Normalized defining polynomial
\( x^{16} + 2 x^{14} - 18 x^{12} - 18 x^{10} + 80 x^{8} + 13 x^{6} - 58 x^{4} - x^{2} + 3 \)
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: | \(189461986215064195173168=2^{4}\cdot 3\cdot 89^{2}\cdot 163^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.50$ | 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: | $2, 3, 89, 163$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{45586} a^{14} + \frac{5113}{22793} a^{12} - \frac{885}{45586} a^{10} + \frac{563}{45586} a^{8} - \frac{1}{2} a^{7} - \frac{10437}{45586} a^{6} - \frac{1}{2} a^{5} + \frac{8951}{45586} a^{4} - \frac{10861}{22793} a^{2} + \frac{9263}{45586}$, $\frac{1}{45586} a^{15} + \frac{5113}{22793} a^{13} - \frac{885}{45586} a^{11} + \frac{563}{45586} a^{9} - \frac{10437}{45586} a^{7} - \frac{1}{2} a^{6} + \frac{8951}{45586} a^{5} + \frac{1071}{45586} a^{3} - \frac{6765}{22793} a - \frac{1}{2}$
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: | \( 996739.025442 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 49152 |
| The 140 conjugacy class representatives for t16n1845 are not computed |
| Character table for t16n1845 is not computed |
Intermediate fields
| 4.4.26569.1, 8.8.62826146729.1 |
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 | R | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.4 | $x^{4} - 5$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.6.0.1 | $x^{6} - x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 2.6.0.1 | $x^{6} - x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $89$ | 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.6.0.1 | $x^{6} - x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $163$ | 163.4.0.1 | $x^{4} - x + 42$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 163.6.4.1 | $x^{6} + 5216 x^{3} + 35363339$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 163.6.4.1 | $x^{6} + 5216 x^{3} + 35363339$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |