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Properties

Label 18.0.91418944539...9328.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 13^{14}\cdot 19^{14}$
Root discriminant $1886.31$
Ramified primes $2, 3, 7, 13, 19$
Class number $18596183472$ (GRH)
Class group $[3, 3, 3, 3, 9, 9, 18, 54, 54, 54]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![948137416523604, -547038123493644, 505306443813876, 23268587608140, -10676966704536, -660557469432, 238912851253, -22405758879, 5420996661, 269577198, -43796637, -8466441, 1727788, -137697, 24423, -1824, 255, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 255*x^16 - 1824*x^15 + 24423*x^14 - 137697*x^13 + 1727788*x^12 - 8466441*x^11 - 43796637*x^10 + 269577198*x^9 + 5420996661*x^8 - 22405758879*x^7 + 238912851253*x^6 - 660557469432*x^5 - 10676966704536*x^4 + 23268587608140*x^3 + 505306443813876*x^2 - 547038123493644*x + 948137416523604)
 
gp: K = bnfinit(x^18 - 9*x^17 + 255*x^16 - 1824*x^15 + 24423*x^14 - 137697*x^13 + 1727788*x^12 - 8466441*x^11 - 43796637*x^10 + 269577198*x^9 + 5420996661*x^8 - 22405758879*x^7 + 238912851253*x^6 - 660557469432*x^5 - 10676966704536*x^4 + 23268587608140*x^3 + 505306443813876*x^2 - 547038123493644*x + 948137416523604, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 255 x^{16} - 1824 x^{15} + 24423 x^{14} - 137697 x^{13} + 1727788 x^{12} - 8466441 x^{11} - 43796637 x^{10} + 269577198 x^{9} + 5420996661 x^{8} - 22405758879 x^{7} + 238912851253 x^{6} - 660557469432 x^{5} - 10676966704536 x^{4} + 23268587608140 x^{3} + 505306443813876 x^{2} - 547038123493644 x + 948137416523604 \)

