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Properties

Label 36.36.4949164676...3125.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{18}\cdot 7^{30}\cdot 13^{33}$
Root discriminant $118.81$
Ramified primes $5, 7, 13$
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![7203054951, -84148737831, -532421593101, 66623171786, 3510763848149, 1798484654352, -9516669187523, -5774232219857, 14716307928191, 8066810405369, -14568115911700, -6544984186949, 9777925862459, 3471389732603, -4628193618151, -1283637936946, 1590272002452, 343756885869, -404867699989, -68170119347, 77372149977, 10123730642, -11167777905, -1128384881, 1216750731, 93783066, -99258341, -5715774, 5956273, 247577, -254655, -7203, 7328, 126, -127, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 127*x^34 + 126*x^33 + 7328*x^32 - 7203*x^31 - 254655*x^30 + 247577*x^29 + 5956273*x^28 - 5715774*x^27 - 99258341*x^26 + 93783066*x^25 + 1216750731*x^24 - 1128384881*x^23 - 11167777905*x^22 + 10123730642*x^21 + 77372149977*x^20 - 68170119347*x^19 - 404867699989*x^18 + 343756885869*x^17 + 1590272002452*x^16 - 1283637936946*x^15 - 4628193618151*x^14 + 3471389732603*x^13 + 9777925862459*x^12 - 6544984186949*x^11 - 14568115911700*x^10 + 8066810405369*x^9 + 14716307928191*x^8 - 5774232219857*x^7 - 9516669187523*x^6 + 1798484654352*x^5 + 3510763848149*x^4 + 66623171786*x^3 - 532421593101*x^2 - 84148737831*x + 7203054951)
 
gp: K = bnfinit(x^36 - x^35 - 127*x^34 + 126*x^33 + 7328*x^32 - 7203*x^31 - 254655*x^30 + 247577*x^29 + 5956273*x^28 - 5715774*x^27 - 99258341*x^26 + 93783066*x^25 + 1216750731*x^24 - 1128384881*x^23 - 11167777905*x^22 + 10123730642*x^21 + 77372149977*x^20 - 68170119347*x^19 - 404867699989*x^18 + 343756885869*x^17 + 1590272002452*x^16 - 1283637936946*x^15 - 4628193618151*x^14 + 3471389732603*x^13 + 9777925862459*x^12 - 6544984186949*x^11 - 14568115911700*x^10 + 8066810405369*x^9 + 14716307928191*x^8 - 5774232219857*x^7 - 9516669187523*x^6 + 1798484654352*x^5 + 3510763848149*x^4 + 66623171786*x^3 - 532421593101*x^2 - 84148737831*x + 7203054951, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 127 x^{34} + 126 x^{33} + 7328 x^{32} - 7203 x^{31} - 254655 x^{30} + 247577 x^{29} + 5956273 x^{28} - 5715774 x^{27} - 99258341 x^{26} + 93783066 x^{25} + 1216750731 x^{24} - 1128384881 x^{23} - 11167777905 x^{22} + 10123730642 x^{21} + 77372149977 x^{20} - 68170119347 x^{19} - 404867699989 x^{18} + 343756885869 x^{17} + 1590272002452 x^{16} - 1283637936946 x^{15} - 4628193618151 x^{14} + 3471389732603 x^{13} + 9777925862459 x^{12} - 6544984186949 x^{11} - 14568115911700 x^{10} + 8066810405369 x^{9} + 14716307928191 x^{8} - 5774232219857 x^{7} - 9516669187523 x^{6} + 1798484654352 x^{5} + 3510763848149 x^{4} + 66623171786 x^{3} - 532421593101 x^{2} - 84148737831 x + 7203054951 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(494916467663075288417691500768863396509709128831881845460867298126220703125=5^{18}\cdot 7^{30}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.81$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(261,·)$, $\chi_{455}(16,·)$, $\chi_{455}(384,·)$, $\chi_{455}(19,·)$, $\chi_{455}(279,·)$, $\chi_{455}(24,·)$, $\chi_{455}(409,·)$, $\chi_{455}(34,·)$, $\chi_{455}(36,·)$, $\chi_{455}(296,·)$, $\chi_{455}(174,·)$, $\chi_{455}(349,·)$, $\chi_{455}(304,·)$, $\chi_{455}(51,·)$, $\chi_{455}(54,·)$, $\chi_{455}(314,·)$, $\chi_{455}(59,·)$, $\chi_{455}(444,·)$, $\chi_{455}(191,·)$, $\chi_{455}(246,·)$, $\chi_{455}(326,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(164,·)$, $\chi_{455}(186,·)$, $\chi_{455}(229,·)$, $\chi_{455}(316,·)$, $\chi_{455}(369,·)$, $\chi_{455}(116,·)$, $\chi_{455}(89,·)$, $\chi_{455}(361,·)$, $\chi_{455}(121,·)$, $\chi_{455}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} + \frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{20} - \frac{1}{3} a^{19} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{25} + \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{19} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{26} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{19} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{27} - \frac{1}{3} a$, $\frac{1}{3} a^{28} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{29} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{30} - \frac{1}{3} a^{4}$, $\frac{1}{156041421} a^{31} - \frac{11997007}{156041421} a^{30} - \frac{2286668}{52013807} a^{29} + \frac{205327}{52013807} a^{28} - \frac{16369225}{156041421} a^{27} + \frac{6733516}{52013807} a^{26} - \frac{8931377}{156041421} a^{25} - \frac{1721356}{156041421} a^{24} - \frac{23506481}{156041421} a^{23} - \frac{10256302}{156041421} a^{22} - \frac{4997074}{52013807} a^{21} + \frac{15308766}{52013807} a^{20} - \frac{12638721}{52013807} a^{19} - \frac{55973099}{156041421} a^{18} + \frac{12263266}{52013807} a^{17} - \frac{23076432}{52013807} a^{16} + \frac{12234971}{156041421} a^{15} + \frac{70582802}{156041421} a^{14} + \frac{24582304}{52013807} a^{13} - \frac{55522922}{156041421} a^{12} - \frac{56437009}{156041421} a^{11} + \frac{52242706}{156041421} a^{10} + \frac{43889330}{156041421} a^{9} - \frac{324937}{52013807} a^{8} - \frac{20566878}{52013807} a^{7} - \frac{22380213}{52013807} a^{6} + \frac{15782954}{52013807} a^{5} + \frac{66580162}{156041421} a^{4} - \frac{18477244}{52013807} a^{3} - \frac{41375125}{156041421} a^{2} + \frac{17614609}{52013807} a + \frac{3575080}{52013807}$, $\frac{1}{28243497201} a^{32} + \frac{67}{28243497201} a^{31} - \frac{2674557769}{28243497201} a^{30} - \frac{1492983214}{28243497201} a^{29} - \frac{740580068}{28243497201} a^{28} - \frac{4655118602}{28243497201} a^{27} + \frac{276551750}{9414499067} a^{26} - \frac{857802307}{28243497201} a^{25} - \frac{25625410}{156041421} a^{24} + \frac{893785246}{9414499067} a^{23} - \frac{882977651}{28243497201} a^{22} + \frac{3888431645}{28243497201} a^{21} - \frac{7081402916}{28243497201} a^{20} - \frac{10988055235}{28243497201} a^{19} + \frac{3401723827}{9414499067} a^{18} - \frac{1147389703}{9414499067} a^{17} + \frac{2602471303}{9414499067} a^{16} - \frac{5854105061}{28243497201} a^{15} + \frac{2306569823}{9414499067} a^{14} + \frac{294078176}{9414499067} a^{13} + \frac{13133891540}{28243497201} a^{12} + \frac{8835028874}{28243497201} a^{11} + \frac{4261519733}{9414499067} a^{10} + \frac{5325935680}{28243497201} a^{9} - \frac{11287992466}{28243497201} a^{8} + \frac{6002875963}{28243497201} a^{7} - \frac{8212698809}{28243497201} a^{6} + \frac{868444583}{28243497201} a^{5} - \frac{10779567989}{28243497201} a^{4} - \frac{10302498104}{28243497201} a^{3} + \frac{2259270196}{9414499067} a^{2} - \frac{585360079}{28243497201} a - \frac{1126212420}{9414499067}$, $\frac{1}{28243497201} a^{33} - \frac{11}{28243497201} a^{31} - \frac{891721517}{9414499067} a^{30} + \frac{425352964}{28243497201} a^{29} - \frac{1143670224}{9414499067} a^{28} - \frac{536272226}{28243497201} a^{27} - \frac{2889927755}{28243497201} a^{26} + \frac{1160274603}{9414499067} a^{25} - \frac{696589354}{9414499067} a^{24} - \frac{1223593016}{9414499067} a^{23} - \frac{300027077}{9414499067} a^{22} + \frac{7197045062}{28243497201} a^{21} + \frac{5677783864}{28243497201} a^{20} - \frac{11446345798}{28243497201} a^{19} - \frac{10189207438}{28243497201} a^{18} + \frac{10598168309}{28243497201} a^{17} - \frac{12685995281}{28243497201} a^{16} + \frac{6387976036}{28243497201} a^{15} - \frac{2732931778}{9414499067} a^{14} - \frac{5003755750}{28243497201} a^{13} + \frac{9691216088}{28243497201} a^{12} + \frac{12595616230}{28243497201} a^{11} + \frac{13472830688}{28243497201} a^{10} - \frac{3414477872}{28243497201} a^{9} + \frac{1910007019}{28243497201} a^{8} - \frac{3780897542}{28243497201} a^{7} + \frac{12465953722}{28243497201} a^{6} + \frac{7846335692}{28243497201} a^{5} - \frac{8557076432}{28243497201} a^{4} + \frac{6969233989}{28243497201} a^{3} + \frac{2245555494}{9414499067} a^{2} - \frac{3318583654}{9414499067} a + \frac{4062339283}{9414499067}$, $\frac{1}{1074354390028839} a^{34} + \frac{4763}{358118130009613} a^{33} + \frac{9460}{1074354390028839} a^{32} + \frac{1049945}{1074354390028839} a^{31} + \frac{82563769895048}{1074354390028839} a^{30} - \frac{121308728815514}{1074354390028839} a^{29} + \frac{29469072961100}{1074354390028839} a^{28} + \frac{55892781884444}{358118130009613} a^{27} - \frac{33090823435622}{358118130009613} a^{26} - \frac{85539399013637}{1074354390028839} a^{25} - \frac{154581673776133}{1074354390028839} a^{24} - \frac{178825636188209}{1074354390028839} a^{23} + \frac{29133670411785}{358118130009613} a^{22} - \frac{26038863125512}{1074354390028839} a^{21} - \frac{503580458128808}{1074354390028839} a^{20} + \frac{267907001932957}{1074354390028839} a^{19} - \frac{469307759450