Normalized defining polynomial
\( x^{16} - x^{15} - 7 x^{14} + 15 x^{13} + 12 x^{12} - 47 x^{11} + 52 x^{10} + 41 x^{9} - 35 x^{8} + 18 x^{7} + 63 x^{6} + 26 x^{5} - 3 x^{4} + 15 x^{3} + 22 x^{2} + 8 x + 1 \)
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: | \(9766449236141015625=3^{12}\cdot 5^{8}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.38$ | 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: | $3, 5, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{1018} a^{14} + \frac{72}{509} a^{13} + \frac{84}{509} a^{12} + \frac{145}{1018} a^{11} + \frac{117}{509} a^{10} - \frac{363}{1018} a^{9} - \frac{232}{509} a^{8} - \frac{15}{509} a^{7} - \frac{59}{1018} a^{6} + \frac{143}{509} a^{5} + \frac{299}{1018} a^{4} + \frac{96}{509} a^{3} + \frac{7}{509} a^{2} - \frac{61}{1018} a + \frac{45}{509}$, $\frac{1}{1118782} a^{15} - \frac{487}{1118782} a^{14} + \frac{45254}{559391} a^{13} - \frac{20795}{159826} a^{12} - \frac{240907}{1118782} a^{11} + \frac{230679}{1118782} a^{10} - \frac{44567}{159826} a^{9} - \frac{30755}{559391} a^{8} + \frac{381279}{1118782} a^{7} - \frac{233273}{1118782} a^{6} - \frac{512035}{1118782} a^{5} - \frac{488787}{1118782} a^{4} + \frac{148630}{559391} a^{3} - \frac{34283}{159826} a^{2} - \frac{90705}{1118782} a - \frac{71660}{559391}$
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: | \( \frac{329}{509} a^{15} - \frac{777}{1018} a^{14} - \frac{2216}{509} a^{13} + \frac{5407}{509} a^{12} + \frac{5565}{1018} a^{11} - \frac{16371}{509} a^{10} + \frac{42719}{1018} a^{9} + \frac{10340}{509} a^{8} - \frac{15748}{509} a^{7} + \frac{23835}{1018} a^{6} + \frac{20389}{509} a^{5} + \frac{10157}{1018} a^{4} - \frac{1633}{509} a^{3} + \frac{6522}{509} a^{2} + \frac{14665}{1018} a + \frac{1736}{509} \) (order $6$) | 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: | \( 1902.09271468 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:D_4$ (as 16T34):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2:D_4$ |
| Character table for $C_2^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.475.1, 4.2.4275.1, 4.0.12825.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.513.1, 8.0.164480625.2, 8.0.164480625.1, 8.0.18275625.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |