Normalized defining polynomial
\( x^{16} - 3 x^{15} - 15 x^{14} + 61 x^{13} - 7 x^{12} - 101 x^{11} - 205 x^{10} + 251 x^{9} + 715 x^{8} + 216 x^{7} - 1848 x^{6} - 192 x^{5} + 1080 x^{4} + 75 x^{3} - 96 x^{2} + 24 x - 3 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-10296087070033187591253552=-\,2^{4}\cdot 3^{7}\cdot 17^{8}\cdot 59^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.58$ | 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: | $2, 3, 17, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{483336903680091067043} a^{15} - \frac{102721040512245097949}{483336903680091067043} a^{14} - \frac{239581431151960858118}{483336903680091067043} a^{13} + \frac{49365485889342134143}{483336903680091067043} a^{12} - \frac{150291678995781230340}{483336903680091067043} a^{11} + \frac{237122129979097494437}{483336903680091067043} a^{10} - \frac{178527402365897956865}{483336903680091067043} a^{9} - \frac{179060134115798162036}{483336903680091067043} a^{8} + \frac{2852783711969240260}{483336903680091067043} a^{7} - \frac{192362376105038576187}{483336903680091067043} a^{6} - \frac{26415375952274311523}{483336903680091067043} a^{5} - \frac{154858245193792768132}{483336903680091067043} a^{4} - \frac{160712466149453622451}{483336903680091067043} a^{3} - \frac{152749723953046308359}{483336903680091067043} a^{2} + \frac{67597179937417506031}{483336903680091067043} a - \frac{233476700119567611873}{483336903680091067043}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 2936154.61191 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 49152 |
| The 104 conjugacy class representatives for t16n1849 are not computed |
| Character table for t16n1849 is not computed |
Intermediate fields
| 4.4.51153.1, 8.8.463143405393.1 |
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 | $16$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $16$ | R |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.12.0.1 | $x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.7.2 | $x^{8} - 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $59$ | 59.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 59.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.8.6.1 | $x^{8} - 59 x^{4} + 55696$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |