Normalized defining polynomial
\( x^{16} - 8 x^{15} + 20 x^{14} - 82 x^{12} + 112 x^{11} - 16 x^{10} + 304 x^{9} - 1364 x^{8} + 1368 x^{7} + 2384 x^{6} - 7816 x^{5} + 8622 x^{4} - 3880 x^{3} - 204 x^{2} + 640 x - 79 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(744861422351789571702784=2^{50}\cdot 17^{4}\cdot 89^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.05$ | 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: | $2, 17, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} - \frac{1}{17} a^{12} + \frac{8}{17} a^{11} - \frac{2}{17} a^{10} - \frac{5}{17} a^{9} + \frac{8}{17} a^{8} + \frac{4}{17} a^{7} - \frac{5}{17} a^{6} - \frac{2}{17} a^{5} + \frac{5}{17} a^{4} + \frac{7}{17} a^{3} + \frac{1}{17} a^{2} - \frac{8}{17}$, $\frac{1}{51} a^{14} + \frac{1}{51} a^{13} - \frac{11}{51} a^{12} + \frac{14}{51} a^{11} + \frac{25}{51} a^{10} - \frac{19}{51} a^{9} - \frac{14}{51} a^{8} + \frac{20}{51} a^{7} - \frac{4}{17} a^{6} + \frac{1}{51} a^{5} + \frac{1}{3} a^{4} - \frac{2}{51} a^{3} + \frac{19}{51} a^{2} - \frac{8}{51} a - \frac{16}{51}$, $\frac{1}{10314174291549} a^{15} + \frac{10787102677}{10314174291549} a^{14} + \frac{129873927676}{10314174291549} a^{13} - \frac{1229418478231}{10314174291549} a^{12} + \frac{5013004768099}{10314174291549} a^{11} - \frac{4371867258217}{10314174291549} a^{10} + \frac{4144730850091}{10314174291549} a^{9} + \frac{726182210633}{10314174291549} a^{8} + \frac{83137226248}{3438058097183} a^{7} - \frac{1598830155785}{10314174291549} a^{6} + \frac{3226206041003}{10314174291549} a^{5} + \frac{2406673208332}{10314174291549} a^{4} + \frac{4732393332820}{10314174291549} a^{3} + \frac{4364807913274}{10314174291549} a^{2} + \frac{3397208603465}{10314174291549} a + \frac{446424681221}{3438058097183}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 1433784.56398 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 62 conjugacy class representatives for t16n781 are not computed |
| Character table for t16n781 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.4352.1, \(\Q(\zeta_{16})^+\), 4.4.34816.1, 8.8.4848615424.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $89$ | 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |