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Properties

Label 36.0.51945655264...9849.1
Degree $36$
Signature $[0, 18]$
Discriminant $7^{30}\cdot 19^{30}$
Root discriminant $58.87$
Ramified primes $7, 19$
Class number $1404$ (GRH)
Class group $[3, 6, 78]$ (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![117649, -84035, 76832, 67571, -166943, 223909, -133314, 391394, -15647, -135047, 571486, -570148, 302152, 340761, 333971, 255997, 167776, 239337, -41684, 86177, 73801, 44304, 31136, 15575, 16514, -9499, 1176, 1449, -626, 128, 137, -20, -16, 11, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - x^34 + 11*x^33 - 16*x^32 - 20*x^31 + 137*x^30 + 128*x^29 - 626*x^28 + 1449*x^27 + 1176*x^26 - 9499*x^25 + 16514*x^24 + 15575*x^23 + 31136*x^22 + 44304*x^21 + 73801*x^20 + 86177*x^19 - 41684*x^18 + 239337*x^17 + 167776*x^16 + 255997*x^15 + 333971*x^14 + 340761*x^13 + 302152*x^12 - 570148*x^11 + 571486*x^10 - 135047*x^9 - 15647*x^8 + 391394*x^7 - 133314*x^6 + 223909*x^5 - 166943*x^4 + 67571*x^3 + 76832*x^2 - 84035*x + 117649)
 
gp: K = bnfinit(x^36 - x^35 - x^34 + 11*x^33 - 16*x^32 - 20*x^31 + 137*x^30 + 128*x^29 - 626*x^28 + 1449*x^27 + 1176*x^26 - 9499*x^25 + 16514*x^24 + 15575*x^23 + 31136*x^22 + 44304*x^21 + 73801*x^20 + 86177*x^19 - 41684*x^18 + 239337*x^17 + 167776*x^16 + 255997*x^15 + 333971*x^14 + 340761*x^13 + 302152*x^12 - 570148*x^11 + 571486*x^10 - 135047*x^9 - 15647*x^8 + 391394*x^7 - 133314*x^6 + 223909*x^5 - 166943*x^4 + 67571*x^3 + 76832*x^2 - 84035*x + 117649, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - x^{34} + 11 x^{33} - 16 x^{32} - 20 x^{31} + 137 x^{30} + 128 x^{29} - 626 x^{28} + 1449 x^{27} + 1176 x^{26} - 9499 x^{25} + 16514 x^{24} + 15575 x^{23} + 31136 x^{22} + 44304 x^{21} + 73801 x^{20} + 86177 x^{19} - 41684 x^{18} + 239337 x^{17} + 167776 x^{16} + 255997 x^{15} + 333971 x^{14} + 340761 x^{13} + 302152 x^{12} - 570148 x^{11} + 571486 x^{10} - 135047 x^{9} - 15647 x^{8} + 391394 x^{7} - 133314 x^{6} + 223909 x^{5} - 166943 x^{4} + 67571 x^{3} + 76832 x^{2} - 84035 x + 117649 \)

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:  \(5194565526429829568692289293990783055042612231547038493482119849=7^{30}\cdot 19^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.87$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(1,·)$, $\chi_{133}(132,·)$, $\chi_{133}(8,·)$, $\chi_{133}(11,·)$, $\chi_{133}(12,·)$, $\chi_{133}(18,·)$, $\chi_{133}(20,·)$, $\chi_{133}(26,·)$, $\chi_{133}(27,·)$, $\chi_{133}(30,·)$, $\chi_{133}(31,·)$, $\chi_{133}(37,·)$, $\chi_{133}(39,·)$, $\chi_{133}(45,·)$, $\chi_{133}(46,·)$, $\chi_{133}(50,·)$, $\chi_{133}(58,·)$, $\chi_{133}(64,·)$, $\chi_{133}(65,·)$, $\chi_{133}(68,·)$, $\chi_{133}(69,·)$, $\chi_{133}(75,·)$, $\chi_{133}(83,·)$, $\chi_{133}(87,·)$, $\chi_{133}(88,·)$, $\chi_{133}(94,·)$, $\chi_{133}(96,·)$, $\chi_{133}(102,·)$, $\chi_{133}(103,·)$, $\chi_{133}(106,·)$, $\chi_{133}(107,·)$, $\chi_{133}(113,·)$, $\chi_{133}(115,·)$, $\chi_{133}(121,·)$, $\chi_{133}(122,·)$, $\chi_{133}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{7} a^{25} + \frac{3}{7} a^{24} + \frac{2}{7} a^{23} + \frac{1}{7} a^{19} + \frac{1}{7} a^{16} + \frac{1}{7} a^{13} + \frac{1}{7} a^{10} + \frac{1}{7} a^{7} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{26} + \frac{1}{7} a^{23} + \frac{1}{7} a^{20} - \frac{3}{7} a^{19} + \frac{1}{7} a^{17} - \frac{3}{7} a^{16} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{4} - \frac{3}{7} a$, $\frac{1}{49} a^{27} + \frac{15}{49} a^{24} - \frac{2}{7} a^{23} - \frac{3}{7} a^{22} + \frac{15}{49} a^{21} - \frac{3}{49} a^{20} - \frac{3}{7} a^{19} - \frac{20}{49} a^{18} - \frac{10}{49} a^{17} - \frac{6}{49} a^{15} - \frac{17}{49} a^{14} + \frac{3}{7} a^{13} + \frac{8}{49} a^{12} - \frac{24}{49} a^{11} - \frac{1}{7} a^{10} + \frac{22}{49} a^{9} + \frac{18}{49} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{11}{49} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{18}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{49} a^{28} + \frac{1}{49} a^{25} - \frac{1}{7} a^{24} + \frac{15}{49} a^{22} - \frac{3}{49} a^{21} - \frac{3}{7} a^{20} + \frac{15}{49} a^{19} - \frac{10}{49} a^{18} - \frac{20}{49} a^{16} - \frac{17}{49} a^{15} + \frac{3}{7} a^{14} - \frac{6}{49} a^{13} - \frac{24}{49} a^{12} - \frac{1}{7} a^{11} + \frac{8}{49} a^{10} + \frac{18}{49} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{11}{49} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{11}{49} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{49} a^{29} + \frac{1}{49} a^{26} + \frac{3}{7} a^{24} - \frac{20}{49} a^{23} - \frac{3}{49} a^{22} - \frac{3}{7} a^{21} + \frac{15}{49} a^{20} - \frac{3}{49} a^{19} - \frac{20}{49} a^{17} - \frac{10}{49} a^{16} + \frac{3}{7} a^{15} - \frac{6}{49} a^{14} - \frac{17}{49} a^{13} - \frac{1}{7} a^{12} + \frac{8}{49} a^{11} - \frac{24}{49} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{18}{49} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{11}{49} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{539} a^{30} + \frac{2}{539} a^{29} - \frac{4}{539} a^{28} - \frac{3}{539} a^{27} - \frac{5}{539} a^{26} + \frac{31}{539} a^{25} - \frac{6}{49} a^{24} - \frac{64}{539} a^{23} - \frac{150}{539} a^{22} - \frac{173}{539} a^{21} + \frac{116}{539} a^{20} + \frac{102}{539} a^{19} - \frac{96}{539} a^{18} + \frac{228}{539} a^{17} + \frac{18}{539} a^{16} - \frac{24}{49} a^{15} + \frac{95}{539} a^{14} + \frac{32}{539} a^{13} - \frac{89}{539} a^{12} - \frac{87}{539} a^{11} + \frac{193}{539} a^{10} - \frac{13}{539} a^{9} + \frac{268}{539} a^{8} - \frac{181}{539} a^{7} + \frac{24}{49} a^{6} - \frac{250}{539} a^{5} - \frac{104}{539} a^{4} + \frac{208}{539} a^{3} + \frac{68}{539} a^{2} + \frac{34}{77} a + \frac{1}{11}$, $\frac{1}{3519153574206283559307839} a^{31} + \frac{2943166106889745624015}{3519153574206283559307839} a^{30} - \frac{32617909136351887506372}{3519153574206283559307839} a^{29} + \frac{26731576922572396258350}{3519153574206283559307839} a^{28} - \frac{1395698641834010833028}{502736224886611937043977} a^{27} - \frac{14415527543882392801626}{3519153574206283559307839} a^{26} - \frac{230442579949187418561902}{3519153574206283559307839} a^{25} - \frac{1684170638053428142225793}{3519153574206283559307839} a^{24} + \frac{1132565576925469677864437}{3519153574206283559307839} a^{23} - \frac{1098943186699742785167117}{3519153574206283559307839} a^{22} - \frac{652914483533396394872145}{3519153574206283559307839} a^{21} - \frac{33630417122554467455031}{319923052200571232664349} a^{20} - \frac{1335724663702684498988914}{3519153574206283559307839} a^{19} + \frac{1161506760197743249336575}{3519153574206283559307839} a^{18} + \frac{1579279659922680783165486}{3519153574206283559307839} a^{17} - \frac{1470896602209084514721552}{3519153574206283559307839} a^{16} - \frac{1501983413047697374793981}{3519153574206283559307839} a^{15} - \frac{645542499655096066583207}{3519153574206283559307839} a^{14} + \frac{523308587136854287111601}{3519153574206283559307839} a^{13} - \frac{592704521764663853739330}{3519153574206283559307839} a^{12} - \frac{23931813508363896233694}{319923052200571232664349} a^{11} - \frac{814834888177202939890398}{3519153574206283559307839} a^{10} - \frac{1584520801269024156705613}{3519153574206283559307839} a^{9} + \frac{31851505216778913136987}{502736224886611937043977} a^{8} - \frac{1016331914917407638018219}{3519153574206283559307839} a^{7} - \frac{345282452023577532213032}{3519153574206283559307839} a^{6} + \frac{166921266079157957105005}{502736224886611937043977} a^{5} - \frac{1366520901967260585884375}{3519153574206283559307839} a^{4} - \frac{182764653085954422979718}{502736224886611937043977} a^{3} - \frac{721179616756034250050692}{3519153574206283559307839} a^{2} - \frac{79519214237894278624666}{502736224886611937043977} a + \frac{21408151155434053913732}{71819460698087419577711}$, $\frac{1}{3519153574206283559307839} a^{32} - \frac{292862483312617584508}{319923052200571232664349} a^{30} + \frac{13556780463721743928723}{3519153574206283559307839} a^{29} - \frac{7113805830844157069546}{3519153574206283559307839} a^{28} + \frac{3721115295711105907586}{502736224886611937043977} a^{27} - \frac{122045821109216527946954}{3519153574206283559307839} a^{26} + \frac{234603150005354152506602}{3519153574206283559307839} a^{25} - \frac{17743431588034388384590}{45703293171510176094907} a^{24} + \frac{74755280501271972111742}{319923052200571232664349} a^{23} - \frac{368396109073459498685347}{3519153574206283559307839} a^{22} - \frac{986215591503263851987242}{3519153574206283559307839} a^{21} + \frac{870976403374219856456590}{3519153574206283559307839} a^{20} + \frac{287342226217663354588822}{3519153574206283559307839} a^{19} - \frac{874859108614900711613835}{3519153574206283559307839} a^{18} - \frac{63950196620159902168759}{3519153574206283559307839} a^{17} - \frac{1563031861784048769660891}{3519153574206283559307839} a^{16} - \frac{383826256425768120836535}{3519153574206283559307839} a^{15} + \frac{1628190492122397081669848}{3519153574206283559307839} a^{14} - \frac{508382387993700145581727}{3519153574206283559307839} a^{13} - \frac{937875766036183261970500}{3519153574206283559307839} a^{12} - \frac{107352064371713566304013}{502736224886611937043977} a^{11} - \frac{4450950357160138452510}{3519153574206283559307839} a^{10} + \frac{1734734546187279876954290}{3519153574206283559307839} a^{9} - \frac{130200113847888516081523}{502736224886611937043977} a^{8} + \frac{1383761623031917556530660}{3519153574206283559307839} a^{7} - \frac{6452406809019466773287}{319923052200571232664349} a^{6} + \frac{9923367036898873467974}{45703293171510176094907} a^{5} - \frac{16352928164034588590590}{71819460698087419577711} a^{4} - \frac{430442130153475356981630}{3519153574206283559307839} a^{3} - \frac{47079023797549325215385}{3519153574206283559307839} a^{2} + \frac{73303976939460103576184}{502736224886611937043977} a - \frac{19370492831543265232769}{71819460698087419577711}$, $\frac{1}{24634075019443984915154873} a^{33} - \frac{1}{24634075019443984915154873} a^{32} - \frac{1}{24634075019443984915154873} a^{31} + \frac{17149056234357301475120}{24634075019443984915154873} a^{30} - \frac{176536436329212442105471}{24634075019443984915154873} a^{29} + \frac{142238323860497839155217}{24634075019443984915154873} a^{28} - \frac{154709226213826480187295}{24634075019443984915154873} a^{27} - \frac{1117552301175318522013085}{24634075019443984915154873} a^{26} - \frac{1719294172106940250203469}{24634075019443984915154873} a^{25} + \frac{14635500343820118968596}{3519153574206283559307839} a^{24} + \frac{233662212808302052528763}{3519153574206283559307839} a^{23} - \frac{158821895463093705576988}{502736224886611937043977} a^{22} + \frac{10282021834895727221284127}{24634075019443984915154873} a^{21} + \frac{172295177359720403259154}{3519153574206283559307839} a^{20} + \frac{4072234155984189944466}{319923052200571232664349} a^{19} - \frac{9541126060382518676652031}{24634075019443984915154873} a^{18} - \frac{1005470658208703363726773}{3519153574206283559307839} a^{17} + \frac{742617743986994279611442}{3519153574206283559307839} a^{16} - \frac{2629101799268601625851840}{24634075019443984915154873} a^{15} - \frac{1389011188423926210987572}{3519153574206283559307839} a^{14} - \frac{499431858946197402379498}{3519153574206283559307839} a^{13} + \frac{1653458072504062747030637}{3519153574206283559307839} a^{12} + \frac{12182777437723474744618667}{24634075019443984915154873} a^{11} + \frac{206462431456485170518687}{2239461365403998628650443} a^{10} + \frac{12061941797137909813595876}{24634075019443984915154873} a^{9} - \frac{9034984278567690485586493}{24634075019443984915154873} a^{8} + \frac{6497775660351502094335844}{24634075019443984915154873} a^{7} - \frac{6457707477077357214085991}{24634075019443984915154873} a^{6} + \frac{7137091407934611233822794}{24634075019443984915154873} a^{5} + \frac{1174193161162402677037857}{24634075019443984915154873} a^{4} + \frac{2184522331556326189653135}{24634075019443984915154873} a^{3} + \frac{1552410966865658115451793}{3519153574206283559307839} a^{2} + \frac{32854210687555905764477}{71819460698087419577711} a - \frac{14092946613613506617463}{71819460698087419577711}$, $\frac{1}{172438525136107894406084111} a^{34} - \frac{1}{172438525136107894406084111} a^{33} - \frac{1}{172438525136107894406084111} a^{32} + \frac{1}{15676229557827990400553101} a^{31} + \frac{46816176387715164356071}{172438525136107894406084111} a^{30} - \frac{421288354235312932659318}{172438525136107894406084111} a^{29} + \frac{327656001459882603947261}{172438525136107894406084111} a^{28} - \frac{714137226702790260477252}{172438525136107894406084111} a^{27} - \frac{3264665433711763244335791}{172438525136107894406084111} a^{26} + \frac{448523584059461754731542}{24634075019443984915154873} a^{25} - \frac{1512638215408664112387357}{3519153574206283559307839} a^{24} - \frac{5815500074065490487316042}{24634075019443984915154873} a^{23} + \frac{3831219273899237341196777}{15676229557827990400553101} a^{22} - \frac{1700437594850401777170151}{24634075019443984915154873} a^{21} - \frac{8645597870045357397978974}{24634075019443984915154873} a^{20} + \frac{39628713614675083209059477}{172438525136107894406084111} a^{19} - \frac{341841590778813213238625}{24634075019443984915154873} a^{18} + \frac{9700727004510190358394746}{24634075019443984915154873} a^{17} + \frac{63634268497838056276351983}{172438525136107894406084111} a^{16} - \frac{10427401240735319259749859}{24634075019443984915154873} a^{15} + \frac{455259588183787932128905}{3519153574206283559307839} a^{14} + \frac{9532990186777340665179717}{24634075019443984915154873} a^{13} + \frac{80209371674160127066850351}{172438525136107894406084111} a^{12} - \frac{39515624435795398715766556}{172438525136107894406084111} a^{11} + \frac{9894165036665821851059313}{172438525136107894406084111} a^{10} + \frac{55984815795618422981852467}{172438525136107894406084111} a^{9} + \frac{1964936344166372535818235}{15676229557827990400553101} a^{8} + \frac{72521537774221270743449207}{172438525136107894406084111} a^{7} - \frac{1978614749439883954748745}{15676229557827990400553101} a^{6} - \frac{12117603627321718907903967}{172438525136107894406084111} a^{5} - \frac{33107400838162943365567311}{172438525136107894406084111} a^{4} + \frac{6714091269213435932525950}{24634075019443984915154873} a^{3} + \frac{386136204862017153705955}{3519153574206283559307839} a^{2} - \frac{4182800084540425420673}{502736224886611937043977} a + \frac{79544168167245827455}{6529041881644310870701}$, $\frac{1}{1207069675952755260842588777} a^{35} - \frac{1}{1207069675952755260842588777} a^{34} - \frac{1}{1207069675952755260842588777} a^{33} + \frac{1}{109733606904795932803871707} a^{32} - \frac{16}{1207069675952755260842588777} a^{31} - \frac{271127271532343173266542}{1207069675952755260842588777} a^{30} - \frac{4921693864081368263683076}{1207069675952755260842588777} a^{29} + \frac{7379514636537083332826676}{1207069675952755260842588777} a^{28} + \frac{294854919366907808407076}{1207069675952755260842588777} a^{27} + \frac{9781687531513651545568970}{172438525136107894406084111} a^{26} + \frac{225359429510033117237165}{24634075019443984915154873} a^{25} + \frac{3027734116852449789103195}{15676229557827990400553101} a^{24} + \frac{249176075076001224363660386}{1207069675952755260842588777} a^{23} - \frac{58676544643334468730462535}{172438525136107894406084111} a^{22} + \frac{51426142904419879379906328}{172438525136107894406084111} a^{21} - \frac{260622348898245725903802212}{1207069675952755260842588777} a^{20} - \frac{15403413195575281989298500}{172438525136107894406084111} a^{19} - \frac{77612658522364242945787793}{172438525136107894406084111} a^{18} - \frac{484174334339617273673729715}{1207069675952755260842588777} a^{17} - \frac{4181281185130800334475063}{172438525136107894406084111} a^{16} - \frac{6267088661923018835585371}{24634075019443984915154873} a^{15} + \frac{26518723417738953831671419}{172438525136107894406084111} a^{14} - \frac{439781188754987385288233894}{1207069675952755260842588777} a^{13} + \frac{16839477142961048721684296}{109733606904795932803871707} a^{12} + \frac{579575773942434610320236919}{1207069675952755260842588777} a^{11} - \frac{128203865055244734888591114}{1207069675952755260842588777} a^{10} - \frac{563405500771877502847931355}{1207069675952755260842588777} a^{9} - \frac{566411581184844995189567904}{1207069675952755260842588777} a^{8} + \frac{119713546731447905359488461}{1207069675952755260842588777} a^{7} - \frac{574438809595664910767737332}{1207069675952755260842588777} a^{6} + \frac{150619830133379939207930034}{1207069675952755260842588777} a^{5} + \frac{35870908877189108598309858}{172438525136107894406084111} a^{4} + \frac{3996259850233498506982913}{24634075019443984915154873} a^{3} - \frac{59137162198177109660969}{502736224886611937043977} a^{2} - \frac{19664427789830662246244}{45703293171510176094907} a + \frac{12074924504285291362777}{71819460698087419577711}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{78}$, which has order $1404$ (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{425328862431808030}{319923052200571232664349} a^{35} - \frac{2126644312159040150}{319923052200571232664349} a^{34} + \frac{1063322156079520075}{319923052200571232664349} a^{33} + \frac{7655919523772544540}{319923052200571232664349} a^{32} - \frac{24881738452260769755}{319923052200571232664349} a^{31} + \frac{13785322333323008751}{319923052200571232664349} a^{30} + \frac{105906886745520199470}{319923052200571232664349} a^{29} - \frac{175022826890689004345}{319923052200571232664349} a^{28} - \frac{79749161705964005625}{45703293171510176094907} a^{27} + \frac{252858008715709873835}{45703293171510176094907} a^{26} - \frac{1491840984979566665225}{319923052200571232664349} a^{25} - \frac{6824188933287143937335}{319923052200571232664349} a^{24} + \frac{23531098259755705808458}{319923052200571232664349} a^{23} - \frac{16043404690927798891600}{319923052200571232664349} a^{22} - \frac{25116732648754343691575}{319923052200571232664349} a^{21} - \frac{4860232911008270358810}{45703293171510176094907} a^{20} - \frac{9462078538089217339395}{319923052200571232664349} a^{19} - \frac{42033762823117076276795}{319923052200571232664349} a^{18} - \frac{12368350655085761608385}{45703293171510176094907} a^{17} + \frac{292902959130880183113205}{319923052200571232664349} a^{16} - \frac{125465209155584459921520}{319923052200571232664349} a^{15} - \frac{143491497003169347848980}{319923052200571232664349} a^{14} - \frac{136724727466310497995695}{319923052200571232664349} a^{13} + \frac{56993216908137412403940}{319923052200571232664349} a^{12} - \frac{284545008966879572070}{319923052200571232664349} a^{11} - \frac{92815051695437932402585}{319923052200571232664349} a^{10} + \frac{1962353479838978777729751}{319923052200571232664349} a^{9} - \frac{79908022036082285924205}{319923052200571232664349} a^{8} - \frac{54884436408200508191200}{319923052200571232664349} a^{7} - \frac{19266334146004824238925}{319923052200571232664349} a^{6} - \frac{1730663141235026874070}{45703293171510176094907} a^{5} + \frac{431283466505853342420}{6529041881644310870701} a^{4} - \frac{626722078793269132205}{6529041881644310870701} a^{3} + \frac{343410507672935838335946}{319923052200571232664349} a^{2} - \frac{145887799814110154290}{6529041881644310870701} a - \frac{510607299349385540015}{6529041881644310870701} \) (order $14$)
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:  \( 1017338455548356.9 \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{-7}) \), 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.361.1, \(\Q(\sqrt{-7}, \sqrt{-19})\), 6.0.5945113699.1, 6.0.5945113699.2, 6.0.16468459.1, 6.0.2476099.1, 6.6.41615795893.2, 6.0.2190305047.1, 6.6.41615795893.1, 6.0.2190305047.2, 6.6.115279213.1, \(\Q(\zeta_{7})\), 6.6.849301957.1, 6.0.44700103.1, 9.9.5534900853769.1, 12.0.1731874467807835667449.1, 12.0.1731874467807835667449.2, 12.0.13289296949899369.1, 12.0.721313814164029849.1, 18.0.210126339255361190328405271099.2, 18.18.72073334364588888282643007986957.1, 18.0.10507848719141112156676338823.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed