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Properties

Label 36.36.8216251102...5625.2
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $128.45$
Ramified primes $5, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![-54664, -996492, 1187472, 56606849, -86946285, -949622917, 2347697989, 3507716589, -12313025417, -5227780689, 30731222743, 2033087095, -45171089573, 4650174763, 42894256903, -8685061710, -27583235170, 7514841399, 12292532381, -4052765184, -3822699612, 1467032372, 823243158, -365203449, -119639181, 62716641, 11033971, -7345079, -539471, 570596, 1248, -27862, 1348, 767, -65, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 65*x^34 + 767*x^33 + 1348*x^32 - 27862*x^31 + 1248*x^30 + 570596*x^29 - 539471*x^28 - 7345079*x^27 + 11033971*x^26 + 62716641*x^25 - 119639181*x^24 - 365203449*x^23 + 823243158*x^22 + 1467032372*x^21 - 3822699612*x^20 - 4052765184*x^19 + 12292532381*x^18 + 7514841399*x^17 - 27583235170*x^16 - 8685061710*x^15 + 42894256903*x^14 + 4650174763*x^13 - 45171089573*x^12 + 2033087095*x^11 + 30731222743*x^10 - 5227780689*x^9 - 12313025417*x^8 + 3507716589*x^7 + 2347697989*x^6 - 949622917*x^5 - 86946285*x^4 + 56606849*x^3 + 1187472*x^2 - 996492*x - 54664)
 
gp: K = bnfinit(x^36 - 9*x^35 - 65*x^34 + 767*x^33 + 1348*x^32 - 27862*x^31 + 1248*x^30 + 570596*x^29 - 539471*x^28 - 7345079*x^27 + 11033971*x^26 + 62716641*x^25 - 119639181*x^24 - 365203449*x^23 + 823243158*x^22 + 1467032372*x^21 - 3822699612*x^20 - 4052765184*x^19 + 12292532381*x^18 + 7514841399*x^17 - 27583235170*x^16 - 8685061710*x^15 + 42894256903*x^14 + 4650174763*x^13 - 45171089573*x^12 + 2033087095*x^11 + 30731222743*x^10 - 5227780689*x^9 - 12313025417*x^8 + 3507716589*x^7 + 2347697989*x^6 - 949622917*x^5 - 86946285*x^4 + 56606849*x^3 + 1187472*x^2 - 996492*x - 54664, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 65 x^{34} + 767 x^{33} + 1348 x^{32} - 27862 x^{31} + 1248 x^{30} + 570596 x^{29} - 539471 x^{28} - 7345079 x^{27} + 11033971 x^{26} + 62716641 x^{25} - 119639181 x^{24} - 365203449 x^{23} + 823243158 x^{22} + 1467032372 x^{21} - 3822699612 x^{20} - 4052765184 x^{19} + 12292532381 x^{18} + 7514841399 x^{17} - 27583235170 x^{16} - 8685061710 x^{15} + 42894256903 x^{14} + 4650174763 x^{13} - 45171089573 x^{12} + 2033087095 x^{11} + 30731222743 x^{10} - 5227780689 x^{9} - 12313025417 x^{8} + 3507716589 x^{7} + 2347697989 x^{6} - 949622917 x^{5} - 86946285 x^{4} + 56606849 x^{3} + 1187472 x^{2} - 996492 x - 54664 \)

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:  \(8216251102044589607143313691057181287626972114083156077958643436431884765625=5^{27}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.45$
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$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(2,·)$, $\chi_{455}(4,·)$, $\chi_{455}(261,·)$, $\chi_{455}(134,·)$, $\chi_{455}(8,·)$, $\chi_{455}(394,·)$, $\chi_{455}(268,·)$, $\chi_{455}(386,·)$, $\chi_{455}(16,·)$, $\chi_{455}(323,·)$, $\chi_{455}(128,·)$, $\chi_{455}(389,·)$, $\chi_{455}(32,·)$, $\chi_{455}(162,·)$, $\chi_{455}(163,·)$, $\chi_{455}(422,·)$, $\chi_{455}(179,·)$, $\chi_{455}(309,·)$, $\chi_{455}(57,·)$, $\chi_{455}(317,·)$, $\chi_{455}(191,·)$, $\chi_{455}(64,·)$, $\chi_{455}(193,·)$, $\chi_{455}(67,·)$, $\chi_{455}(324,·)$, $\chi_{455}(197,·)$, $\chi_{455}(326,·)$, $\chi_{455}(333,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(228,·)$, $\chi_{455}(358,·)$, $\chi_{455}(114,·)$, $\chi_{455}(382,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{19} - \frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{20} - \frac{1}{8} a^{19} - \frac{1}{8} a^{18} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{20} - \frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{24} - \frac{1}{8} a^{17} - \frac{1}{8} a^{10} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{25} - \frac{1}{16} a^{21} - \frac{1}{8} a^{20} - \frac{1}{16} a^{19} - \frac{1}{16} a^{17} - \frac{1}{8} a^{15} + \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} + \frac{3}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{3}{16} a^{7} + \frac{1}{4} a^{6} + \frac{5}{16} a^{5} - \frac{1}{2} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{26} - \frac{1}{16} a^{22} - \frac{1}{16} a^{20} - \frac{1}{8} a^{19} + \frac{1}{16} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} + \frac{1}{8} a^{14} + \frac{3}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{3}{8} a^{7} + \frac{1}{16} a^{6} - \frac{3}{8} a^{5} + \frac{1}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{27} - \frac{1}{16} a^{23} - \frac{1}{16} a^{21} - \frac{1}{8} a^{20} + \frac{1}{16} a^{19} - \frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} + \frac{1}{8} a^{15} + \frac{3}{16} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{3}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{28} - \frac{1}{32} a^{27} - \frac{1}{32} a^{25} + \frac{1}{32} a^{24} + \frac{1}{32} a^{23} - \frac{1}{32} a^{22} + \frac{1}{32} a^{20} - \frac{1}{16} a^{19} - \frac{1}{8} a^{18} - \frac{1}{32} a^{17} + \frac{1}{32} a^{14} + \frac{5}{32} a^{13} - \frac{1}{16} a^{12} + \frac{7}{32} a^{11} - \frac{3}{32} a^{10} + \frac{3}{32} a^{9} - \frac{7}{32} a^{8} - \frac{5}{16} a^{7} + \frac{15}{32} a^{6} + \frac{1}{4} a^{5} - \frac{7}{16} a^{4} - \frac{5}{32} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{29} - \frac{1}{64} a^{27} + \frac{1}{64} a^{26} - \frac{1}{32} a^{25} + \frac{1}{32} a^{24} - \frac{1}{16} a^{23} - \frac{3}{64} a^{22} - \frac{1}{64} a^{21} - \frac{3}{64} a^{20} + \frac{1}{16} a^{19} - \frac{3}{64} a^{18} + \frac{5}{64} a^{17} + \frac{1}{16} a^{16} + \frac{9}{64} a^{15} - \frac{5}{32} a^{14} - \frac{9}{64} a^{13} - \frac{5}{64} a^{12} - \frac{3}{32} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} + \frac{1}{64} a^{8} + \frac{31}{64} a^{7} - \frac{11}{64} a^{6} - \frac{1}{4} a^{5} - \frac{17}{64} a^{4} + \frac{29}{64} a^{3} + \frac{1}{8} a^{2} - \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{320} a^{30} + \frac{1}{320} a^{29} + \frac{3}{320} a^{28} + \frac{1}{40} a^{27} - \frac{1}{320} a^{26} + \frac{1}{40} a^{25} + \frac{9}{160} a^{24} + \frac{17}{320} a^{23} - \frac{1}{20} a^{22} + \frac{3}{80} a^{21} - \frac{19}{320} a^{20} - \frac{7}{320} a^{19} - \frac{19}{160} a^{18} - \frac{39}{320} a^{17} - \frac{11}{320} a^{16} - \frac{1}{320} a^{15} + \frac{9}{320} a^{14} + \frac{1}{160} a^{13} + \frac{37}{320} a^{12} - \frac{13}{160} a^{11} + \frac{9}{80} a^{10} - \frac{11}{320} a^{9} - \frac{13}{80} a^{8} + \frac{37}{80} a^{7} - \frac{63}{320} a^{6} + \frac{103}{320} a^{5} - \frac{21}{80} a^{4} + \frac{93}{320} a^{3} - \frac{9}{80} a^{2} - \frac{21}{80} a + \frac{7}{40}$, $\frac{1}{320} a^{31} + \frac{1}{160} a^{29} - \frac{1}{64} a^{28} + \frac{1}{320} a^{27} + \frac{9}{320} a^{26} - \frac{11}{320} a^{24} - \frac{3}{320} a^{23} - \frac{1}{160} a^{22} - \frac{11}{320} a^{21} - \frac{19}{160} a^{20} - \frac{31}{320} a^{19} + \frac{39}{320} a^{18} + \frac{9}{160} a^{17} + \frac{1}{32} a^{16} - \frac{3}{32} a^{15} - \frac{57}{320} a^{14} - \frac{11}{64} a^{13} - \frac{43}{320} a^{12} + \frac{3}{80} a^{11} + \frac{23}{320} a^{10} - \frac{31}{320} a^{9} + \frac{3}{32} a^{8} - \frac{11}{320} a^{7} - \frac{1}{5} a^{6} - \frac{87}{320} a^{5} + \frac{77}{320} a^{4} + \frac{61}{320} a^{3} - \frac{1}{40} a^{2} + \frac{1}{16} a + \frac{3}{40}$, $\frac{1}{320} a^{32} - \frac{1}{160} a^{29} - \frac{1}{64} a^{28} + \frac{1}{40} a^{27} + \frac{7}{320} a^{26} + \frac{3}{320} a^{25} + \frac{11}{320} a^{24} + \frac{1}{80} a^{23} + \frac{3}{160} a^{22} - \frac{7}{320} a^{21} + \frac{1}{10} a^{20} + \frac{13}{320} a^{19} - \frac{1}{320} a^{18} + \frac{33}{320} a^{17} - \frac{7}{80} a^{16} + \frac{3}{32} a^{15} + \frac{37}{320} a^{14} - \frac{1}{10} a^{13} + \frac{33}{320} a^{12} + \frac{1}{64} a^{11} + \frac{37}{320} a^{10} + \frac{13}{80} a^{9} + \frac{29}{160} a^{8} - \frac{21}{64} a^{7} + \frac{13}{40} a^{6} + \frac{51}{320} a^{5} - \frac{1}{20} a^{4} - \frac{89}{320} a^{3} + \frac{33}{80} a^{2} - \frac{7}{80} a - \frac{19}{40}$, $\frac{1}{314995840} a^{33} - \frac{228419}{157497920} a^{32} + \frac{207787}{157497920} a^{31} - \frac{3399}{157497920} a^{30} + \frac{443617}{78748960} a^{29} + \frac{175213}{31499584} a^{28} + \frac{226577}{157497920} a^{27} - \frac{320241}{19687240} a^{26} + \frac{4371749}{314995840} a^{25} + \frac{45179}{19687240} a^{24} - \frac{1030615}{31499584} a^{23} + \frac{2140897}{78748960} a^{22} - \frac{7538473}{314995840} a^{21} + \frac{3120271}{78748960} a^{20} + \frac{13641383}{314995840} a^{19} - \frac{2001751}{31499584} a^{18} - \frac{19853651}{314995840} a^{17} + \frac{2016489}{31499584} a^{16} + \frac{23725779}{157497920} a^{15} + \frac{3235909}{15749792} a^{14} + \frac{108629}{39374480} a^{13} - \frac{9125357}{78748960} a^{12} - \frac{9918409}{314995840} a^{11} - \frac{2038539}{157497920} a^{10} + \frac{7433917}{157497920} a^{9} - \frac{206511}{157497920} a^{8} - \frac{87950993}{314995840} a^{7} - \frac{31773747}{157497920} a^{6} - \frac{2799801}{15749792} a^{5} + \frac{17454569}{78748960} a^{4} + \frac{1987551}{62999168} a^{3} + \frac{3491915}{15749792} a^{2} + \frac{20847409}{78748960} a + \frac{12113517}{39374480}$, $\frac{1}{864978576640} a^{34} + \frac{113}{864978576640} a^{33} - \frac{83485273}{108122322080} a^{32} + \frac{352762381}{432489288320} a^{31} - \frac{12250341}{43248928832} a^{30} - \frac{55257901}{10812232208} a^{29} + \frac{966802223}{108122322080} a^{28} + \frac{5821654321}{216244644160} a^{27} - \frac{25042560383}{864978576640} a^{26} + \frac{12768888883}{864978576640} a^{25} - \frac{11126376673}{216244644160} a^{24} - \frac{4873231749}{86497857664} a^{23} - \frac{23021986617}{864978576640} a^{22} + \frac{6473697679}{172995715328} a^{21} + \frac{42478043609}{864978576640} a^{20} + \frac{67893854687}{864978576640} a^{19} + \frac{57045833473}{864978576640} a^{18} + \frac{36618051327}{864978576640} a^{17} + \frac{43840168749}{432489288320} a^{16} + \frac{21312352127}{216244644160} a^{15} + \frac{564599267}{2703058052} a^{14} - \frac{74446538933}{432489288320} a^{13} + \frac{4568682967}{172995715328} a^{12} + \frac{41495589627}{172995715328} a^{11} - \frac{67890257583}{432489288320} a^{10} - \frac{29702243791}{216244644160} a^{9} + \frac{28127470173}{172995715328} a^{8} - \frac{55081205983}{864978576640} a^{7} - \frac{101223537667}{216244644160} a^{6} - \frac{17462027127}{108122322080} a^{5} + \frac{87839945913}{864978576640} a^{4} + \frac{292845122221}{864978576640} a^{3} - \frac{11953822937}{54061161040} a^{2} - \frac{18892824607}{216244644160} a - \frac{1619548157}{21624464416}$, $\frac{1}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{35} - \frac{24832236397425341935828168347807928916907351032761072741289934876321043455781673490008383}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{34} - \frac{1834797815684579685857292900402910701790462044584425832441728369390217442756565994883021506747}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{33} + \frac{944966261723778609836203206918413865434760334767396070452692275618156969544093409396398258147296131}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{32} + \frac{617086860993960522806809953036135632226490361870117241143046572491439971019999054926424326262381507}