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Properties

Label 36.0.17515031822...3125.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{48}\cdot 5^{18}\cdot 13^{33}$
Root discriminant $101.57$
Ramified primes $3, 5, 13$
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![254886323381, -163796723415, 227273426838, -140326122819, 232594560831, -188488040397, 186530283006, -179069923941, 193914680868, -164805518158, 133566120075, -87157837233, 56950578528, -31141822035, 17439706425, -8261897447, 3898113045, -1550727096, 611810329, -222454680, 84771678, -30772058, 10489914, -3273684, 1104025, -415143, 161433, -47895, 13113, -3546, 1779, -567, 153, -13, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 + 9*x^34 - 13*x^33 + 153*x^32 - 567*x^31 + 1779*x^30 - 3546*x^29 + 13113*x^28 - 47895*x^27 + 161433*x^26 - 415143*x^25 + 1104025*x^24 - 3273684*x^23 + 10489914*x^22 - 30772058*x^21 + 84771678*x^20 - 222454680*x^19 + 611810329*x^18 - 1550727096*x^17 + 3898113045*x^16 - 8261897447*x^15 + 17439706425*x^14 - 31141822035*x^13 + 56950578528*x^12 - 87157837233*x^11 + 133566120075*x^10 - 164805518158*x^9 + 193914680868*x^8 - 179069923941*x^7 + 186530283006*x^6 - 188488040397*x^5 + 232594560831*x^4 - 140326122819*x^3 + 227273426838*x^2 - 163796723415*x + 254886323381)
 
gp: K = bnfinit(x^36 - 3*x^35 + 9*x^34 - 13*x^33 + 153*x^32 - 567*x^31 + 1779*x^30 - 3546*x^29 + 13113*x^28 - 47895*x^27 + 161433*x^26 - 415143*x^25 + 1104025*x^24 - 3273684*x^23 + 10489914*x^22 - 30772058*x^21 + 84771678*x^20 - 222454680*x^19 + 611810329*x^18 - 1550727096*x^17 + 3898113045*x^16 - 8261897447*x^15 + 17439706425*x^14 - 31141822035*x^13 + 56950578528*x^12 - 87157837233*x^11 + 133566120075*x^10 - 164805518158*x^9 + 193914680868*x^8 - 179069923941*x^7 + 186530283006*x^6 - 188488040397*x^5 + 232594560831*x^4 - 140326122819*x^3 + 227273426838*x^2 - 163796723415*x + 254886323381, 1)
 

Normalized defining polynomial

\( x^{36} - 3 x^{35} + 9 x^{34} - 13 x^{33} + 153 x^{32} - 567 x^{31} + 1779 x^{30} - 3546 x^{29} + 13113 x^{28} - 47895 x^{27} + 161433 x^{26} - 415143 x^{25} + 1104025 x^{24} - 3273684 x^{23} + 10489914 x^{22} - 30772058 x^{21} + 84771678 x^{20} - 222454680 x^{19} + 611810329 x^{18} - 1550727096 x^{17} + 3898113045 x^{16} - 8261897447 x^{15} + 17439706425 x^{14} - 31141822035 x^{13} + 56950578528 x^{12} - 87157837233 x^{11} + 133566120075 x^{10} - 164805518158 x^{9} + 193914680868 x^{8} - 179069923941 x^{7} + 186530283006 x^{6} - 188488040397 x^{5} + 232594560831 x^{4} - 140326122819 x^{3} + 227273426838 x^{2} - 163796723415 x + 254886323381 \)

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:  \(1751503182280629962324239137560707045640770185903918052149707794189453125=3^{48}\cdot 5^{18}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.57$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(514,·)$, $\chi_{585}(391,·)$, $\chi_{585}(16,·)$, $\chi_{585}(19,·)$, $\chi_{585}(406,·)$, $\chi_{585}(409,·)$, $\chi_{585}(154,·)$, $\chi_{585}(544,·)$, $\chi_{585}(34,·)$, $\chi_{585}(166,·)$, $\chi_{585}(424,·)$, $\chi_{585}(556,·)$, $\chi_{585}(304,·)$, $\chi_{585}(181,·)$, $\chi_{585}(184,·)$, $\chi_{585}(571,·)$, $\chi_{585}(316,·)$, $\chi_{585}(61,·)$, $\chi_{585}(574,·)$, $\chi_{585}(319,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(211,·)$, $\chi_{585}(214,·)$, $\chi_{585}(349,·)$, $\chi_{585}(229,·)$, $\chi_{585}(361,·)$, $\chi_{585}(109,·)$, $\chi_{585}(499,·)$, $\chi_{585}(376,·)$, $\chi_{585}(121,·)$, $\chi_{585}(379,·)$, $\chi_{585}(124,·)$, $\chi_{585}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{233} a^{20} + \frac{37}{233} a^{19} + \frac{61}{233} a^{18} - \frac{34}{233} a^{17} + \frac{5}{233} a^{16} - \frac{101}{233} a^{15} + \frac{86}{233} a^{14} - \frac{69}{233} a^{13} - \frac{77}{233} a^{12} - \frac{113}{233} a^{11} - \frac{6}{233} a^{10} + \frac{9}{233} a^{9} + \frac{67}{233} a^{8} - \frac{22}{233} a^{7} - \frac{85}{233} a^{6} - \frac{27}{233} a^{5} + \frac{74}{233} a^{4} - \frac{32}{233} a^{3} + \frac{41}{233} a^{2} + \frac{19}{233} a - \frac{67}{233}$, $\frac{1}{233} a^{21} + \frac{90}{233} a^{19} + \frac{39}{233} a^{18} + \frac{98}{233} a^{17} - \frac{53}{233} a^{16} + \frac{95}{233} a^{15} + \frac{11}{233} a^{14} - \frac{87}{233} a^{13} - \frac{60}{233} a^{12} - \frac{19}{233} a^{11} - \frac{2}{233} a^{10} - \frac{33}{233} a^{9} + \frac{62}{233} a^{8} + \frac{30}{233} a^{7} + \frac{89}{233} a^{6} - \frac{92}{233} a^{5} + \frac{26}{233} a^{4} + \frac{60}{233} a^{3} - \frac{100}{233} a^{2} - \frac{71}{233} a - \frac{84}{233}$, $\frac{1}{233} a^{22} - \frac{29}{233} a^{19} - \frac{33}{233} a^{18} - \frac{22}{233} a^{17} + \frac{111}{233} a^{16} + \frac{14}{233} a^{15} + \frac{95}{233} a^{14} + \frac{92}{233} a^{13} - \frac{79}{233} a^{12} - \frac{84}{233} a^{11} + \frac{41}{233} a^{10} - \frac{49}{233} a^{9} + \frac{58}{233} a^{8} - \frac{28}{233} a^{7} + \frac{102}{233} a^{6} - \frac{107}{233} a^{5} - \frac{76}{233} a^{4} - \frac{16}{233} a^{3} - \frac{33}{233} a^{2} + \frac{70}{233} a - \frac{28}{233}$, $\frac{1}{233} a^{23} + \frac{108}{233} a^{19} + \frac{116}{233} a^{18} + \frac{57}{233} a^{17} - \frac{74}{233} a^{16} - \frac{38}{233} a^{15} + \frac{23}{233} a^{14} + \frac{17}{233} a^{13} + \frac{13}{233} a^{12} + \frac{26}{233} a^{11} + \frac{10}{233} a^{10} + \frac{86}{233} a^{9} + \frac{51}{233} a^{8} - \frac{70}{233} a^{7} - \frac{9}{233} a^{6} + \frac{73}{233} a^{5} + \frac{33}{233} a^{4} - \frac{29}{233} a^{3} + \frac{94}{233} a^{2} + \frac{57}{233} a - \frac{79}{233}$, $\frac{1}{233} a^{24} + \frac{81}{233} a^{19} - \frac{7}{233} a^{18} + \frac{103}{233} a^{17} - \frac{112}{233} a^{16} - \frac{20}{233} a^{15} + \frac{49}{233} a^{14} + \frac{9}{233} a^{13} - \frac{46}{233} a^{12} + \frac{98}{233} a^{11} + \frac{35}{233} a^{10} + \frac{11}{233} a^{9} - \frac{83}{233} a^{8} + \frac{37}{233} a^{7} - \frac{67}{233} a^{6} - \frac{80}{233} a^{5} - \frac{99}{233} a^{4} + \frac{55}{233} a^{3} + \frac{56}{233} a^{2} - \frac{34}{233} a + \frac{13}{233}$, $\frac{1}{233} a^{25} + \frac{25}{233} a^{19} + \frac{55}{233} a^{18} + \frac{79}{233} a^{17} + \frac{41}{233} a^{16} + \frac{75}{233} a^{15} + \frac{33}{233} a^{14} - \frac{49}{233} a^{13} + \frac{44}{233} a^{12} + \frac{101}{233} a^{11} + \frac{31}{233} a^{10} - \frac{113}{233} a^{9} - \frac{31}{233} a^{8} + \frac{84}{233} a^{7} + \frac{48}{233} a^{6} - \frac{9}{233} a^{5} - \frac{114}{233} a^{4} + \frac{85}{233} a^{3} - \frac{93}{233} a^{2} + \frac{105}{233} a + \frac{68}{233}$, $\frac{1}{233} a^{26} + \frac{62}{233} a^{19} - \frac{48}{233} a^{18} - \frac{41}{233} a^{17} - \frac{50}{233} a^{16} - \frac{5}{233} a^{15} - \frac{102}{233} a^{14} - \frac{95}{233} a^{13} - \frac{71}{233} a^{12} + \frac{60}{233} a^{11} + \frac{37}{233} a^{10} - \frac{23}{233} a^{9} + \frac{40}{233} a^{8} - \frac{101}{233} a^{7} + \frac{19}{233} a^{6} + \frac{95}{233} a^{5} + \frac{99}{233} a^{4} + \frac{8}{233} a^{3} + \frac{12}{233} a^{2} + \frac{59}{233} a + \frac{44}{233}$, $\frac{1}{233} a^{27} - \frac{12}{233} a^{19} - \frac{95}{233} a^{18} - \frac{39}{233} a^{17} - \frac{82}{233} a^{16} + \frac{102}{233} a^{15} - \frac{68}{233} a^{14} + \frac{13}{233} a^{13} - \frac{59}{233} a^{12} + \frac{53}{233} a^{11} + \frac{116}{233} a^{10} - \frac{52}{233} a^{9} - \frac{61}{233} a^{8} - \frac{15}{233} a^{7} + \frac{6}{233} a^{6} - \frac{91}{233} a^{5} + \frac{80}{233} a^{4} - \frac{101}{233} a^{3} + \frac{80}{233} a^{2} + \frac{31}{233} a - \frac{40}{233}$, $\frac{1}{233} a^{28} + \frac{116}{233} a^{19} - \frac{6}{233} a^{18} - \frac{24}{233} a^{17} - \frac{71}{233} a^{16} - \frac{115}{233} a^{15} + \frac{113}{233} a^{14} + \frac{45}{233} a^{13} + \frac{61}{233} a^{12} - \frac{75}{233} a^{11} + \frac{109}{233} a^{10} + \frac{47}{233} a^{9} + \frac{90}{233} a^{8} - \frac{25}{233} a^{7} + \frac{54}{233} a^{6} - \frac{11}{233} a^{5} + \frac{88}{233} a^{4} - \frac{71}{233} a^{3} + \frac{57}{233} a^{2} - \frac{45}{233} a - \frac{105}{233}$, $\frac{1}{233} a^{29} - \frac{104}{233} a^{19} - \frac{110}{233} a^{18} - \frac{88}{233} a^{17} + \frac{4}{233} a^{16} - \frac{54}{233} a^{15} + \frac{88}{233} a^{14} - \frac{90}{233} a^{13} + \frac{3}{233} a^{12} - \frac{64}{233} a^{11} + \frac{44}{233} a^{10} - \frac{22}{233} a^{9} - \frac{108}{233} a^{8} + \frac{43}{233} a^{7} + \frac{63}{233} a^{6} - \frac{42}{233} a^{5} - \frac{34}{233} a^{4} + \frac{41}{233} a^{3} + \frac{92}{233} a^{2} + \frac{21}{233} a + \frac{83}{233}$, $\frac{1}{233} a^{30} + \frac{10}{233} a^{19} - \frac{35}{233} a^{18} - \frac{37}{233} a^{17} + \frac{69}{233} a^{15} + \frac{50}{233} a^{13} + \frac{83}{233} a^{12} - \frac{58}{233} a^{11} + \frac{53}{233} a^{10} - \frac{104}{233} a^{9} + \frac{21}{233} a^{8} + \frac{105}{233} a^{7} - \frac{28}{233} a^{6} - \frac{46}{233} a^{5} + \frac{48}{233} a^{4} + \frac{26}{233} a^{3} + \frac{91}{233} a^{2} - \frac{38}{233} a + \frac{22}{233}$, $\frac{1}{233} a^{31} + \frac{61}{233} a^{19} + \frac{52}{233} a^{18} + \frac{107}{233} a^{17} + \frac{19}{233} a^{16} + \frac{78}{233} a^{15} - \frac{111}{233} a^{14} + \frac{74}{233} a^{13} + \frac{13}{233} a^{12} + \frac{18}{233} a^{11} - \frac{44}{233} a^{10} - \frac{69}{233} a^{9} - \frac{99}{233} a^{8} - \frac{41}{233} a^{7} + \frac{105}{233} a^{6} + \frac{85}{233} a^{5} - \frac{15}{233} a^{4} - \frac{55}{233} a^{3} + \frac{18}{233} a^{2} + \frac{65}{233} a - \frac{29}{233}$, $\frac{1}{233} a^{32} - \frac{108}{233} a^{19} + \frac{114}{233} a^{18} - \frac{4}{233} a^{17} + \frac{6}{233} a^{16} - \frac{8}{233} a^{15} - \frac{46}{233} a^{14} + \frac{28}{233} a^{13} + \frac{55}{233} a^{12} + \frac{92}{233} a^{11} + \frac{64}{233} a^{10} + \frac{51}{233} a^{9} + \frac{66}{233} a^{8} + \frac{49}{233} a^{7} - \frac{89}{233} a^{6} + \frac{1}{233} a^{5} + \frac{91}{233} a^{4} + \frac{106}{233} a^{3} - \frac{106}{233} a^{2} - \frac{23}{233} a - \frac{107}{233}$, $\frac{1}{233} a^{33} - \frac{84}{233} a^{19} + \frac{60}{233} a^{18} + \frac{62}{233} a^{17} + \frac{66}{233} a^{16} - \frac{3}{233} a^{15} - \frac{4}{233} a^{14} + \frac{59}{233} a^{13} - \frac{69}{233} a^{12} - \frac{24}{233} a^{11} + \frac{102}{233} a^{10} + \frac{106}{233} a^{9} + \frac{62}{233} a^{8} + \frac{98}{233} a^{7} - \frac{92}{233} a^{6} - \frac{29}{233} a^{5} - \frac{57}{233} a^{4} - \frac{67}{233} a^{3} - \frac{22}{233} a^{2} + \frac{81}{233} a - \frac{13}{233}$, $\frac{1}{233} a^{34} - \frac{94}{233} a^{19} + \frac{60}{233} a^{18} + \frac{6}{233} a^{17} - \frac{49}{233} a^{16} - \frac{100}{233} a^{15} + \frac{60}{233} a^{14} - \frac{40}{233} a^{13} + \frac{32}{233} a^{12} - \frac{70}{233} a^{11} + \frac{68}{233} a^{10} - \frac{114}{233} a^{9} - \frac{99}{233} a^{8} - \frac{76}{233} a^{7} + \frac{54}{233} a^{6} + \frac{5}{233} a^{5} + \frac{91}{233} a^{4} + \frac{86}{233} a^{3} + \frac{30}{233} a^{2} - \frac{48}{233} a - \frac{36}{233}$, $\frac{1}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{35} - \frac{608992732688588808029003018301883050008597926070189154970734858687793269625820833916293750975718481828946335965941439731581054071460154056761209731297395015931397683263734773371719507867}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{34} - \frac{411067404753142552108965493638953456012926636622603257067478550774974722431270582412017985081114391527744968345715143657695833763313359788796768096195291112111595282947331792004593866139}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{33} + \frac{124613598300390836666427994145537581852724175403556908236805882820701331604953512632384533346946245971095637799534533079480892772229823646969323910571386070102091604747069641755000075820}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{32} + \frac{545353007832292592783788120833635136486221930545495073979258974040977454362482447754768138353355549010824266659600995218843983655739452364348912553789214317809096875501485620606075878892}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{31} - \frac{119123376188529324030066541513525330886452960299231115317341362386577376985867081619097675353267956531732372596661843367126296061565885029232791207690059773283424229873880200452062912585}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{30} + \frac{541699705815272851273511961457421104565769062152121768161503532654268070034215135533060237511230123836261154230998953158720825690944590506836584973577308771133535905519283809688910777236}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{29} + \frac{858557309459590148580964200074348581187098858522447239360943076809881082460468329507569955956250432225344183319645575811857759244761271618091995817347466715249929153487494744580804330869}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{28} + \frac{636679304086916017394443179261986065942056215674619179649182348180466861615777153973287818815272321017687914609967890018422348306252609651627586471531492343100061997457790819710633676780}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{27} + \frac{403933249956026547448911219268550442475060643561616952718007615761952813899915435226403062617682011298995187411265750582324253291973857720717157829508169271438721564427996219201147311438}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{26} - \frac{161447663848054851981960193101699771021125973589756194311899432897468313949020806785478845738245182231579481882889702117805334327760484591786041866268493253901640555632431846031758439151}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{25} + \frac{675178430344831793509562432141736330303263478135037447319505880038887090723905160183898544815524993240240236875151521264731483936655328168499163082501572597020382817674457591761948725083}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