot 3^{12}\cdot 5^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.43$ | 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 a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{112} a^{13} - \frac{1}{16} a^{12} - \frac{3}{112} a^{9} - \frac{1}{16} a^{8} + \frac{1}{14} a^{7} - \frac{13}{112} a^{5} - \frac{3}{16} a^{4} + \frac{5}{14} a^{3} - \frac{1}{112} a + \frac{5}{16}$, $\frac{1}{1408641808} a^{14} - \frac{137835}{28747792} a^{12} + \frac{24479669}{1408641808} a^{10} + \frac{93992585}{1408641808} a^{8} - \frac{116847653}{1408641808} a^{6} + \frac{221156535}{1408641808} a^{4} - \frac{81305393}{1408641808} a^{2} + \frac{5836635}{28747792}$, $\frac{1}{9860492656} a^{15} - \frac{137835}{201234544} a^{13} - \frac{503761009}{9860492656} a^{11} + \frac{974393715}{9860492656} a^{9} + \frac{587473251}{9860492656} a^{7} - \frac{1}{4} a^{6} - \frac{483164369}{9860492656} a^{5} - \frac{1}{4} a^{4} + \frac{1151256189}{9860492656} a^{3} + \frac{1}{4} a^{2} + \frac{2243161}{201234544} a + \frac{1}{4}$
Class group and class number
$C_{2}\times C_{6}$, which has order $12$ (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: | \( 45687.5845647 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:Q_8$ (as 16T31):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2:Q_8$ |
| Character table for $C_2^2:Q_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.0.92160.3, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.92160.6, 8.0.33973862400.12, 8.0.19110297600.3, 8.8.12230590464.1 |
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 16 sibling: | 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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $3$ | 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
| $5$ | 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |