Normalized defining polynomial
\( x^{18} - 9 x^{17} + 90 x^{16} - 516 x^{15} + 3204 x^{14} - 14112 x^{13} + 66792 x^{12} - 238122 x^{11} + 919602 x^{10} - 2697362 x^{9} + 8749269 x^{8} - 20953674 x^{7} + 57705369 x^{6} - 108970803 x^{5} + 254607615 x^{4} - 347840628 x^{3} + 682386453 x^{2} - 523610901 x + 844617131 \)
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: | \(-26036970568557781217070611380223151=-\,3^{44}\cdot 31^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $81.65$ | 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: | $3, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(837=3^{3}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{837}(1,·)$, $\chi_{837}(712,·)$, $\chi_{837}(652,·)$, $\chi_{837}(526,·)$, $\chi_{837}(466,·)$, $\chi_{837}(340,·)$, $\chi_{837}(280,·)$, $\chi_{837}(154,·)$, $\chi_{837}(94,·)$, $\chi_{837}(805,·)$, $\chi_{837}(745,·)$, $\chi_{837}(619,·)$, $\chi_{837}(559,·)$, $\chi_{837}(433,·)$, $\chi_{837}(373,·)$, $\chi_{837}(247,·)$, $\chi_{837}(187,·)$, $\chi_{837}(61,·)$$\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}{49006804076455695203825880950855693951956618347963166849} a^{17} - \frac{12854088837248721726619199629288122706685839219701808131}{49006804076455695203825880950855693951956618347963166849} a^{16} + \frac{15844059022775427114922734975298635917000718168921660335}{49006804076455695203825880950855693951956618347963166849} a^{15} - \frac{22817334270813962571236561375504622486699370664480581302}{49006804076455695203825880950855693951956618347963166849} a^{14} - \frac{14319472845372431359054551014847453395134875635259528498}{49006804076455695203825880950855693951956618347963166849} a^{13} - \frac{16159183003525939078874167585384257324896952876751527951}{49006804076455695203825880950855693951956618347963166849} a^{12} - \frac{9979626143420035221890463335296550438360559681802743976}{49006804076455695203825880950855693951956618347963166849} a^{11} - \frac{4132716440995097476854743264381159126446395132304651222}{49006804076455695203825880950855693951956618347963166849} a^{10} - \frac{22771686191817786970608820146995232900059801674279058044}{49006804076455695203825880950855693951956618347963166849} a^{9} - \frac{12604551579597532722083359751119723782698191293881355319}{49006804076455695203825880950855693951956618347963166849} a^{8} + \frac{7603590055637101479343070655211730542721006826873956599}{49006804076455695203825880950855693951956618347963166849} a^{7} + \frac{1264683491145716794634482609745219238447058834759613023}{49006804076455695203825880950855693951956618347963166849} a^{6} + \frac{1063275589093932076988420075149496826780160087542418050}{49006804076455695203825880950855693951956618347963166849} a^{5} - \frac{20854797219727248637452162250371155138231123935344083745}{49006804076455695203825880950855693951956618347963166849} a^{4} - \frac{21280762786004902312062800841269289686360503186622376880}{49006804076455695203825880950855693951956618347963166849} a^{3} - \frac{17956934516924026683980392049310348487612047361807857}{82642165390313145369014976308356988114598007332146993} a^{2} - \frac{3821474405290871721265337024871879259401543576631205087}{49006804076455695203825880950855693951956618347963166849} a + \frac{5470319409390405179133691026977264037157524653970168580}{49006804076455695203825880950855693951956618347963166849}$
Class group and class number
$C_{93339}$, which has order $93339$ (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: | \( 40934.0329443 \) (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
| \(\Q(\sqrt{-31}) \), \(\Q(\zeta_{9})^+\), 6.0.195458751.1, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 31 | Data not computed | ||||||