Normalized defining polynomial
\( x^{16} - 256 x^{14} - 3250812 x^{12} + 560738536 x^{10} + 1565106740490 x^{8} + 40185108913616 x^{6} - 118702711888152884 x^{4} - 1229247489897179560 x^{2} + 280665274177700436025 \)
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: | \(35717732754345647446813752640158274945024000000=2^{56}\cdot 5^{6}\cdot 17^{2}\cdot 41^{6}\cdot 137^{2}\cdot 35089^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $812.00$ | 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: | $2, 5, 17, 41, 137, 35089$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} + \frac{1}{4} a^{4} - \frac{1}{8}$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{5} - \frac{1}{8} a$, $\frac{1}{8} a^{10} + \frac{1}{4} a^{6} - \frac{1}{8} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{120} a^{12} + \frac{7}{120} a^{10} - \frac{1}{20} a^{8} + \frac{19}{60} a^{6} - \frac{41}{120} a^{4} + \frac{11}{40} a^{2} - \frac{1}{3}$, $\frac{1}{120} a^{13} + \frac{7}{120} a^{11} - \frac{1}{20} a^{9} + \frac{19}{60} a^{7} - \frac{41}{120} a^{5} + \frac{11}{40} a^{3} - \frac{1}{3} a$, $\frac{1}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{14} - \frac{7732855948642199545339800050377405027397287136690942644775432442063107}{1929492641388643188022723422079997144373156978186391576837316232534823600} a^{12} - \frac{324533442202189618439389300608394799974273892801021953443872102594560357}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{10} + \frac{161827004484403115316346777898649334403443253287102789189469464356939521}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{8} - \frac{58485780002426577271053816751913867058293017041737418592247487692767949}{165385083547597987544804864749714040946270598130262135157484248502984880} a^{6} + \frac{1410505946013320837837893208677213997919413472061816402107218090851877111}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{4} - \frac{1612536450894233328755352536704236185716580095942333446610281171389089}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{2} - \frac{136577811267212466848081536424727740383592646635770054274630591}{345517493311991536255020697520786458565686121969133827208234960}$, $\frac{1}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{15} - \frac{7732855948642199545339800050377405027397287136690942644775432442063107}{1929492641388643188022723422079997144373156978186391576837316232534823600} a^{13} - \frac{324533442202189618439389300608394799974273892801021953443872102594560357}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{11} + \frac{161827004484403115316346777898649334403443253287102789189469464356939521}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{9} - \frac{58485780002426577271053816751913867058293017041737418592247487692767949}{165385083547597987544804864749714040946270598130262135157484248502984880} a^{7} + \frac{1410505946013320837837893208677213997919413472061816402107218090851877111}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{5} - \frac{1612536450894233328755352536704236185716580095942333446610281171389089}{5788477924165929564068170266239991433119470934559174730511948697604470800} a^{3} - \frac{136577811267212466848081536424727740383592646635770054274630591}{345517493311991536255020697520786458565686121969133827208234960} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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: | \( 12341574495500000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 58 conjugacy class representatives for t16n1127 are not computed |
| Character table for t16n1127 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.44066406400.1 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | Data not computed | ||||||
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.2 | $x^{4} - 17 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 17.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $41$ | 41.4.2.2 | $x^{4} - 41 x^{2} + 20172$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $137$ | 137.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.4.2.2 | $x^{4} - 137 x^{2} + 112614$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 137.8.0.1 | $x^{8} - x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 35089 | Data not computed | ||||||