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:  \(-91418944539476011360498909897881806021621599527281394659328=-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 13^{14}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1886.31$
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:  $2, 3, 7, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{57} a^{9} - \frac{4}{19} a^{8} + \frac{9}{19} a^{7} + \frac{3}{19} a^{6} - \frac{6}{19} a^{5} - \frac{6}{19} a^{4} - \frac{1}{57} a^{3}$, $\frac{1}{171} a^{10} - \frac{1}{57} a^{8} - \frac{22}{57} a^{7} - \frac{8}{57} a^{6} + \frac{17}{57} a^{5} + \frac{11}{171} a^{4} - \frac{4}{57} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{171} a^{11} + \frac{23}{57} a^{8} + \frac{1}{3} a^{7} + \frac{26}{57} a^{6} - \frac{43}{171} a^{5} - \frac{22}{57} a^{4} - \frac{20}{57} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{995904} a^{12} - \frac{79}{38304} a^{11} - \frac{25}{12768} a^{10} - \frac{157}{25536} a^{9} + \frac{6067}{27664} a^{8} + \frac{447}{1064} a^{7} - \frac{4411}{10944} a^{6} + \frac{10249}{38304} a^{5} + \frac{16609}{165984} a^{4} + \frac{3767}{8512} a^{3} - \frac{85}{336} a^{2} - \frac{1}{56} a + \frac{939}{2912}$, $\frac{1}{995904} a^{13} + \frac{59}{38304} a^{11} + \frac{101}{76608} a^{10} - \frac{307}{55328} a^{9} + \frac{397}{798} a^{8} + \frac{10739}{76608} a^{7} + \frac{65}{2128} a^{6} - \frac{195743}{497952} a^{5} + \frac{15907}{76608} a^{4} - \frac{1669}{4256} a^{3} + \frac{1}{28} a^{2} - \frac{199}{8736} a + \frac{37}{112}$, $\frac{1}{1991808} a^{14} - \frac{1}{1991808} a^{13} - \frac{13}{153216} a^{11} - \frac{3251}{1991808} a^{10} + \frac{47}{20748} a^{9} + \frac{6155}{153216} a^{8} - \frac{5}{8064} a^{7} - \frac{23435}{165984} a^{6} + \frac{48547}{104832} a^{5} + \frac{34175}{153216} a^{4} - \frac{2209}{6384} a^{3} + \frac{991}{5824} a^{2} - \frac{317}{2496} a - \frac{55}{112}$, $\frac{1}{290833845120} a^{15} + \frac{2305}{19388923008} a^{14} + \frac{54667}{145416922560} a^{13} + \frac{1465}{4474366848} a^{12} + \frac{312909349}{290833845120} a^{11} + \frac{186380333}{145416922560} a^{10} + \frac{1083116693}{290833845120} a^{9} - \frac{344278811}{1065325440} a^{8} + \frac{62223186803}{145416922560} a^{7} - \frac{51787432001}{290833845120} a^{6} + \frac{33476674787}{290833845120} a^{5} + \frac{795344869}{1597988160} a^{4} + \frac{273260081}{692461536} a^{3} + \frac{122394929}{2551174080} a^{2} + \frac{586233029}{1275587040} a + \frac{11930661}{32707360}$, $\frac{1}{739595951705243035847866230166700160} a^{16} + \frac{1533134587676672336137}{1416850482193952175953766724457280} a^{15} + \frac{9323249558105347909152486883}{56891996285018695065220479243592320} a^{14} - \frac{40357374450626409994043910253}{246531983901747678615955410055566720} a^{13} - \frac{146444236848697364712251182693}{369797975852621517923933115083350080} a^{12} - \frac{486069342623012298035361543658081}{246531983901747678615955410055566720} a^{11} + \frac{5304216093290213127320372805901}{1961792975345472243628292387710080} a^{10} - \frac{1030560048803676743859754826249191}{123265991950873839307977705027783360} a^{9} + \frac{233923566891287358513621476402566409}{739595951705243035847866230166700160} a^{8} - \frac{12935238001389775711510019562517991}{27392442655749742068439490006174080} a^{7} + \frac{2506104564644449840253323537947013}{28445998142509347532610239621796160} a^{6} - \frac{3198818082934569629486873165354367}{27392442655749742068439490006174080} a^{5} - \frac{1235548934612989524213339837852179}{2934904570258900935904231072090080} a^{4} - \frac{15707566922852101547013457053269477}{61632995975436919653988852513891680} a^{3} + \frac{6083515293499868459406185307493}{23764409475780574379791344713280} a^{2} - \frac{35004252105174296040153951751587}{72085375409867742285367078963616} a - \frac{14047542906067439498597702789657}{77234330796286866734321870318160}$, $\frac{1}{59952295666257636992657587365493485641374382033280} a^{17} - \frac{25741956973573}{59952295666257636992657587365493485641374382033280} a^{16} - \frac{644915343265484530768906908932422457}{1498807391656440924816439684137337141034359550832} a^{15} + \frac{6779573660513168116314554324298269705482193}{59952295666257636992657587365493485641374382033280} a^{14} + \frac{1125873189999452424154329186078293694081101}{2606621550706853782289460320238847201798886175360} a^{13} + \frac{8803382721961343236033592294817305359410097}{29976147833128818496328793682746742820687191016640} a^{12} + \frac{72348573616855321399749132525077155122769317449}{59952295666257636992657587365493485641374382033280} a^{11} + \frac{52296415805273654055643092500339450867978238019}{59952295666257636992657587365493485641374382033280} a^{10} + \frac{603235371941858177070064327906759771925200253}{1303310775353426891144730160119423600899443087680} a^{9} + \frac{29436525940740182264057627258637391573324756720183}{59952295666257636992657587365493485641374382033280} a^{8} - \frac{23645663646046231904579440753632904210246220217379}{59952295666257636992657587365493485641374382033280} a^{7} - \frac{6499291803898201683682510259489971708597544952577}{29976147833128818496328793682746742820687191016640} a^{6} - \frac{211017257258215371347657071060310787538111718863}{2498012319427401541360732806895561901723932584720} a^{5} - \frac{54832861301937649964034675312498484419193169033}{138778462190411196742262933716420105651329588040} a^{4} - \frac{3408625005065191943304671015219268953650022305271}{9992049277709606165442931227582247606895730338880} a^{3} + \frac{3958293291680586835339474764919563914841431119}{12521365009661160608324475222534144870796654560} a^{2} - \frac{32650305650312047200445187130683043888849966431}{87649555067628124258271326557739014095576581920} a + \frac{1345867076751728773019456481181723602708347323}{87649555067628124258271326557739014095576581920}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{18}\times C_{54}\times C_{54}\times C_{54}$, which has order $18596183472$ (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:  \( -\frac{112284034646975309}{2979658902119740614030588086089440} a^{17} - \frac{315992436376391371}{224879917141112499172119855553920} a^{16} - \frac{17371203226484653513}{1489829451059870307015294043044720} a^{15} - \frac{56160165290490089451}{165536605673318923001699338116080} a^{14} + \frac{4984227264200708337}{14394487449853819391452116357920} a^{13} - \frac{142168536878316401744617}{2979658902119740614030588086089440} a^{12} + \frac{689620514842733060202743}{1489829451059870307015294043044720} a^{11} - \frac{36214132571070071797827983}{5959317804239481228061176172178880} a^{10} + \frac{2223126623414386157070541}{32387596762171093630767261805320} a^{9} - \frac{65643157605547019486250629}{248304908509978384502549007174120} a^{8} + \frac{2172581623001003812441027431}{331073211346637846003398676232160} a^{7} - \frac{34667208369123802171212376783}{1489829451059870307015294043044720} a^{6} + \frac{74543828298093710332519850299}{2979658902119740614030588086089440} a^{5} - \frac{655162128658898799143577230449}{1324292845386551384013594704928640} a^{4} - \frac{3627058506025022297564579333}{16553660567331892300169933811608} a^{3} - \frac{39510750716031395880875164331}{52274717581048080947905054141920} a^{2} + \frac{24759931968924168771289668078997}{34849811720698720631936702761280} a + \frac{53822756454218836608219919323}{435622646508734007899208784516} \) (order $6$)
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:  \( 34782068743257.96 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.8968323.1 x3, 3.1.186732.1, 6.0.104606519472.3, 6.0.241292452296987.1, 9.1.174565121124349094450651855424.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$