{24} + \frac{261431492841480725711016056446076791484575633695265195588226345230543292965834184929724646075157081152681155769629026040773360990684456057910479909869794125774285206076073749845521406573}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{23} + \frac{818078434228964162623941766723034889792803884829221656465248518144775569955492450388017324975756357347616815991514053438913228600206173753006829283326067548007581865252534630408708874038}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{22} + \frac{123642707685066745460242657559812990019362177603042605437979212396406141741096934172230128094783113843227299428721489439951566191331776340200386574151713696844850591171115002782656150316}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{21} + \frac{200269097171645431026492283298650202867486801606338684035920690739837249591210153473376929770706968487059995411277800636714707788395289484616371223880675306004550108296949057479443652556}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{20} - \frac{12177111250584689068166137526106247099590050509349369621114205608165849561303992225828355492245488411328663044388724195080571618660894042593790166370218036583337158980170844530503930243600}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{19} - \frac{20299058963969853979530890256474197127312310892981529171607634555114879870891431560223758173370494233879608261268091999508787932718462576025757639508540820289758796129394481588416866135174}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{18} - \frac{182984598930668097143965185888909554145383406006112204591781565197332882776406516712613877353146860668142396336462458932460359942860584256249932069443331363621448322367207612464602623560339}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{17} + \frac{177257089606287394977013087612151400175149953275900451441786736891569739700784849678910127426805035046445766531488335581021051357222352738017709932890639067161063243665708562495219071678374}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{16} - \frac{75264244994354166069276345379603389807715881626182496543185452094315346706842765202798576570231185704375342445679948306159919937055711965340874091601787746942194109786574488443351022426123}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{15} + \frac{174830888045331205070923456598181426027188856340510258232838047748848876064896650035713300643648654006408141175239015380824014690193910985846060296732065866356020104380856028241859581392046}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{14} + \frac{75048130510933092684503130908366431941693544313163596382942085670853404190553128426498647044980781937430642439903360389989388275862195361686426747360663700779694124006787734134940381212260}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{13} + \frac{151160140980469630615602020418479048095453520896906107541945435911991991825919493539045057333247395680691694925998646111453305789961615915399605462254889701326546106357016005287758018913030}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{12} + \frac{129636998525256457871279377525054953136443244457576214251626308801847299258468496988503097116065117177387234676776579109176286983696888073182595712914759531161884135349622340541746957088006}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{11} - \frac{120389532352340089369675212131736531997211127728413562957960956516