Normalized defining polynomial
\( x^{16} - x^{15} + 137 x^{14} - 137 x^{13} + 7753 x^{12} - 7753 x^{11} + 234057 x^{10} - 234057 x^{9} + 4063817 x^{8} - 4063817 x^{7} + 40829513 x^{6} - 40829513 x^{5} + 228000329 x^{4} - 228000329 x^{3} + 655819337 x^{2} - 655819337 x + 941032009 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4025736448695106798281569680113=3^{8}\cdot 11^{8}\cdot 17^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $81.81$ | 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: | $3, 11, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(561=3\cdot 11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{561}(1,·)$, $\chi_{561}(131,·)$, $\chi_{561}(197,·)$, $\chi_{561}(65,·)$, $\chi_{561}(329,·)$, $\chi_{561}(331,·)$, $\chi_{561}(463,·)$, $\chi_{561}(529,·)$, $\chi_{561}(67,·)$, $\chi_{561}(100,·)$, $\chi_{561}(164,·)$, $\chi_{561}(166,·)$, $\chi_{561}(296,·)$, $\chi_{561}(298,·)$, $\chi_{561}(428,·)$, $\chi_{561}(362,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{164643641} a^{9} + \frac{21415099}{164643641} a^{8} + \frac{72}{164643641} a^{7} + \frac{53417208}{164643641} a^{6} + \frac{1728}{164643641} a^{5} + \frac{80482314}{164643641} a^{4} + \frac{15360}{164643641} a^{3} - \frac{77630298}{164643641} a^{2} + \frac{36864}{164643641} a - \frac{77630298}{164643641}$, $\frac{1}{164643641} a^{10} + \frac{80}{164643641} a^{8} - \frac{6677151}{164643641} a^{7} + \frac{2240}{164643641} a^{6} - \frac{44633174}{164643641} a^{5} + \frac{25600}{164643641} a^{4} - \frac{55556220}{164643641} a^{3} + \frac{102400}{164643641} a^{2} - \frac{57581239}{164643641} a + \frac{65536}{164643641}$, $\frac{1}{164643641} a^{11} - \frac{73448661}{164643641} a^{8} - \frac{3520}{164643641} a^{7} - \frac{37275148}{164643641} a^{6} - \frac{112640}{164643641} a^{5} - \frac{73039341}{164643641} a^{4} - \frac{1126400}{164643641} a^{3} + \frac{61027884}{164643641} a^{2} - \frac{2883584}{164643641} a - \frac{46034518}{164643641}$, $\frac{1}{164643641} a^{12} - \frac{4224}{164643641} a^{8} - \frac{17568068}{164643641} a^{7} - \frac{157696}{164643641} a^{6} + \frac{70643297}{164643641} a^{5} - \frac{2027520}{164643641} a^{4} - \frac{70410929}{164643641} a^{3} - \frac{8650752}{164643641} a^{2} + \frac{728341}{164643641} a - \frac{5767168}{164643641}$, $\frac{1}{164643641} a^{13} + \frac{50451199}{164643641} a^{8} + \frac{146432}{164643641} a^{7} - \frac{21501922}{164643641} a^{6} + \frac{5271552}{164643641} a^{5} + \frac{62408383}{164643641} a^{4} + \frac{56229888}{164643641} a^{3} + \frac{60482461}{164643641} a^{2} - \frac{14697273}{164643641} a + \frac{59754120}{164643641}$, $\frac{1}{164643641} a^{14} + \frac{186368}{164643641} a^{8} - \frac{31828148}{164643641} a^{7} + \frac{7827456}{164643641} a^{6} - \frac{20777400}{164643641} a^{5} - \frac{57295673}{164643641} a^{4} - \frac{56959633}{164643641} a^{3} - \frac{16828843}{164643641} a^{2} + \frac{41322920}{164643641} a - \frac{2131570}{164643641}$, $\frac{1}{164643641} a^{15} + \frac{5502901}{164643641} a^{8} - \frac{5591040}{164643641} a^{7} + \frac{63398762}{164643641} a^{6} - \frac{50052295}{164643641} a^{5} - \frac{19872803}{164643641} a^{4} - \frac{80499426}{164643641} a^{3} - \frac{50608650}{164643641} a^{2} + \frac{42631400}{164643641} a + \frac{72712071}{164643641}$
Class group and class number
$C_{2}\times C_{28066}$, which has order $56132$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 3640.012213375973 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 16 |
| The 16 conjugacy class representatives for $C_{16}$ |
| Character table for $C_{16}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | $16$ | $16$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||
| 17 | Data not computed | ||||||