Normalized defining polynomial
\( x^{18} - 2 x^{17} + 102 x^{16} - 178 x^{15} + 5032 x^{14} - 7646 x^{13} + 155495 x^{12} - 203272 x^{11} + 3292921 x^{10} - 3626002 x^{9} + 49341285 x^{8} - 44214104 x^{7} + 521749394 x^{6} - 358900172 x^{5} + 3749466651 x^{4} - 1771316602 x^{3} + 16612972693 x^{2} - 4071582408 x + 34608450449 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-801839982660831770253640955071823872=-\,2^{18}\cdot 13^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $98.78$ | 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: | $2, 13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(988=2^{2}\cdot 13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{988}(1,·)$, $\chi_{988}(519,·)$, $\chi_{988}(207,·)$, $\chi_{988}(467,·)$, $\chi_{988}(727,·)$, $\chi_{988}(729,·)$, $\chi_{988}(157,·)$, $\chi_{988}(415,·)$, $\chi_{988}(935,·)$, $\chi_{988}(937,·)$, $\chi_{988}(365,·)$, $\chi_{988}(625,·)$, $\chi_{988}(883,·)$, $\chi_{988}(885,·)$, $\chi_{988}(311,·)$, $\chi_{988}(313,·)$, $\chi_{988}(571,·)$, $\chi_{988}(833,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{899305918516152974213000422394094414327875617334558656169637401} a^{17} - \frac{57985419662963667466090220414650834868902689274754845457564892}{899305918516152974213000422394094414327875617334558656169637401} a^{16} - \frac{91467323654105673614643154865459018178024653977203039866564236}{899305918516152974213000422394094414327875617334558656169637401} a^{15} - \frac{248611827103197624134122583861450655732798154162940956925043508}{899305918516152974213000422394094414327875617334558656169637401} a^{14} + \frac{142247151921000435815892152315875056029348339367278505906540415}{899305918516152974213000422394094414327875617334558656169637401} a^{13} + \frac{78749583607847570754771257394660753687478582545029807100109705}{899305918516152974213000422394094414327875617334558656169637401} a^{12} - \frac{413417231788056587570940633184890942896352362239948763626163263}{899305918516152974213000422394094414327875617334558656169637401} a^{11} + \frac{276285562075172351149660099891520842053239796281947619970318408}{899305918516152974213000422394094414327875617334558656169637401} a^{10} + \frac{49938563482478291216308306816091215563722830085118229884466978}{899305918516152974213000422394094414327875617334558656169637401} a^{9} - \frac{271143171583567584063760330898463438158249363510921900134188730}{899305918516152974213000422394094414327875617334558656169637401} a^{8} - \frac{28835588012819625722359020483002783922447155668497662679306808}{899305918516152974213000422394094414327875617334558656169637401} a^{7} - \frac{164644020481774620014977936710291759466422149740762692190857241}{899305918516152974213000422394094414327875617334558656169637401} a^{6} + \frac{332635871989595163751656206191735287038299027735843771362331793}{899305918516152974213000422394094414327875617334558656169637401} a^{5} + \frac{119331080990966300424645085324259785261574810593187472032755058}{899305918516152974213000422394094414327875617334558656169637401} a^{4} + \frac{447467670595758576261290963474757366896237669600791075834556305}{899305918516152974213000422394094414327875617334558656169637401} a^{3} + \frac{163720293362115353934220268184886814667235116708105481427794829}{899305918516152974213000422394094414327875617334558656169637401} a^{2} + \frac{432306278197085076761863738486707141751886810648179617494581131}{899305918516152974213000422394094414327875617334558656169637401} a - \frac{356080768135620923261924881584276378883409914091327669355396708}{899305918516152974213000422394094414327875617334558656169637401}$
Class group and class number
$C_{9}\times C_{88578}$, which has order $797202$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 22305.8950792 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-13}) \), 3.3.361.1, 6.0.18324175168.4, \(\Q(\zeta_{19})^+\) |
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 | R | $18$ | $18$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | $18$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 | ||||||
| 13 | Data not computed | ||||||
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |