Normalized defining polynomial
\( x^{16} - 2 x^{15} + 8 x^{14} - 24 x^{13} + 70 x^{12} - 194 x^{11} + 423 x^{10} - 722 x^{9} + 1006 x^{8} - 1150 x^{7} + 1075 x^{6} - 840 x^{5} + 570 x^{4} - 330 x^{3} + 145 x^{2} - 40 x + 5 \)
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: | \(960400000000000000=2^{16}\cdot 5^{14}\cdot 7^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.30$ | 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: | $2, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{12} - \frac{2}{11} a^{11} - \frac{5}{11} a^{10} + \frac{5}{11} a^{9} - \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{1}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2} - \frac{5}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} + \frac{2}{11} a^{9} + \frac{4}{11} a^{8} - \frac{5}{11} a^{7} + \frac{3}{11} a^{6} - \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{233064469} a^{15} + \frac{1600240}{233064469} a^{14} + \frac{7186469}{233064469} a^{13} + \frac{10619286}{233064469} a^{12} - \frac{2160915}{233064469} a^{11} + \frac{6848695}{21187679} a^{10} - \frac{158489}{12266551} a^{9} - \frac{891323}{12266551} a^{8} + \frac{16387271}{233064469} a^{7} - \frac{9601646}{233064469} a^{6} + \frac{782789}{21187679} a^{5} - \frac{109728098}{233064469} a^{4} + \frac{107452614}{233064469} a^{3} - \frac{33255846}{233064469} a^{2} - \frac{96293497}{233064469} a + \frac{4406644}{21187679}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{947733903}{233064469} a^{15} - \frac{1429040916}{233064469} a^{14} + \frac{6848670402}{233064469} a^{13} - \frac{1759964844}{21187679} a^{12} + \frac{56612875035}{233064469} a^{11} - \frac{14144888184}{21187679} a^{10} + \frac{16996328803}{12266551} a^{9} - \frac{27465571150}{12266551} a^{8} + \frac{689939332479}{233064469} a^{7} - \frac{740737932051}{233064469} a^{6} + \frac{642684620742}{233064469} a^{5} - \frac{468942172425}{233064469} a^{4} + \frac{301365008034}{233064469} a^{3} - \frac{159338061489}{233064469} a^{2} + \frac{56002486965}{233064469} a - \frac{9210680307}{233064469} \) (order $20$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2187.29813817 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$OD_{16}:C_2$ (as 16T16):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $(C_8:C_2):C_2$ |
| Character table for $(C_8:C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{20})\) |
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 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $5$ | 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $7$ | 7.8.4.2 | $x^{8} + 49 x^{4} - 1029 x^{2} + 12005$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 7.8.0.1 | $x^{8} - x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |