Normalized defining polynomial
\( x^{16} - 4 x^{15} + 15 x^{14} - 23 x^{13} + 24 x^{12} + 20 x^{11} - 36 x^{10} + 52 x^{9} + 64 x^{8} - 56 x^{7} + 57 x^{6} + 2 x^{5} + 165 x^{4} - 80 x^{3} + 45 x^{2} + 26 x + 169 \)
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: | \(53374971077213779099861=19^{8}\cdot 61^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.33$ | 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: | $19, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{117} a^{14} + \frac{5}{39} a^{13} + \frac{14}{117} a^{12} + \frac{1}{13} a^{11} + \frac{1}{9} a^{10} - \frac{2}{39} a^{9} - \frac{7}{117} a^{8} + \frac{4}{13} a^{7} - \frac{58}{117} a^{6} + \frac{4}{39} a^{5} + \frac{25}{117} a^{4} + \frac{1}{13} a^{3} + \frac{11}{117} a^{2} + \frac{17}{39} a - \frac{2}{9}$, $\frac{1}{344345039725869} a^{15} + \frac{518819823064}{344345039725869} a^{14} + \frac{246543329387}{344345039725869} a^{13} - \frac{1927711903009}{344345039725869} a^{12} - \frac{53264089385123}{344345039725869} a^{11} - \frac{12307404468326}{344345039725869} a^{10} - \frac{33709044120118}{344345039725869} a^{9} + \frac{831691010786}{49192148532267} a^{8} + \frac{120526154033216}{344345039725869} a^{7} + \frac{71925397449935}{344345039725869} a^{6} - \frac{4530895686287}{344345039725869} a^{5} + \frac{62038510809055}{344345039725869} a^{4} - \frac{81034677951352}{344345039725869} a^{3} + \frac{31995039332135}{344345039725869} a^{2} + \frac{60406461636760}{344345039725869} a + \frac{9392472138232}{26488079978913}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 141692.453566 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $D_{16}$ |
| Character table for $D_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 4.0.22021.1, 8.0.29580390901.1 |
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 | $16$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $16$ | $16$ | $16$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $61$ | 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.4.2.1 | $x^{4} + 183 x^{2} + 14884$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 183 x^{2} + 14884$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 183 x^{2} + 14884$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |