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Properties

Label 36.0.19446261184...9361.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{24}\cdot 13^{30}$
Root discriminant $53.73$
Ramified primes $3, 7, 13$
Class number $21888$ (GRH)
Class group $[2, 2, 2, 2, 2, 6, 114]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 81, -92, 2093, -1065, 22097, -6767, 162475, -7791, 708581, -79641, 2109469, -223893, 4237386, -685062, 6093363, -998749, 6143602, -1126928, 4427149, -629685, 2165880, -233544, 782564, -58763, 199953, -12295, 37992, -1832, 5049, -236, 482, -17, 28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 28*x^34 - 17*x^33 + 482*x^32 - 236*x^31 + 5049*x^30 - 1832*x^29 + 37992*x^28 - 12295*x^27 + 199953*x^26 - 58763*x^25 + 782564*x^24 - 233544*x^23 + 2165880*x^22 - 629685*x^21 + 4427149*x^20 - 1126928*x^19 + 6143602*x^18 - 998749*x^17 + 6093363*x^16 - 685062*x^15 + 4237386*x^14 - 223893*x^13 + 2109469*x^12 - 79641*x^11 + 708581*x^10 - 7791*x^9 + 162475*x^8 - 6767*x^7 + 22097*x^6 - 1065*x^5 + 2093*x^4 - 92*x^3 + 81*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 28*x^34 - 17*x^33 + 482*x^32 - 236*x^31 + 5049*x^30 - 1832*x^29 + 37992*x^28 - 12295*x^27 + 199953*x^26 - 58763*x^25 + 782564*x^24 - 233544*x^23 + 2165880*x^22 - 629685*x^21 + 4427149*x^20 - 1126928*x^19 + 6143602*x^18 - 998749*x^17 + 6093363*x^16 - 685062*x^15 + 4237386*x^14 - 223893*x^13 + 2109469*x^12 - 79641*x^11 + 708581*x^10 - 7791*x^9 + 162475*x^8 - 6767*x^7 + 22097*x^6 - 1065*x^5 + 2093*x^4 - 92*x^3 + 81*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 28 x^{34} - 17 x^{33} + 482 x^{32} - 236 x^{31} + 5049 x^{30} - 1832 x^{29} + 37992 x^{28} - 12295 x^{27} + 199953 x^{26} - 58763 x^{25} + 782564 x^{24} - 233544 x^{23} + 2165880 x^{22} - 629685 x^{21} + 4427149 x^{20} - 1126928 x^{19} + 6143602 x^{18} - 998749 x^{17} + 6093363 x^{16} - 685062 x^{15} + 4237386 x^{14} - 223893 x^{13} + 2109469 x^{12} - 79641 x^{11} + 708581 x^{10} - 7791 x^{9} + 162475 x^{8} - 6767 x^{7} + 22097 x^{6} - 1065 x^{5} + 2093 x^{4} - 92 x^{3} + 81 x^{2} + 6 x + 1 \)

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:  \(194462611843897382024510486606832173718103280006113355559179361=3^{18}\cdot 7^{24}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.73$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(107,·)$, $\chi_{273}(4,·)$, $\chi_{273}(134,·)$, $\chi_{273}(263,·)$, $\chi_{273}(142,·)$, $\chi_{273}(16,·)$, $\chi_{273}(22,·)$, $\chi_{273}(23,·)$, $\chi_{273}(25,·)$, $\chi_{273}(155,·)$, $\chi_{273}(29,·)$, $\chi_{273}(170,·)$, $\chi_{273}(43,·)$, $\chi_{273}(172,·)$, $\chi_{273}(179,·)$, $\chi_{273}(53,·)$, $\chi_{273}(191,·)$, $\chi_{273}(64,·)$, $\chi_{273}(74,·)$, $\chi_{273}(205,·)$, $\chi_{273}(79,·)$, $\chi_{273}(211,·)$, $\chi_{273}(212,·)$, $\chi_{273}(88,·)$, $\chi_{273}(218,·)$, $\chi_{273}(92,·)$, $\chi_{273}(95,·)$, $\chi_{273}(100,·)$, $\chi_{273}(233,·)$, $\chi_{273}(235,·)$, $\chi_{273}(113,·)$, $\chi_{273}(116,·)$, $\chi_{273}(121,·)$, $\chi_{273}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{26} a^{30} + \frac{5}{13} a^{29} + \frac{1}{13} a^{28} + \frac{3}{26} a^{27} + \frac{6}{13} a^{26} - \frac{9}{26} a^{25} - \frac{1}{26} a^{24} - \frac{5}{13} a^{23} + \frac{5}{26} a^{22} - \frac{5}{13} a^{21} + \frac{7}{26} a^{20} - \frac{3}{13} a^{19} + \frac{2}{13} a^{18} - \frac{11}{26} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} + \frac{1}{26} a^{14} - \frac{5}{13} a^{13} + \frac{5}{13} a^{12} - \frac{3}{13} a^{11} - \frac{3}{13} a^{10} - \frac{2}{13} a^{9} - \frac{9}{26} a^{8} + \frac{1}{13} a^{7} + \frac{9}{26} a^{6} + \frac{4}{13} a^{5} - \frac{5}{13} a^{4} - \frac{2}{13} a^{3} - \frac{7}{26} a^{2} + \frac{1}{26} a + \frac{1}{26}$, $\frac{1}{338} a^{31} + \frac{3}{169} a^{30} - \frac{71}{169} a^{29} - \frac{5}{338} a^{28} + \frac{5}{13} a^{27} + \frac{47}{338} a^{26} + \frac{139}{338} a^{25} - \frac{55}{169} a^{24} + \frac{19}{338} a^{23} + \frac{11}{169} a^{22} + \frac{21}{338} a^{21} - \frac{4}{169} a^{20} - \frac{64}{169} a^{19} - \frac{53}{338} a^{18} + \frac{57}{338} a^{17} + \frac{7}{26} a^{16} + \frac{79}{338} a^{15} + \frac{45}{169} a^{14} - \frac{40}{169} a^{13} - \frac{23}{169} a^{12} + \frac{61}{169} a^{11} + \frac{36}{169} a^{10} + \frac{137}{338} a^{9} - \frac{46}{169} a^{8} - \frac{129}{338} a^{7} + \frac{25}{169} a^{6} - \frac{21}{169} a^{5} + \frac{57}{169} a^{4} + \frac{9}{338} a^{3} + \frac{159}{338} a^{2} + \frac{127}{338} a - \frac{41}{169}$, $\frac{1}{1014} a^{32} + \frac{2}{507} a^{30} - \frac{125}{338} a^{29} - \frac{76}{507} a^{28} + \frac{163}{338} a^{27} - \frac{25}{78} a^{26} + \frac{230}{507} a^{25} + \frac{53}{338} a^{24} + \frac{227}{507} a^{23} + \frac{41}{338} a^{22} - \frac{44}{169} a^{21} + \frac{30}{169} a^{20} + \frac{23}{78} a^{19} + \frac{89}{1014} a^{18} + \frac{113}{1014} a^{17} + \frac{209}{1014} a^{16} - \frac{64}{169} a^{15} + \frac{119}{507} a^{14} - \frac{62}{169} a^{13} + \frac{95}{507} a^{12} - \frac{200}{507} a^{11} - \frac{35}{1014} a^{10} + \frac{193}{507} a^{9} + \frac{475}{1014} a^{8} + \frac{29}{169} a^{7} + \frac{47}{169} a^{6} + \frac{235}{507} a^{5} + \frac{209}{1014} a^{4} - \frac{95}{338} a^{3} + \frac{265}{1014} a^{2} + \frac{176}{507} a - \frac{1}{507}$, $\frac{1}{6531097950} a^{33} + \frac{772948}{3265548975} a^{32} + \frac{3881383}{6531097950} a^{31} - \frac{115208611}{6531097950} a^{30} - \frac{537270337}{3265548975} a^{29} + \frac{609553177}{3265548975} a^{28} + \frac{2287386083}{6531097950} a^{27} + \frac{261073193}{6531097950} a^{26} + \frac{152870219}{653109795} a^{25} - \frac{37178170}{130621959} a^{24} - \frac{732070358}{3265548975} a^{23} + \frac{37943723}{217703265} a^{22} + \frac{507030877}{2177032650} a^{21} - \frac{163707649}{6531097950} a^{20} + \frac{70280971}{435406530} a^{19} - \frac{6312412}{43540653} a^{18} + \frac{82338747}{362838775} a^{17} - \frac{1964990111}{6531097950} a^{16} - \frac{2128234337}{6531097950} a^{15} + \frac{1022181778}{3265548975} a^{14} + \frac{47734016}{3265548975} a^{13} + \frac{592759463}{3265548975} a^{12} - \frac{1653782629}{6531097950} a^{11} + \frac{722484713}{3265548975} a^{10} - \frac{1326206027}{3265548975} a^{9} - \frac{1373572172}{3265548975} a^{8} + \frac{290330327}{725677550} a^{7} - \frac{67070842}{653109795} a^{6} - \frac{513626393}{2177032650} a^{5} - \frac{2943653579}{6531097950} a^{4} - \frac{109265203}{251196075} a^{3} - \frac{67640537}{2177032650} a^{2} + \frac{200131213}{2177032650} a - \frac{162037919}{3265548975}$, $\frac{1}{452732007955993633957050} a^{34} - \frac{14810111393447}{452732007955993633957050} a^{33} + \frac{17529764639159872831}{90546401591198726791410} a^{32} + \frac{38998758942108131572}{45273200795599363395705} a^{31} - \frac{1059682984220015411813}{226366003977996816978525} a^{30} - \frac{12954708144833604459757}{226366003977996816978525} a^{29} + \frac{6287242615284476116987}{150910669318664544652350} a^{28} + \frac{58238810515505767208933}{150910669318664544652350} a^{27} - \frac{32107687747547293387853}{150910669318664544652350} a^{26} + \frac{20932486080888804376741}{90546401591198726791410} a^{25} + \frac{4434464288402368666909}{452732007955993633957050} a^{24} + \frac{49583600678396067843389}{226366003977996816978525} a^{23} + \frac{16390953871394199220381}{75455334659332272326175} a^{22} + \frac{7285394980800627032818}{17412769536768985921425} a^{21} - \frac{147146321511448100936453}{452732007955993633957050} a^{20} - \frac{6918285219813843919}{89561228082293498310} a^{19} + \frac{29034100929013098464066}{75455334659332272326175} a^{18} - \frac{86449164740237918870257}{226366003977996816978525} a^{17} - \frac{22088092553906666219071}{50303556439554848217450} a^{16} + \frac{58652017057129550982011}{226366003977996816978525} a^{15} + \frac{77264907085352742981449}{452732007955993633957050} a^{14} - \frac{79087844988388004465}{3018213386373290893047} a^{13} + \frac{107730181324806834463303}{452732007955993633957050} a^{12} + \frac{206068670126581470632873}{452732007955993633957050} a^{11} - \frac{94458231233558259912661}{226366003977996816978525} a^{10} + \frac{2699036930184818996482}{8383926073259141369575} a^{9} - \frac{330042822047576895278}{3482553907353797184285} a^{8} - \frac{91579734885757724033569}{452732007955993633957050} a^{7} - \frac{76708380796839744100972}{226366003977996816978525} a^{6} - \frac{6638390830209127688341}{226366003977996816978525} a^{5} + \frac{43207537263041360082073}{150910669318664544652350} a^{4} + \frac{16785067608445836970811}{34825539073537971842850} a^{3} + \frac{51458462743873343943679}{150910669318664544652350} a^{2} - \frac{10719699241108029066679}{45273200795599363395705} a + \frac{214470762848951435849809}{452732007955993633957050}$, $\frac{1}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{35} - \frac{4857495292930397023835129569811551592681878}{101036913930142404056212777578870463962370531981540854017664859431675} a^{34} - \frac{4977343946439768730201096321293583137976792731347233767626}{101036913930142404056212777578870463962370531981540854017664859431675} a^{33} + \frac{219474244425267209764315055717221843025032289400349671898119280346}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{32} + \frac{108687974289133315397913666471036633039756335657390123416178109341}{105078390487348100218461288682025282520865353260802488178371453808942} a^{31} - \frac{671542831620529457071959827903251174910970979849427339619229635509}{262695976218370250546153221705063206302163383152006220445928634522355} a^{30} + \frac{31130137539639684168383361538141141672313160849141462061573802747272}{437826627030617084243588702841772010503605638586677034076547724203925} a^{29} + \frac{428304153909137860366435222820374896092202792380337350019331639518427}{875653254061234168487177405683544021007211277173354068153095448407850} a^{28} + \frac{2178398217182718109381882838232450833982914178302175339728257968309}{97294806006803796498575267298171557889690141908150452017010605378650} a^{27} + \frac{24340189797016010742343658253822620315256894677081389937650045583782}{52539195243674050109230644341012641260432676630401244089185726904471} a^{26} - \frac{31906998605785599778430807845197764877495400264981723038718960716191}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{25} - \frac{590301705725709348053987912737143577988228185281448096826871718902763}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{24} - \frac{319627810841039046956110518415000771115238127476095451764290406397683}{875653254061234168487177405683544021007211277173354068153095448407850} a^{23} + \frac{3895824967889471818191060815173677897468163439829758841595469471861}{7795132825470927315909591148518196032705144900653003574063164229150} a^{22} - \frac{511135298050179506433952583240364537938902541533760961238047432161482}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{21} + \frac{4457227726176794255191844921848164943637312600794121799457903002783}{9729480600680379649857526729817155788969014190815045201701060537865} a^{20} - \frac{183172194586034874755337151971998926888998582417912955786283837937259}{437826627030617084243588702841772010503605638586677034076547724203925} a^{19} + \frac{114761280917761103227870895984043528524180834043464648534597704227256}{262695976218370250546153221705063206302163383152006220445928634522355} a^{18} + \frac{110344833402374986983706691644235659584121491022094723875680210141683}{291884418020411389495725801894514673669070425724451356051031816135950} a^{17} - \frac{99890919475236533576377242701251809220329143126183294929152915467477}{202073827860284808112425555157740927924741063963081708035329718863350} a^{16} - \frac{385167263273755863587982945960217382862227843535666799476985465340