Normalized defining polynomial
\( x^{16} - x^{15} - 4 x^{14} + 6 x^{13} + 3 x^{12} - 26 x^{11} + 24 x^{10} + 93 x^{9} - 212 x^{8} + 54 x^{7} + 316 x^{6} - 497 x^{5} + 373 x^{4} - 227 x^{3} + 154 x^{2} - 67 x + 11 \)
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: | \(9007501562500000000=2^{8}\cdot 5^{14}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.30$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{22} a^{12} - \frac{1}{2} a^{11} - \frac{3}{22} a^{10} + \frac{9}{22} a^{9} + \frac{5}{22} a^{8} - \frac{4}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{22} a^{4} + \frac{5}{11} a^{3} + \frac{1}{22} a^{2} - \frac{3}{22} a - \frac{1}{2}$, $\frac{1}{22} a^{13} + \frac{4}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{3}{22} a^{8} + \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{22} a^{5} - \frac{1}{22} a^{4} + \frac{1}{22} a^{3} + \frac{4}{11} a^{2} - \frac{1}{2}$, $\frac{1}{22} a^{14} - \frac{1}{11} a^{11} - \frac{2}{11} a^{10} - \frac{3}{22} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{3}{22} a^{6} - \frac{9}{22} a^{5} - \frac{3}{22} a^{4} - \frac{3}{11} a^{3} - \frac{4}{11} a^{2} - \frac{9}{22} a$, $\frac{1}{721317542} a^{15} - \frac{532863}{360658771} a^{14} - \frac{9202855}{721317542} a^{13} - \frac{4593266}{360658771} a^{12} + \frac{84451293}{360658771} a^{11} - \frac{325706699}{721317542} a^{10} - \frac{32808682}{360658771} a^{9} - \frac{96664179}{721317542} a^{8} + \frac{237876885}{721317542} a^{7} + \frac{231497547}{721317542} a^{6} + \frac{69285030}{360658771} a^{5} - \frac{72867711}{721317542} a^{4} - \frac{334775999}{721317542} a^{3} + \frac{118082339}{721317542} a^{2} + \frac{141129048}{360658771} a + \frac{15283621}{65574322}$
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: | \( \frac{1163091227}{721317542} a^{15} - \frac{791695599}{721317542} a^{14} - \frac{2502716833}{360658771} a^{13} + \frac{2658670304}{360658771} a^{12} + \frac{2771678693}{360658771} a^{11} - \frac{28448566849}{721317542} a^{10} + \frac{18353520857}{721317542} a^{9} + \frac{57919457543}{360658771} a^{8} - \frac{208755662661}{721317542} a^{7} - \frac{6435799719}{360658771} a^{6} + \frac{184705775673}{360658771} a^{5} - \frac{450403757995}{721317542} a^{4} + \frac{135502844154}{360658771} a^{3} - \frac{165162634097}{721317542} a^{2} + \frac{120984869521}{721317542} a - \frac{1493797046}{32787161} \) (order $10$) | 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: | \( 3973.7989882 \) | 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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{5})\), 4.4.6125.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 8.0.37515625.1 |
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 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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||