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Properties

Label 36.0.19865676195...6352.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $108.66$
Ramified primes $2, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3561903327529, -8434770285524, 15548809804884, -16012210942028, 15742746932912, -11556021388336, 9425727731754, -6986616644348, 5602128086021, -4158681286228, 3090474193456, -2150175684852, 1470275270855, -945622774200, 588444243076, -343875076752, 192953738019, -101716601500, 51310501256, -24295546348, 10982223270, -4653122096, 1879297474, -709486284, 254822270, -85116720, 26975312, -7884176, 2177147, -546200, 128546, -26740, 5157, -828, 120, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 + 120*x^34 - 828*x^33 + 5157*x^32 - 26740*x^31 + 128546*x^30 - 546200*x^29 + 2177147*x^28 - 7884176*x^27 + 26975312*x^26 - 85116720*x^25 + 254822270*x^24 - 709486284*x^23 + 1879297474*x^22 - 4653122096*x^21 + 10982223270*x^20 - 24295546348*x^19 + 51310501256*x^18 - 101716601500*x^17 + 192953738019*x^16 - 343875076752*x^15 + 588444243076*x^14 - 945622774200*x^13 + 1470275270855*x^12 - 2150175684852*x^11 + 3090474193456*x^10 - 4158681286228*x^9 + 5602128086021*x^8 - 6986616644348*x^7 + 9425727731754*x^6 - 11556021388336*x^5 + 15742746932912*x^4 - 16012210942028*x^3 + 15548809804884*x^2 - 8434770285524*x + 3561903327529)
 
gp: K = bnfinit(x^36 - 12*x^35 + 120*x^34 - 828*x^33 + 5157*x^32 - 26740*x^31 + 128546*x^30 - 546200*x^29 + 2177147*x^28 - 7884176*x^27 + 26975312*x^26 - 85116720*x^25 + 254822270*x^24 - 709486284*x^23 + 1879297474*x^22 - 4653122096*x^21 + 10982223270*x^20 - 24295546348*x^19 + 51310501256*x^18 - 101716601500*x^17 + 192953738019*x^16 - 343875076752*x^15 + 588444243076*x^14 - 945622774200*x^13 + 1470275270855*x^12 - 2150175684852*x^11 + 3090474193456*x^10 - 4158681286228*x^9 + 5602128086021*x^8 - 6986616644348*x^7 + 9425727731754*x^6 - 11556021388336*x^5 + 15742746932912*x^4 - 16012210942028*x^3 + 15548809804884*x^2 - 8434770285524*x + 3561903327529, 1)
 

Normalized defining polynomial

\( x^{36} - 12 x^{35} + 120 x^{34} - 828 x^{33} + 5157 x^{32} - 26740 x^{31} + 128546 x^{30} - 546200 x^{29} + 2177147 x^{28} - 7884176 x^{27} + 26975312 x^{26} - 85116720 x^{25} + 254822270 x^{24} - 709486284 x^{23} + 1879297474 x^{22} - 4653122096 x^{21} + 10982223270 x^{20} - 24295546348 x^{19} + 51310501256 x^{18} - 101716601500 x^{17} + 192953738019 x^{16} - 343875076752 x^{15} + 588444243076 x^{14} - 945622774200 x^{13} + 1470275270855 x^{12} - 2150175684852 x^{11} + 3090474193456 x^{10} - 4158681286228 x^{9} + 5602128086021 x^{8} - 6986616644348 x^{7} + 9425727731754 x^{6} - 11556021388336 x^{5} + 15742746932912 x^{4} - 16012210942028 x^{3} + 15548809804884 x^{2} - 8434770285524 x + 3561903327529 \)

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:  \(19865676195397711398725987827960270735894630080373118478444419023321956352=2^{54}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.66$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(645,·)$, $\chi_{728}(641,·)$, $\chi_{728}(9,·)$, $\chi_{728}(141,·)$, $\chi_{728}(529,·)$, $\chi_{728}(149,·)$, $\chi_{728}(25,·)$, $\chi_{728}(541,·)$, $\chi_{728}(197,·)$, $\chi_{728}(289,·)$, $\chi_{728}(37,·)$, $\chi_{728}(113,·)$, $\chi_{728}(557,·)$, $\chi_{728}(669,·)$, $\chi_{728}(393,·)$, $\chi_{728}(569,·)$, $\chi_{728}(317,·)$, $\chi_{728}(709,·)$, $\chi_{728}(417,·)$, $\chi_{728}(333,·)$, $\chi_{728}(81,·)$, $\chi_{728}(85,·)$, $\chi_{728}(121,·)$, $\chi_{728}(93,·)$, $\chi_{728}(421,·)$, $\chi_{728}(225,·)$, $\chi_{728}(613,·)$, $\chi_{728}(337,·)$, $\chi_{728}(361,·)$, $\chi_{728}(109,·)$, $\chi_{728}(625,·)$, $\chi_{728}(501,·)$, $\chi_{728}(233,·)$, $\chi_{728}(673,·)$, $\chi_{728}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{4} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{3}{8} a$, $\frac{1}{8} a^{20} - \frac{1}{8} a^{14} + \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{3}{8} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{15} + \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{3}{8} a$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{3}{8} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{3}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{24} + \frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{3}{16}$, $\frac{1}{16} a^{25} + \frac{1}{16} a^{17} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} + \frac{1}{8} a^{3} + \frac{3}{16} a$, $\frac{1}{16} a^{26} - \frac{1}{16} a^{18} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{3}{16} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{27} - \frac{1}{16} a^{19} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{3}{16} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{28} - \frac{1}{16} a^{20} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{3}{16} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{29} - \frac{1}{16} a^{21} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{30} - \frac{1}{32} a^{28} - \frac{1}{32} a^{24} + \frac{1}{32} a^{22} + \frac{1}{32} a^{20} + \frac{3}{32} a^{16} - \frac{1}{16} a^{14} - \frac{1}{8} a^{12} - \frac{1}{16} a^{8} - \frac{3}{32} a^{6} + \frac{7}{32} a^{4} - \frac{7}{16} a^{2} - \frac{15}{32}$, $\frac{1}{32} a^{31} - \frac{1}{32} a^{29} - \frac{1}{32} a^{25} + \frac{1}{32} a^{23} + \frac{1}{32} a^{21} + \frac{3}{32} a^{17} - \frac{1}{16} a^{15} - \frac{1}{8} a^{13} - \frac{1}{16} a^{9} - \frac{3}{32} a^{7} + \frac{7}{32} a^{5} - \frac{7}{16} a^{3} - \frac{15}{32} a$, $\frac{1}{32} a^{32} - \frac{1}{32} a^{28} - \frac{1}{32} a^{26} - \frac{1}{16} a^{22} + \frac{1}{32} a^{20} - \frac{1}{32} a^{18} + \frac{1}{32} a^{16} - \frac{1}{16} a^{14} - \frac{1}{16} a^{10} + \frac{7}{32} a^{8} + \frac{1}{8} a^{6} + \frac{1}{32} a^{4} - \frac{1}{32} a^{2} - \frac{3}{32}$, $\frac{1}{32} a^{33} - \frac{1}{32} a^{29} - \frac{1}{32} a^{27} - \frac{1}{16} a^{23} + \frac{1}{32} a^{21} - \frac{1}{32} a^{19} + \frac{1}{32} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{11} + \frac{7}{32} a^{9} + \frac{1}{8} a^{7} + \frac{1}{32} a^{5} - \frac{1}{32} a^{3} - \frac{3}{32} a$, $\frac{1}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{34} + \frac{8553352289911052496796930274588366065473766609485495446228115784593085123666978805129585}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{33} - \frac{137611807150842850192426736315607508017383285740680791470628743591302540789035002141095}{758243712227281600125678863613942869254047804428131594244599302603534345661033851526271816} a^{32} - \frac{9641123877989984383530783655635954443067281226838417440976230478490312233627681440184005}{758243712227281600125678863613942869254047804428131594244599302603534345661033851526271816} a^{31} + \frac{2554942882425539934834073427707236073692965862410323575037519789338220069613072339913817}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{30} - \frac{40156568526856124485917344747276829086559121658310036739768084630160640462758515092136123}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{29} + \frac{480949596485977206644733329443214315019863929732383845789895275752093365774454461177441}{16756767121044897240346494223512549596774537114433847386620979063061532500796328210525344} a^{28} - \frac{32749539408741713895330113374527219951550483812061536402618587834481839419014895254395383}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{27} - \frac{31788466154594518749048208351775224608109399454385245021670902297884384075023489405734553}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{26} + \frac{41520871112694584366409633423225338330151416840874054102584435571178478084976447725962267}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{25} - \frac{1535918673457541295873191168464556129100525551530896991687309421852989234229231802792871}{379121856113640800062839431806971434627023902214065797122299651301767172830516925763135908} a^{24} - \frac{28475972316099805523272818819706794140044620249891237320788833600584376871557509253605421}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{23} - \frac{187359531997465529739258691254366329568363280246486081296760136970196696786872925708387921}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{22} - \frac{148307088697542755995200649272898234230475440822788503327454489656574757788644956930169465}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{21} + \frac{138581045953364842507245553952416639594999660924749938540583461027766428199533594336916321}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{20} - \frac{66304092123881490455649343919098162783369860299813983806596856664089864361687424782934239}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{19} + \frac{102122695761881348729219147289445412568514139872083421690316000203364526443138051037447203}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{18} - \frac{1872374439029857828909585842509975645152365969613773048729580467570727220284535146331717}{16756767121044897240346494223512549596774537114433847386620979063061532500796328210525344} a^{17} - \frac{8564187039491201458024166664509917078559527653784131404201631896661006052888785149749777}{379121856113640800062839431806971434627023902214065797122299651301767172830516925763135908} a^{16} + \frac{176614334930571190630469940423718194627612012650045978367192299154667932021985648338567975}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{15} - \frac{89126402075339537851943937679439340657427209707038977137072151237957064376724502475260447}{758243712227281600125678863613942869254047804428131594244599302603534345661033851526271816} a^{14} - \frac{11280556630263677532819043513470150712311384352040668044900035462482633770415784894160363}{94780464028410200015709857951742858656755975553516449280574912825441793207629231440783977} a^{13} - \frac{179135557798470205501843320541621832412492428506442096293861250304767108205691786364312261}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{12} - \frac{277086939615565675283361136478057403791424757417422152269844708289729475171937265619209757}{1516487424454563200251357727227885738508095608856263188489198605207068691322067703052543632} a^{11} + \frac{333532146529496481591438990518029364049260397609301313524470552191796991488269597100603563}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{10} - \frac{162130796333961250623810722693827467509082252889972031324947110853234812610010911426588501}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{9} + \frac{45805072310886823200424061510927876518230949739971922759659222587553624199094990158509377}{379121856113640800062839431806971434627023902214065797122299651301767172830516925763135908} a^{8} - \frac{23451512333491588234097391086814626350709028941512899487315889259187538196572425324581215}{189560928056820400031419715903485717313511951107032898561149825650883586415258462881567954} a^{7} - \frac{746394530724115355284484364191414450971865462551673533088345689559858709427431812910320369}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{6} + \frac{1267805212183705860263969611876980264847732127142388490247330163557488116111101648052362187}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{5} + \frac{482265130740807182218447339515786611035119279808502239113591527535074853078212701410957545}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{4} + \frac{162818157991333028339922654529960651973981012306610732917849392671393445156128864032411469}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{3} - \frac{677864109278449686964070215454534771763524665392769688391124565247377950420929552098690041}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a^{2} - \frac{355845753411920789575185738727971895690161471079645127394727293522260795647398699887303149}{3032974848909126400502715454455771477016191217712526376978397210414137382644135406105087264} a - \frac{475457260823513548811863093141106779878599072128201205276511913207689704224957846464405}{8378383560522448620173247111756274798387268557216923693310489531530766250398164105262672}$, $\frac{1}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{35} - \frac{5077776800089233194591}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{34} + \frac{217690062799101404161573219820458747184051122703435152864945194988991324452049370650414511884982154887389669957}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{33} - \frac{544998359805158600028556385050832556611748081682837079518020696472047460761847489578529034824444757270376376753}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{32} - \frac{117426374127266312957150479413168590792525973102691511476060536073620406199494744520457677925623719316258863641}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{31} + \frac{589213386736547053809548100925579064576220102027169941540918144402901372382968501727163706247396652587633507439}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{30} + \frac{1056646265187717759475611909145178415436723661084342229612582324710509121255028910124496636402924117077899258121}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{29} + \frac{1086159096721427632381744295283259382630674299192873782957540382439506966076841232450150975104455359837790593117}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{28} + \frac{60950184751318416116802839313710669807348679605669696402694499620850609901012570646597730955771901333735206857}{4560189574512852648855678021160046577732196390040505512502128474967616249501809689933523738825500161470986960369} a^{27} + \frac{208503928136709613041032925432249141385816338850962883666456622914656594558249347086132890805253506078469547281}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{26} + \frac{1356004144953223325325537125933289841744391530669918640369612951632358381781422646539822817651706062092505834421}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{25} + \frac{862414861475065380496523615930802504957231640117028547719774806653962933961726368080198588179234242489431286211}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{24} + \frac{583694293108557140907000604176943211079713193673694477763756597205326428185747358312592360317006718772069903609}{18240758298051410595422712084640186310928785560162022050008513899870464998007238759734094955302000645883947841476} a^{23} + \frac{1314836722563465747610627595955466633791625996908318613537703450066209331565244613660534943668537815448374828087}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{22} - \frac{717198104540552070139865825022387715905586565466451821019432939778053131820649216990617002191254972491616756197}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{21} - \frac{2698240193989557964163852950424205889520057501484632686475434804188375303922618849063835757204764098561962845665}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{20} + \frac{6701139573970718398704285214210142861000015640189338536585042968300031222629535545426623252006610398645511066101}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{19} + \frac{2002354349840967452326931832797220333176294402813381310902981194661033254351869792593550424007367215973013547303}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{18} - \frac{18015329604453494692978888374778025416057197953010998205154259591178180777313262123790548669311669900258687051489}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{17} + \frac{1607774315842293500463003115379317764819901734343973749294710898958105159191945887641545765453375427000592479161}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{16} - \frac{8386743118562575632052268042140830361110930990053125796841829710846616167926618922862032748033864117850260889487}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{15} + \frac{809488308587085203456355461037151442201573382253620523343970626730529561577165985494320580682747744835776342171}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{14} - \frac{1697609366346207012710285980644594869300680628962416051472718281918479108225757334519326198401872871108660360847}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{13} - \frac{2022096153951343402056686548246954826513872228720038785197018986949290953218806262620725819489986334037383608471}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{12} + \frac{11441429813098847812807133913668128280939188061236106360659169681610272241570222155084004641697234461370996724891}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{11} - \frac{888103856265042126742838108578942145477842080596889168867773002146639827362004797331953216352965411145961154136}{4560189574512852648855678021160046577732196390040505512502128474967616249501809689933523738825500161470986960369} a^{10} + \frac{916070299008838941054264198961850300732546366139970768524293589048758414601371887602345115693911893598736567969}{36481516596102821190845424169280372621857571120324044100017027799740929996014477519468189910604001291767895682952} a^{9} - \frac{3285963577918578556698559356393479546024314807968494733015040541147981015903284410454653329899209020557276446251}{18240758298051410595422712084640186310928785560162022050008513899870464998007238759734094955302000645883947841476} a^{8} + \frac{607103045813835392899251672860662165401561115987585014131133531834232710884762838812243543811340740221361494269}{9120379149025705297711356042320093155464392780081011025004256949935232499003619379867047477651000322941973920738} a^{7} - \frac{16235626033947217577375394692046701707103228278304090069098910644335738659633553807576863825367571653129121200767}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{6} + \frac{8435685393200696456760102860837145016548310173411271961014787614357414390939654265284872496272906932337048168733}{18240758298051410595422712084640186310928785560162022050008513899870464998007238759734094955302000645883947841476} a^{5} - \frac{12602463289583526368257366002084280160882861870646540238375296241581209795900216216944658954055381760041101713339}{72963033192205642381690848338560745243715142240648088200034055599481859992028955038936379821208002583535791365904} a^{4} + \frac{67997188500064522076562977392169634053768385845220714678352084394920173762220514593903923740337500747943307535567}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a^{3} - \frac{2260015738573357263446956018998480392229091437986028745493943889497535635431553598100029268197838933317711429267}{9120379149025705297711356042320093155464392780081011025004256949935232499003619379867047477651000322941973920738} a^{2} + \frac{41223702256191616137997662330185191315782544035365979592205782128526981433901815748482970867387543428335246372179}{145926066384411284763381696677121490487430284481296176400068111198963719984057910077872759642416005167071582731808} a - \frac{2364095470348182484254269739205014430077042946428115594281524893337061475178612295375203583234949929533539555}{201555340309960338070969194305416423325179950941016818232138275136690220972455676903139170776817686694850252392}$

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:  \( -1 \) (order $2$)
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

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 4.0.140608.2, \(\Q(\zeta_{13})^+\), 6.6.891474493.2, 6.6.5274997.1, 6.6.891474493.1, 9.9.567869252041.1, 12.0.469804094334435328.1, 12.0.2708327112823247113289728.1, 12.0.16025604217889036173312.1, 12.0.2708327112823247113289728.2, 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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$2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$