566}{1074354390028839} a^{18} + \frac{401926573735604}{1074354390028839} a^{17} + \frac{181107059065640}{1074354390028839} a^{16} - \frac{108516513218179}{1074354390028839} a^{15} + \frac{138585930905174}{358118130009613} a^{14} - \frac{496430863635080}{1074354390028839} a^{13} + \frac{1891337237613}{358118130009613} a^{12} + \frac{104537155145219}{358118130009613} a^{11} - \frac{128213138378701}{358118130009613} a^{10} - \frac{105926472941306}{1074354390028839} a^{9} + \frac{29170465986967}{358118130009613} a^{8} + \frac{488452893875177}{1074354390028839} a^{7} - \frac{435144847146070}{1074354390028839} a^{6} - \frac{243744710088770}{1074354390028839} a^{5} - \frac{324573267338926}{1074354390028839} a^{4} + \frac{404053482370745}{1074354390028839} a^{3} - \frac{388067879871550}{1074354390028839} a^{2} - \frac{68653869848384}{1074354390028839} a + \frac{105143673291489}{358118130009613}$, $\frac{1}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{35} - \frac{750903843109258807458135589692238105536928152235920962290574879714386339109845646148815522893768496910004399}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{34} + \frac{47250651128294453634082582717422784105920997085827301256280290395974263096681312119095367786117799022654880646587}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{33} - \frac{5148883443731360953815319684477192501473760702161637541790867729824740595369642295218391188941660327477290328921}{398416563468251606661101560852418086117533295817088252344203183073090948624764456648694793996558706107085645714960790987509} a^{32} + \frac{8180628302734777867616860764965245075118955666239522259629825764568457962094800810146208125256544575407355579896568}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{31} + \frac{74126006899085725546535253679740685917807825104085061581893391548481355129030381308014900315493620662048302632282986738963}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{30} + \frac{86571764650166699000932147786470078911400151170039130211302790210685874657225207736353711205893469369504889252429135349388}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{29} + \frac{474028129913932275673456278980458790869431623673340570503915969380526144435986014384289768160151932396070204284769380076217}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{28} + \frac{33313420855074948911492632269069127328917441642112101024370627552129199329515437248930321729576210619475383697144724360041}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{27} - \frac{69326499619837022752747384655821060183317420057125013082714432574811201965649262542390141878350773024206443109315658713108}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{26} + \frac{302293769971635236082904842215601415814450271854190465419559400310594153297428745053972060351642806757994332497277012839490}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{25} - \frac{78395453698835820499099085317710219055685499090560417986740586426517925970728976474695311708923880655207926818418782076000}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{24} + \frac{51152428383006738713997415263998694557032352620684558018689425946970883554722255046677029473624517559971990865352755899032}{398416563468251606661101560852418086117533295817088252344203183073090948624764456648694793996558706107085645714960790987509} a^{23} - \frac{914462019140627615485037787349031652080559969813649191934220286616619796242344405549127834373986208919503321687299339876}{19810768349250079889226044462274932458882871062728145144186898605844301312833591767062172077176952237368899510688658115401} a^{22} - \frac{90809247797396207388713251224487339102031491888750217323527926405793190750178601407149176305775643951289151771958038502703}{398416563468251606661101560852418086117533295817088252344203183073090948624764456648694793996558706107085645714960790987509} a^{21} + \frac{84539430709749541218230231076977997772442883826938852120986713069766490663674610031087512264470210051857963281369911003495}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{20} + \frac{507655733834154903021144564045614987761829298854284952583242613819584747268569022759063060977730088428550325232371936604803}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{19} + \frac{42256823001857770212793021031685678758798443075855829777561486341037111123548486011937438279759366633066