Normalized defining polynomial
\( x^{16} - 12 x^{14} - 12 x^{13} + 30 x^{12} + 152 x^{11} + 92 x^{10} - 436 x^{9} - 790 x^{8} + 1540 x^{6} + 1580 x^{5} - 342 x^{4} - 1544 x^{3} - 1020 x^{2} - 268 x - 23 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(58443643655519503122432=2^{40}\cdot 3^{8}\cdot 17^{4}\cdot 97\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.48$ | 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: | $2, 3, 17, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{14} a^{9} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{4} - \frac{5}{14} a^{3} + \frac{3}{14} a^{2} - \frac{1}{2} a - \frac{1}{7}$, $\frac{1}{182} a^{13} + \frac{3}{182} a^{12} - \frac{5}{182} a^{11} + \frac{37}{182} a^{10} + \frac{5}{91} a^{9} + \frac{9}{91} a^{8} + \frac{32}{91} a^{7} - \frac{24}{91} a^{6} - \frac{75}{182} a^{5} + \frac{5}{26} a^{4} - \frac{31}{182} a^{3} - \frac{37}{182} a^{2} - \frac{43}{91} a - \frac{25}{91}$, $\frac{1}{1274} a^{14} + \frac{3}{1274} a^{13} - \frac{9}{637} a^{12} + \frac{116}{637} a^{11} + \frac{57}{637} a^{10} - \frac{121}{637} a^{9} + \frac{155}{1274} a^{8} + \frac{41}{637} a^{7} - \frac{517}{1274} a^{6} + \frac{61}{1274} a^{5} - \frac{44}{91} a^{4} - \frac{11}{91} a^{3} - \frac{290}{637} a^{2} + \frac{248}{637} a + \frac{23}{98}$, $\frac{1}{16562} a^{15} - \frac{3}{16562} a^{14} - \frac{29}{16562} a^{13} + \frac{226}{8281} a^{12} + \frac{3}{98} a^{11} - \frac{1395}{16562} a^{10} + \frac{61}{637} a^{9} - \frac{2633}{16562} a^{8} - \frac{5293}{16562} a^{7} - \frac{6819}{16562} a^{6} - \frac{2963}{16562} a^{5} + \frac{563}{1183} a^{4} + \frac{7953}{16562} a^{3} + \frac{843}{2366} a^{2} - \frac{1958}{8281} a + \frac{859}{16562}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 675189.030825 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4096 |
| The 106 conjugacy class representatives for t16n1590 are not computed |
| Character table for t16n1590 is not computed |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1 |
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 | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 97 | Data not computed | ||||||