↑

Properties

Label 36.0.61939692939...1889.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 7^{24}\cdot 11^{18}$
Root discriminant $63.06$
Ramified primes $3, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![128100283921, 0, 0, 7417705475, 0, 0, -4945223862, 0, 0, -934302955, 0, 0, 191238344, 0, 0, 25621008, 0, 0, 3859957, 0, 0, 72093, 0, 0, -11512, 0, 0, -3470, 0, 0, 9, 0, 0, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 2*x^33 + 9*x^30 - 3470*x^27 - 11512*x^24 + 72093*x^21 + 3859957*x^18 + 25621008*x^15 + 191238344*x^12 - 934302955*x^9 - 4945223862*x^6 + 7417705475*x^3 + 128100283921)
 
gp: K = bnfinit(x^36 - 2*x^33 + 9*x^30 - 3470*x^27 - 11512*x^24 + 72093*x^21 + 3859957*x^18 + 25621008*x^15 + 191238344*x^12 - 934302955*x^9 - 4945223862*x^6 + 7417705475*x^3 + 128100283921, 1)
 

Normalized defining polynomial

\( x^{36} - 2 x^{33} + 9 x^{30} - 3470 x^{27} - 11512 x^{24} + 72093 x^{21} + 3859957 x^{18} + 25621008 x^{15} + 191238344 x^{12} - 934302955 x^{9} - 4945223862 x^{6} + 7417705475 x^{3} + 128100283921 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(61939692939787751759835204451502424263870718887575541286872471889=3^{54}\cdot 7^{24}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.06$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(1,·)$, $\chi_{693}(386,·)$, $\chi_{693}(683,·)$, $\chi_{693}(263,·)$, $\chi_{693}(142,·)$, $\chi_{693}(527,·)$, $\chi_{693}(529,·)$, $\chi_{693}(274,·)$, $\chi_{693}(659,·)$, $\chi_{693}(23,·)$, $\chi_{693}(155,·)$, $\chi_{693}(32,·)$, $\chi_{693}(296,·)$, $\chi_{693}(298,·)$, $\chi_{693}(43,·)$, $\chi_{693}(428,·)$, $\chi_{693}(562,·)$, $\chi_{693}(571,·)$, $\chi_{693}(65,·)$, $\chi_{693}(67,·)$, $\chi_{693}(452,·)$, $\chi_{693}(197,·)$, $\chi_{693}(331,·)$, $\chi_{693}(463,·)$, $\chi_{693}(340,·)$, $\chi_{693}(604,·)$, $\chi_{693}(221,·)$, $\chi_{693}(100,·)$, $\chi_{693}(485,·)$, $\chi_{693}(232,·)$, $\chi_{693}(617,·)$, $\chi_{693}(109,·)$, $\chi_{693}(494,·)$, $\chi_{693}(373,·)$, $\chi_{693}(505,·)$, $\chi_{693}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{18} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{19} - \frac{1}{2} a$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{20} - \frac{1}{2} a^{2}$, $\frac{1}{129448595712742} a^{30} - \frac{7310675833877}{129448595712742} a^{27} - \frac{25051473094306}{64724297856371} a^{24} - \frac{53501000169051}{129448595712742} a^{21} - \frac{50804809027809}{129448595712742} a^{18} - \frac{9967054891810}{64724297856371} a^{15} + \frac{4369078997463}{64724297856371} a^{12} + \frac{11034726749462}{64724297856371} a^{9} + \frac{386813949247}{1578641411131} a^{6} - \frac{25517780561597}{129448595712742} a^{3} - \frac{48912882147447}{129448595712742}$, $\frac{1}{9190850295604682} a^{31} - \frac{1949039611525007}{9190850295604682} a^{28} - \frac{348672962376161}{4595425147802341} a^{25} - \frac{1736332744434697}{9190850295604682} a^{22} + \frac{1373129743812353}{9190850295604682} a^{19} - \frac{980831522737375}{4595425147802341} a^{16} - \frac{837046793135360}{4595425147802341} a^{13} + \frac{1176072088164140}{4595425147802341} a^{10} + \frac{43010132049784}{112083540190301} a^{7} + \frac{103930815151145}{9190850295604682} a^{4} + \frac{1504470266405457}{9190850295604682} a$, $\frac{1}{652550370987932422} a^{32} - \frac{21653932970873038}{326275185493966211} a^{29} - \frac{119829726805237027}{326275185493966211} a^{26} + \frac{200462373758868307}{652550370987932422} a^{23} - \frac{139473902136065224}{326275185493966211} a^{20} - \frac{5576256670539716}{326275185493966211} a^{17} - \frac{88150124601379839}{326275185493966211} a^{14} - \frac{76946155424475657}{326275185493966211} a^{11} + \frac{3293432797568513}{7957931353511371} a^{8} + \frac{27676481701965191}{652550370987932422} a^{5} + \frac{154698977584581152}{326275185493966211} a^{2}$, $\frac{1}{42184487737839498563233654737802910346562192433378} a^{33} - \frac{10167369085138129602687880588263434}{21092243868919749281616827368901455173281096216689} a^{30} - \frac{86701796967798254868650715344487435846151044513}{42184487737839498563233654737802910346562192433378} a^{27} - \frac{417216547380648518757576208884799382934046237051}{1028889944825353623493503774092753910891760791058} a^{24} + \frac{9536608039826812938267132645078858563559625244891}{21092243868919749281616827368901455173281096216689} a^{21} + \frac{4290995590944715318663085547677847364451155497915}{42184487737839498563233654737802910346562192433378} a^{18} + \frac{26637619305997283876504393779428993101297789494}{514444972412676811746751887046376955445880395529} a^{15} - \frac{5387994874118354928799339777515580503273518294807}{21092243868919749281616827368901455173281096216689} a^{12} + \frac{960825755531742027719338404476148922538878224958}{21092243868919749281616827368901455173281096216689} a^{9} - \frac{17673883929073943262623664005655988085010990898455}{42184487737839498563233654737802910346562192433378} a^{6} + \frac{555858085931792530076425562447050033909076826601}{21092243868919749281616827368901455173281096216689} a^{3} - \frac{18782924586960543412493912050336845429069161}{117863065784062234922183600777296340002297198}$, $\frac{1}{2995098629386604397989589486384006634605915662769838} a^{34} - \frac{10167369085138129602687880588263434}{1497549314693302198994794743192003317302957831384919} a^{31} + \frac{210835736892229694561299622973670064296964811122377}{2995098629386604397989589486384006634605915662769838} a^{28} + \frac{12958352735348948586657972854321001458658844046703}{73051186082600107268038767960585527673315016165118} a^{25} + \frac{494658217024981046415454162129812327549024838228738}{1497549314693302198994794743192003317302957831384919} a^{22} - \frac{375369394049610771750439807092548345754608576402487}{2995098629386604397989589486384006634605915662769838} a^{19} + \frac{7743312205496149460077782699475083324789503722429}{36525593041300053634019383980292763836657508082559} a^{16} + \frac{163349956077239639324135279173696060882975251438705}{1497549314693302198994794743192003317302957831384919} a^{13} + \frac{296252239920408231970354921569096521348474225258604}{1497549314693302198994794743192003317302957831384919} a^{10} - \frac{945732614161542911653764068237320015709379224432771}{2995098629386604397989589486384006634605915662769838} a^{7} - \frac{358012287685703945257409639708877687911869558857112}{1497549314693302198994794743192003317302957831384919} a^{4} + \frac{2927793720014595329642096107382071654628360789}{8368277670668418679475035655188040140163101058} a$, $\frac{1}{212652002686448912257260853533264471057020012056658498} a^{35} - \frac{10167369085138129602687880588263434}{106326001343224456128630426766632235528510006028329249} a^{32} + \frac{30161822030758273674457194486813736410356121438820757}{212652002686448912257260853533264471057020012056658498} a^{29} + \frac{451265469230949592194890580617834167498548941037411}{5186634211864607616030752525201572464805366147723378} a^{26} + \frac{25952996566811118429326964796393868721699307971772361}{106326001343224456128630426766632235528510006028329249} a^{23} + \frac{20590321011656620014176686597595498096486801062986379}{212652002686448912257260853533264471057020012056658498} a^{20} + \frac{1286139068650998026650756222009721817607802286611994}{2593317105932303808015376262600786232402683073861689} a^{17} - \frac{17807241820242386748613401639130343746752518725180323}{106326001343224456128630426766632235528510006028329249} a^{14} - \frac{31152283368638937946920334685462973142013640233824695}{106326001343224456128630426766632235528510006028329249} a^{11} + \frac{52966042714797336252158846686674799407197102705424313}{212652002686448912257260853533264471057020012056658498} a^{8} - \frac{31806547896245050124148099246740947351273984017940411}{106326001343224456128630426766632235528510006028329249} a^{5} + \frac{212134735486725062316517987487083075158705887239}{594147714617457726242727531518350849951580175118} a^{2}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{832586803845365017627085503572003000}{414489154357404428174590297036258875533616892163} a^{34} + \frac{554638973144463931969324545125793839}{20218983139385581862175136440793115879688628886} a^{31} - \frac{450887019376537579969011625144749094805}{828978308714808856349180594072517751067233784326} a^{28} - \frac{2076591941151950584096255650554816147845}{414489154357404428174590297036258875533616892163} a^{25} - \frac{128476164661073630741762224997662491720775}{828978308714808856349180594072517751067233784326} a^{22} + \frac{1347804113315114185757735286282274517480795}{828978308714808856349180594072517751067233784326} a^{19} + \frac{5392752704728647850682868649506474306034935}{414489154357404428174590297036258875533616892163} a^{16} + \frac{76587538860576402832642007426472296088908575}{414489154357404428174590297036258875533616892163} a^{13} - \frac{287068648385920482670036572140147467428450390}{414489154357404428174590297036258875533616892163} a^{10} - \frac{3265303264270640753750421431748561083285495895}{414489154357404428174590297036258875533616892163} a^{7} - \frac{106481302111343896331353240865684607449230705805}{828978308714808856349180594072517751067233784326} a^{4} + \frac{5852907936175899880968408561904729316195}{32621939049006984478877276716661040687046} a \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.64827.1, 6.0.47258883.2, 6.0.8732691.1, 6.6.26198073.1, 6.0.20967191091.5, 6.6.62901573273.3, 6.0.3195731.1, 6.6.86284737.1, 6.0.20967191091.4, 6.6.62901573273.1, 9.9.62523502209.1, 12.0.686339028913329.1, 12.0.3956607920218587932529.2, 12.0.7445055839159169.1, 12.0.3956607920218587932529.1, 18.0.105548084868928352751387.1, 18.0.9217661592820801741280239766571.4, 18.18.248876863006161647014566473697417.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
7Data not computed
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$