051}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{15} + \frac{16338401488470097444405722753101286283484810941013926065601886245327}{67357942620094936037475185052580309308247021321027236011776572954450} a^{14} + \frac{1073905969707775526630744174087466857056981219796622924494658871843237}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{13} - \frac{652884197425836945674885838316988530154798798268382677460829963768879}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{12} + \frac{29930938068491014914487321694406972901702146522108809538097096047667}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{11} - \frac{253862805603173604065634982299055263529918793846173008235566009777851}{875653254061234168487177405683544021007211277173354068153095448407850} a^{10} - \frac{451329953070917702224577032026994100347354313753810968384424601337181}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{9} + \frac{618339512949299395034575877758997913549273433977559182609357402950209}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{8} + \frac{361998923008583302835012901336525698553664601120080450983578619117518}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{7} + \frac{490881378968787968148503664007296114736319357871346117021590356847446}{1313479881091851252730766108525316031510816915760031102229643172611775} a^{6} - \frac{193884178824893320158062508968348770334150789220679638762634985100179}{875653254061234168487177405683544021007211277173354068153095448407850} a^{5} - \frac{965176899855797015847592921744935524458567771977022573142712966729319}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{4} - \frac{149013699906357301245767220828932348746156898380495481095706938101347}{437826627030617084243588702841772010503605638586677034076547724203925} a^{3} - \frac{1274032007186588668277838274043993219028025465706083757767683734293369}{2626959762183702505461532217050632063021633831520062204459286345223550} a^{2} - \frac{646502953214626182589035245280985626109068776297237261074483309511104}{1313479881091851252730766108525316031510816915760031102229643172611775} a + \frac{9412628658738270158949731614832982813564242885018767184014302461731}{175130650812246833697435481136708804201442255434670813630619089681570}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{114}$, which has order $21888$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{56903885588240738313226097213682322763623512441922757}{485851798891454577438471572140111742998088472216820275} a^{35} + \frac{8399574653656639993341457829875776633419277213140347}{64780239852193943658462876285348232399745129628909370} a^{34} - \frac{8307579821660499492481420844950297229797827650601331771}{2526429354235563802680052175128581063590060055527465430} a^{33} + \frac{29535133854701254238684462800802705827595662576356439577}{12632146771177819013400260875642905317950300277637327150} a^{32} - \frac{54962131201469251875554864942283228276411215493400020269}{971703597782909154876943144280223485996176944433640550} a^{31} + \frac{84915776966215102935837239328650467195019934593574080599}{2526429354235563802680052175128581063590060055527465430} a^{30} - \frac{1247494734353817184124591100169563400878317141610588683539}{2105357795196303168900043479273817552991716712939554525} a^{29} + \frac{1748377949645077667012502849356024536746211067661086717258}{6316073385588909506700130437821452658975150138818663575} a^{28} - \frac{28132500502769752782299612962005473749818341542845009710911}{6316073385588909506700130437821452658975150138818663575} a^{27} + \frac{12048738765337851244956334979750971526985054851101445609131}{6316073385588909506700130437821452658975150138818663575} a^{26} - \frac{296019664952231608926113578709483205295323812638620108517003}{12632146771177819013400260875642905317950300277637327150} a^{25} + \frac{58923757390839793421756144561881268946232766738327064310414}{6316073385588909506700130437821452658975150138818663575} a^{24} - \frac{1157886193820950338898124868059307623087483140114807132389421}{12632146771177819013400260875642905317950300277637327150} a^{23} + \frac{18630219293244980777622844378060188747421367417544559123471}{505285870847112760536010435025716212718012011105493086} a^{22} - \frac{3205443330053454142656594821581737871784407271211333911467949}{12632146771177819013400260875642905317950300277637327150} a^{21} + \frac{630002155127189066205418313408162537217324846722180759826502}{6316073385588909506700130437821452658975150138818663575} a^{20} - \frac{2183544596298436976643587928366978063246307269255421284074379}{4210715590392606337800086958547635105983433425879109050} a^{19} + \frac{2329146229840703248926793349329630782826039704755508063067829}{12632146771177819013400260875642905317950300277637327150} a^{18} - \frac{4533366693906838971575013411962785700121388373486158049231572}{6316073385588909506700130437821452658975150138818663575} a^{17} + \frac{95150286524849098115304553892886057338526534022639742776759}{505285870847112760536010435025716212718012011105493086} a^{16} - \frac{889796511397595956208659063217153755260974709134658682350584}{1263214677117781901340026087564290531795030027763732715} a^{15} + \frac{317877060425574773048683371231835887643791763505489827062891}{2105357795196303168900043479273817552991716712939554525} a^{14} - \frac{3055701769479956934563205351949575982508957771258650165845796}{6316073385588909506700130437821452658975150138818663575} a^{13} + \frac{946618854386771546398945287073003777312849405326952464081469}{12632146771177819013400260875642905317950300277637327150} a^{12} - \frac{2977025581971955036161041203487875086605132230666915038206581}{12632146771177819013400260875642905317950300277637327150} a^{11} + \frac{71062135766522708662799705277596730492684224421330524765964}{2105357795196303168900043479273817552991716712939554525} a^{10} - \frac{13010345798417512263527026888238785015707156083118114548951}{168428623615704253512003478341905404239337337035164362} a^{9} + \frac{1505157710338971136524531991255952624565799278944742195013}{168428623615704253512003478341905404239337337035164362} a^{8} - \frac{21470541214027762000222287462812238662719500733343895326738}{1263214677117781901340026087564290531795030027763732715} a^{7} + \frac{103195280466650188364172023779636547783216044784210906467}{38868143911316366195077725771208939439847077777345622} a^{6} - \frac{14070398330403435260745224442513473063708698558455707102149}{6316073385588909506700130437821452658975150138818663575} a^{5} + \frac{293911682431215619839797480080302578815177685949744407353}{842143118078521267560017391709527021196686685175821810} a^{4} - \frac{1293510461326664053046808715537946242260593755549993505776}{6316073385588909506700130437821452658975150138818663575} a^{3} + \frac{90988776481785739773627819420603809535646445677710576299}{2526429354235563802680052175128581063590060055527465430} a^{2} - \frac{7775942081431236721494747286178258347552111617620949452}{1263214677117781901340026087564290531795030027763732715} a + \frac{3992065153836352835578097879007859510864316196161170369}{6316073385588909506700130437821452658975150138818663575} \) (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:  \( 4866030378143.887 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, \(\Q(\sqrt{-3}, \sqrt{13})\), 6.0.771147.1, 6.0.1851523947.1, 6.0.64827.1, 6.0.1851523947.2, 6.0.10024911.1, \(\Q(\zeta_{13})^+\), 6.0.24069811311.2, 6.6.891474493.2, 6.0.142424919.1, 6.6.5274997.1, 6.0.24069811311.1, 6.6.891474493.1, 9.9.567869252041.1, 12.0.100498840557921.1, 12.0.579355816547143538721.1, 12.0.20284857552156561.1, 12.0.579355816547143538721.2, 18.0.6347285018761982937208599123.3, 18.0.13944985186220076513047292273231.1, 18.18.708478645847689707516501157.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$