142084820786478241558482017833751673262035808698909315290454278050835340423108224185125755962522340382501967683790183026626}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{10} + \frac{63750401789435691283377377550275026348922171145165026236298239477531584844126042763295850164710275414363349736821970451494903015145488346469683574991667596922934971527510966763427005026899}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{9} - \frac{200851540464584280821789362163086159786147772134420231690392744738095136898872368840649476507558319808050705971789636937804286484790364164780414879258537385237562956432870609263269433827531}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{8} - \frac{77446042406133579036305140842766508571803275221125440153993328822359049279164937073159076574039048506881964009173667448494276507142550734881635658691264505116096836055930147255861415645952}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{7} - \frac{164860791810250654438489172630707314311660706491361726845456801488097787426273091790410348018160143306907027504753409882440432331405601793931897675800746064261645178511573333354434702816154}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{6} + \frac{122388343418727632819986396777383635963436334376548189768403959566256851423674481565796790869605602772595388196594370259265914899323593237900446419601465864344315129221554925244851980155034}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{5} + \frac{121215162290874281566603367701268881100617578344614995396177635850053138944434282610339084816712716707956725493510424287733025405878587771054666642548491897450867634364763691482199061590413}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{4} - \frac{98925962034654276434021308325982212449405822958219787558844599002179448909223263014740752413545765923530597284023117842392911784785625732759104381843695736031756013206627279750251141934313}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{3} + \frac{6363664068398616572273676947117669663003211960240528948895459680612272945136738177122411624396357919997449129969830228951953839292307471271543964206700167006694716272523595672707004694139}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a^{2} - \frac{71604935570660871964171313280866473810956801172571860333118384110218170284054031044093323667330907330266007303279487432515344496538339598642066491303526409436323523310815874426692833933008}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271} a - \frac{197215751978805168182200774144793490467319810997665199892938221883285550221345293426751395254388591098851254949191824411207991501849903694875585653934653929372181534435583393041460028524036}{442372932215609464688643282576544641667714075082438091077594886465970880971785020428150013560075457061496674681036767847101511710464431761512864655288271016941373968117744398973515132973271}$

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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, 4.0.54925.1, 6.6.14414517.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 6.6.2436053373.1, 9.9.2565164201769.1, 12.0.7132639466471967703125.1, 12.0.28002506156828125.1, 12.0.1205416069833762541828125.2, 12.0.1205416069833762541828125.1, 18.18.14456408038335708501176406117.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
3Data not computed
$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}$
13Data not computed