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Properties

Label 18.0.22126735189...4032.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 727^{15}$
Root discriminant $1187.73$
Ramified primes $2, 3, 727$
Class number $25819166016$ (GRH)
Class group $[2, 2, 6, 6, 2922, 61362]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19989274074918912, 176904804349440, 1724517820231056, 257270466657168, 57576694070148, 5574455143176, 708210563500, 62716099132, 1059157049, 414420542, 381518397, 80449872, 11671470, 811404, 35862, -2764, -475, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 475*x^16 - 2764*x^15 + 35862*x^14 + 811404*x^13 + 11671470*x^12 + 80449872*x^11 + 381518397*x^10 + 414420542*x^9 + 1059157049*x^8 + 62716099132*x^7 + 708210563500*x^6 + 5574455143176*x^5 + 57576694070148*x^4 + 257270466657168*x^3 + 1724517820231056*x^2 + 176904804349440*x + 19989274074918912)
 
gp: K = bnfinit(x^18 - 2*x^17 - 475*x^16 - 2764*x^15 + 35862*x^14 + 811404*x^13 + 11671470*x^12 + 80449872*x^11 + 381518397*x^10 + 414420542*x^9 + 1059157049*x^8 + 62716099132*x^7 + 708210563500*x^6 + 5574455143176*x^5 + 57576694070148*x^4 + 257270466657168*x^3 + 1724517820231056*x^2 + 176904804349440*x + 19989274074918912, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 475 x^{16} - 2764 x^{15} + 35862 x^{14} + 811404 x^{13} + 11671470 x^{12} + 80449872 x^{11} + 381518397 x^{10} + 414420542 x^{9} + 1059157049 x^{8} + 62716099132 x^{7} + 708210563500 x^{6} + 5574455143176 x^{5} + 57576694070148 x^{4} + 257270466657168 x^{3} + 1724517820231056 x^{2} + 176904804349440 x + 19989274074918912 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(-22126735189344705358145337335147732804466351023987884032=-\,2^{27}\cdot 3^{9}\cdot 727^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1187.73$
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:  $2, 3, 727$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{6} - \frac{1}{4} a^{5} + \frac{1}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{7} - \frac{1}{6} a^{5} - \frac{5}{12} a^{3} - \frac{1}{2} a$, $\frac{1}{72} a^{8} + \frac{1}{36} a^{7} - \frac{1}{36} a^{6} - \frac{1}{18} a^{5} - \frac{17}{72} a^{4} + \frac{1}{36} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{144} a^{9} - \frac{1}{144} a^{8} - \frac{1}{72} a^{7} - \frac{1}{36} a^{6} - \frac{35}{144} a^{5} + \frac{11}{144} a^{4} + \frac{5}{12} a^{3} + \frac{1}{24} a^{2} + \frac{1}{4} a$, $\frac{1}{432} a^{10} + \frac{1}{432} a^{9} + \frac{1}{216} a^{8} + \frac{1}{27} a^{7} - \frac{7}{432} a^{6} - \frac{35}{432} a^{5} + \frac{7}{108} a^{4} + \frac{25}{72} a^{3} - \frac{1}{36} a^{2} - \frac{1}{2} a$, $\frac{1}{864} a^{11} + \frac{1}{864} a^{9} + \frac{1}{432} a^{8} + \frac{25}{864} a^{7} + \frac{1}{27} a^{6} + \frac{25}{288} a^{5} + \frac{73}{432} a^{4} - \frac{37}{144} a^{3} + \frac{7}{72} a^{2} + \frac{1}{3} a$, $\frac{1}{1728} a^{12} + \frac{1}{1728} a^{10} + \frac{1}{864} a^{9} - \frac{11}{1728} a^{8} + \frac{1}{54} a^{7} - \frac{23}{576} a^{6} - \frac{35}{864} a^{5} - \frac{43}{288} a^{4} - \frac{11}{144} a^{3} - \frac{5}{24} a^{2} + \frac{1}{4} a$, $\frac{1}{5184} a^{13} - \frac{1}{5184} a^{12} + \frac{1}{5184} a^{11} + \frac{1}{5184} a^{10} - \frac{13}{5184} a^{9} - \frac{29}{5184} a^{8} - \frac{101}{5184} a^{7} + \frac{143}{5184} a^{6} - \frac{47}{1296} a^{5} + \frac{3}{32} a^{4} - \frac{91}{432} a^{3} - \frac{7}{72} a^{2} - \frac{1}{12} a$, $\frac{1}{31104} a^{14} - \frac{1}{31104} a^{13} - \frac{5}{31104} a^{12} - \frac{17}{31104} a^{11} + \frac{17}{31104} a^{10} - \frac{59}{31104} a^{9} - \frac{179}{31104} a^{8} + \frac{1013}{31104} a^{7} - \frac{445}{15552} a^{6} - \frac{613}{2592} a^{5} - \frac{41}{324} a^{4} + \frac{161}{864} a^{3} + \frac{1}{3} a^{2} - \frac{1}{24} a$, $\frac{1}{45225216} a^{15} + \frac{13}{1884384} a^{14} + \frac{149}{7537536} a^{13} - \frac{2089}{11306304} a^{12} - \frac{2147}{3768768} a^{11} - \frac{355}{471096} a^{10} + \frac{40165}{22612608} a^{9} + \frac{3853}{1256256} a^{8} + \frac{186533}{15075072} a^{7} - \frac{33659}{2826576} a^{6} + \frac{25}{2908} a^{5} - \frac{452573}{1884384} a^{4} + \frac{21059}{139584} a^{3} + \frac{10003}{34896} a^{2} + \frac{847}{34896} a - \frac{329}{727}$, $\frac{1}{135675648} a^{16} - \frac{485}{33918912} a^{14} - \frac{3331}{67837824} a^{13} + \frac{16223}{67837824} a^{12} - \frac{15985}{67837824} a^{11} + \frac{209}{16959456} a^{10} - \frac{22429}{67837824} a^{9} - \frac{712691}{135675648} a^{8} - \frac{232253}{22612608} a^{7} - \frac{577811}{33918912} a^{6} - \frac{625895}{5653152} a^{5} - \frac{1928431}{11306304} a^{4} - \frac{410743}{1884384} a^{3} + \frac{41297}{314064} a^{2} - \frac{2329}{17448} a + \frac{47}{727}$, $\frac{1}{1059962667134354071594747209158077535083066176325586324853317835636758525931694166021745686016} a^{17} + \frac{3060503233029723998500595239955000269976541240281751123203201268379954156114950085091}{1059962667134354071594747209158077535083066176325586324853317835636758525931694166021745686016} a^{16} - \frac{83574673571098659700202838010782329541515558712104091499913916384082748527144053281}{16561916673974282368667925143094961485672909005087286325833091181824351967682721344089776344} a^{15} + \frac{527385266157393753537896387567816792156024830225480926983632448858692911582526739332601}{176660444522392345265791201526346255847177696054264387475552972606126420988615694336957614336} a^{14} + \frac{2559182885826573362556260303693500988490844066654749135940742065568316313877516339301835}{88330222261196172632895600763173127923588848027132193737776486303063210494307847168478807168} a^{13} + \frac{7454936231640508680527234522343905863283692107052886535471258151402544118878900028759623}{88330222261196172632895600763173127923588848027132193737776486303063210494307847168478807168} a^{12} - \frac{11668595525683844509881474004402939751586141132065071609491209393021421681435658283716663}{58886814840797448421930400508782085282392565351421462491850990868708806996205231445652538112} a^{11} + \frac{19278338334584381133880838504070662593581710552523872419492337674041447307975624097082623}{19628938280265816140643466836260695094130855117140487497283663622902935665401743815217512704} a^{10} - \frac{828322584687962408471761935188066835071118550154556227853342646656024858675741153137301803}{353320889044784690531582403052692511694355392108528774951105945212252841977231388673915228672} a^{9} - \frac{1698649961249348680975649668540402825379981938047290561131463887215210611786152147788149563}{1059962667134354071594747209158077535083066176325586324853317835636758525931694166021745686016} a^{8} - \frac{7509210313704103659761408317617672784668533211259954298864300066575477015444239190699008157}{529981333567177035797373604579038767541533088162793162426658917818379262965847083010872843008} a^{7} - \frac{1044717042678218939397072764566139130797489657539957038100933214257918515894053040802658147}{33123833347948564737335850286189922971345818010174572651666182363648703935365442688179552688} a^{6} - \frac{194794387240689653029267052080796401069778600934337477831541880252054555334755795106053007}{9814469140132908070321733418130347547065427558570243748641831811451467832700871907608756352} a^{5} + \frac{8699549969891064183301099244503305295035909931085029269715598022922522644697459787020742719}{88330222261196172632895600763173127923588848027132193737776486303063210494307847168478807168} a^{4} - \frac{5083392221931758304093321842897408373749004986366627635862382455708055715375341880921912547}{14721703710199362105482600127195521320598141337855365622962747717177201749051307861413134528} a^{3} + \frac{455230994100701755682226241531540062976371956244606807079801802264618442332553169736732245}{2453617285033227017580433354532586886766356889642560937160457952862866958175217976902189088} a^{2} + \frac{18911544914795998345428884387153088396172395798320040007178602160189323197959299235564099}{136312071390734834310024075251810382598130938313475607620025441825714831009734332050121616} a + \frac{778503504227505193423466101734327486046784181886412714546216493253697030783969214775342}{2839834820640309048125501567746049637461061214864075158750530038035725646036131917710867}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{2922}\times C_{61362}$, which has order $25819166016$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 49077055635172.75 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-4362}) \), 3.1.17448.1 x3, 3.3.528529.1, 6.0.5311741819392.1, Deg 6, Deg 6 x2, 9.3.1483797384282637845084672.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
727Data not computed