Normalized defining polynomial
\( x^{18} - x^{17} + 205 x^{16} - 206 x^{15} + 15660 x^{14} - 15867 x^{13} + 567199 x^{12} - 549752 x^{11} + 10183475 x^{10} - 9611761 x^{9} + 89713575 x^{8} - 122956128 x^{7} + 446565912 x^{6} - 1040646774 x^{5} + 1226326338 x^{4} - 3721651392 x^{3} + 3392343195 x^{2} - 3279342495 x + 13375615217 \)
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: | \(-822204469544577674838182026941616159571=-\,23^{9}\cdot 37^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $145.19$ | 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: | $23, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(851=23\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{851}(1,·)$, $\chi_{851}(645,·)$, $\chi_{851}(70,·)$, $\chi_{851}(321,·)$, $\chi_{851}(781,·)$, $\chi_{851}(206,·)$, $\chi_{851}(530,·)$, $\chi_{851}(599,·)$, $\chi_{851}(344,·)$, $\chi_{851}(737,·)$, $\chi_{851}(804,·)$, $\chi_{851}(231,·)$, $\chi_{851}(620,·)$, $\chi_{851}(850,·)$, $\chi_{851}(47,·)$, $\chi_{851}(114,·)$, $\chi_{851}(507,·)$, $\chi_{851}(252,·)$$\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}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{17} - \frac{24959961662275374323411547927273464498198759682491618042305611224036153114547028958969}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{16} + \frac{4686353067887499103454723604182528792609527281324824806082595147471624001703029847095}{13449978370194262398109583799539480352789959504781617998207712876209405135862058564523} a^{15} - \frac{454370362032805198409295459423298788751578047951268392758995315079531824239928181290381}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{14} - \frac{384761691759314884192877608469633524449775990478750630400347826904405639687561881767964}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{13} - \frac{387966064221204850911220429009175742791502410217379153972386237835278535471420712767319}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{12} + \frac{294490869989128669256650158877340083863919176785821671539764596716950918858501831240889}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{11} + \frac{9705633987236471707047664355971733555164526326350905069821797773469842173293430125805}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{10} + \frac{437698870767190129057946257118278870683256613816746361079360102641133082714636719727960}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{9} + \frac{222586643079759196970607592358646831809792159784005436100801135343299543336296133531960}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{8} + \frac{4317141271653021247279502853292574222759087215776139660522128389652297177174487872994}{13449978370194262398109583799539480352789959504781617998207712876209405135862058564523} a^{7} - \frac{393462656652214279716605129571784491696972506594008318298375471838503509581647661364366}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{6} + \frac{319528987014952842259097071632037571713278596028751152048187796270189886383965106359604}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{5} - \frac{422534246743510078197951630792834720347915083868018893881491734954984432165070113653831}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{4} + \frac{53202441901448805264997825238908580351377545997379852342078672484615085067937540566434}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{3} - \frac{110856310036598798599660841141344383160954088433655588700372910397222327565673555552527}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a^{2} + \frac{393329145304437486734572736154705222213259789651812831511280589298405472930225666218123}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179} a - \frac{447639440534542283471270098823850707338057676786030616812578707421634508288724769007493}{981848421024181155061999617366382065753667043849058113869163039963286574917930275210179}$
Class group and class number
$C_{2}\times C_{4}\times C_{276860}$, which has order $2214880$ (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: | \( 409151.3102125697 \) (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
| \(\Q(\sqrt{-851}) \), 3.3.1369.1, 6.0.843707924819.1, 9.9.3512479453921.1 |
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 | $18$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | $18$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.6.3.1 | $x^{6} - 46 x^{4} + 529 x^{2} - 194672$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 23.6.3.1 | $x^{6} - 46 x^{4} + 529 x^{2} - 194672$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 23.6.3.1 | $x^{6} - 46 x^{4} + 529 x^{2} - 194672$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37 | Data not computed | ||||||