730532880018542106}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{18} - \frac{1012750717168199517160288345344812009000179958880045137301457293258004045676692934163386455916407184582388536865346562486782}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{17} + \frac{433807118674801024010350216161763372516996866754378804087377876109960519529485871594072818150399677347485799952769742558008}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{16} - \frac{359119755485967939304815564919430142756926905167265810476167305148172682258542873034342521836189345859478981369600706136962}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a^{15} + \frac{203409878216241397177774912684229261265749610168347555184262766093593089310611308907981827268127261752362824832351066403989}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{14} - \frac{1709372792602506804253432363922335935084234360399251942129815403146582890799348860289366869040209871746037317401422764587242}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{13} + \frac{940910825139822814857451088759831456655054025293883447962581101883738786919798129181474751356190945589219700848232759814303}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{12} - \frac{535546745351266220746853251857994518368436583515626098427330868377424298493043908479314071776161019628061858835291899238165}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{11} - \frac{1775559522658457177453209973468060028760200042950604767585865829638319431114413882311474250031520178323678764275522943187211}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{10} + \frac{1736429039371245873702625871136925324867929786288703804830988587916410515746400636031532782474915147244512112920318623126330}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{9} - \frac{822598072232346963562806952840549638286298908138494736951640668234603171796637944448984691751625175007135790118830694615351}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{8} + \frac{218994666150463224229314195102464455652903691854155051312888869713327712585666441810290096040342679742756454438229847038673}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{7} + \frac{1642803300884054833838964025549122129404936664168795776089084538953455204879288968422126039793379762303695511468999250406578}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{6} + \frac{1515385029718344882442785945188817608221384943970965960097764628831561698206063132096703249346861503305701606280772829335680}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{5} - \frac{94264583639285984463260559504351804266038554947330438049717325009845040334092100230582947847799114335604387389841978322454}{398416563468251606661101560852418086117533295817088252344203183073090948624764456648694793996558706107085645714960790987509} a^{4} + \frac{736990768580952572599333653259066099133415656738251353211673668073667713500023129269953997927575758563092838804480533341491}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{3} + \frac{726088737837339127179591688218856434876686545644219415111400439420690454777212604175322902731974234481408976418110229383483}{3585749071214264459949914047671762775057799662353794271097828647657818537622880109838253145969028354963770811434647118887581} a^{2} - \frac{350591303469921796047539597535106432416504241092792696259799853875495142025463041102677980093206690318542114997212714047767}{1195249690404754819983304682557254258352599887451264757032609549219272845874293369946084381989676118321256937144882372962527} a + \frac{362886839240704608477617129349342825542446183431160791953637740232701442543952934753157701839024290930461766847205}{1493428453180044798446206125995065774026662563921537242940503834422775266825621684683821237990592287082402466917393}$

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:  $35$
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, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 4.4.2691325.1, \(\Q(\zeta_{13})^+\), 6.6.5274997.1, 6.6.891474493.1, 6.6.891474493.2, 9.9.567869252041.1, 12.12.3294466846844672078125.1, 12.12.46804821889195607453125.1, 12.12.7910014899274057659578125.2, 12.12.7910014899274057659578125.1, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$5$5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
7Data not computed
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$