Normalized defining polynomial
\( x^{16} - 992 x^{14} - 208 x^{13} + 356704 x^{12} + 87600 x^{11} - 57863576 x^{10} - 9972320 x^{9} + 4455644214 x^{8} + 1188103936 x^{7} - 167097841240 x^{6} - 67051400272 x^{5} + 2855439882632 x^{4} + 1097830640448 x^{3} - 16905504082608 x^{2} + 4093471289648 x + 17362718655679 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(124072986059627688411885857786694729728=2^{66}\cdot 113^{3}\cdot 1039^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $240.36$ | 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: | $2, 113, 1039$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{6}{31} a^{13} - \frac{1}{31} a^{12} - \frac{2}{31} a^{11} + \frac{12}{31} a^{10} - \frac{15}{31} a^{9} - \frac{6}{31} a^{8} - \frac{10}{31} a^{6} + \frac{15}{31} a^{5} - \frac{12}{31} a^{4} - \frac{3}{31} a^{3} - \frac{12}{31} a^{2} - \frac{12}{31} a - \frac{14}{31}$, $\frac{1}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{15} - \frac{374210535068871162029890402715432789153355147221479018669376309020988637297774445172690900897420}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{14} - \frac{80716018774430432737666184983789358629855082027037699327530682761536312220374898257215177901341613}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{13} - \frac{2106436036924400055806045169827995152045752672264958932316088303496301855503356736231321454768516}{12049381561867272111331649877863265725791602878288991610653846893417747757189661551023969589427183} a^{12} - \frac{68419644114358436056738613965494938103291197184431504434379555619754678299913182297370712513035100}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{11} + \frac{131029464140285254544072467965766494910881007023063139998561324668355359590733284247507594346324530}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{10} - \frac{161600874135039115294979534562691573231042601880672062733034047651211006722021245706322573072161759}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{9} + \frac{30269161029373503516908522387229747584435161475183678050257108718835133527705238118874685893429788}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{8} + \frac{86936096333887972825489543921427850761023882532167902463921242803310419456968482898922520329391189}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{7} + \frac{80805534393573824427158200654485348205798192674375922224205533617611032842241952671003277837691517}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{6} - \frac{162773528865981408749055994925455020583240748877315256125044692331395013948968665558018479638443936}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{5} - \frac{68754922874204503720894363475640890553452521021363870808100739749499151465267734521891619557472070}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{4} + \frac{41443759524731254319817171636980523009317885562090984055675262905900038436495344075420092570404401}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{3} + \frac{92323333618905123462702166984891362250518170590047472529078896670445653465798153547613014493692536}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a^{2} - \frac{120144147176679338365419365931663923192994572050672554457031343295408362052778387334985558018867056}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673} a + \frac{151947632320471528575583213799408739126809629872529398954498505404535004669386450667678077090598130}{373530828417885435451281146213761237499539689226958739930269253695950180472879508081743057272242673}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 124173895398000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 32 conjugacy class representatives for t16n859 |
| Character table for t16n859 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7583301632.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | $16$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | $16$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | $16$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | $16$ |
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 | ||||||
| $113$ | 113.2.0.1 | $x^{2} - x + 10$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 113.4.2.1 | $x^{4} + 2147 x^{2} + 1276900$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 113.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 1039 | Data not computed | ||||||