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{31} + \frac{185737755634652640326749387602368374936088556151186296471058901891956227599692378055217321385511823}{533150930502678189135956923045449670563040968151656173500032898392752166159913384789313057791436798720} a^{30} - \frac{533431747936172501813069325584071217757297251524831996189625372723145595525908495010019947639884819}{106630186100535637827191384609089934112608193630331234700006579678550433231982676957862611558287359744} a^{29} - \frac{74873004855525581107137665831966032619345290941862380400370891711361334727065422452625411001848137}{13328773262566954728398923076136241764076024203791404337500822459818804153997834619732826444785919968} a^{28} + \frac{13420330657713470979892116273883404673899593851967887186498525906706872094903196481774377390730646193}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{27} - \frac{11184395742818644639165363983113264613482702143762622928340725877577319278656370235771483281261590923}{533150930502678189135956923045449670563040968151656173500032898392752166159913384789313057791436798720} a^{26} + \frac{47181936514751421534242284337799766832737277808948877115074048428165576761935945660176305188476713087}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{25} - \frac{51031245628992822127500436570280840940021943181806354217949826068570052404800841697772428899673387277}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{24} + \frac{78537462934688127971991269167878302380235665705266006126937334235335288790292264836639703723841001157}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{23} + \frac{13169570938583121795771275312110340180793654269557084651572469761066979941958903300595743519267766035}{213260372201071275654382769218179868225216387260662469400013159357100866463965353915725223116574719488} a^{22} - \frac{5233968426141498183066285565184333938238611408107269183673114658360560977385331199069686250604521633}{133287732625669547283989230761362417640760242037914043375008224598188041539978346197328264447859199680} a^{21} + \frac{603675242197501207067292558872210728979854074325958993288280541949376788519359032548969901286779357}{106630186100535637827191384609089934112608193630331234700006579678550433231982676957862611558287359744} a^{20} + \frac{2186601400584009955977310054084293550563571843737662762315680687414660274298038455293353929713600791}{133287732625669547283989230761362417640760242037914043375008224598188041539978346197328264447859199680} a^{19} + \frac{1474303088971143402393814195813768327487342346757937949484153594003104317588812345409816458702099277}{16660966578208693410498653845170302205095030254739255421876028074773505192497293274666033055982399960} a^{18} - \frac{235691249703785554122742542402597916768247014941311435169289802813728299737688343256052709091490066659}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{17} + \frac{78278082316319015675602480266462721657682668934917183991616501801204638131140029573263045653868628907}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{16} - \frac{15618833747714653009437804350119358365474191122691388143365076016969365508447187238259449806974925767}{66643866312834773641994615380681208820380121018957021687504112299094020769989173098664132223929599840} a^{15} + \frac{28135289147309000478828131039052076452750301257039649250056446524082284776309437937414554087962933037}{213260372201071275654382769218179868225216387260662469400013159357100866463965353915725223116574719488} a^{14} - \frac{64912342227896328876802149592165162903094911731789671266497136269384282235106614232846704157217703587}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{13} + \frac{77897864829522274632017281009780677857080386448910253410118221507443003021897945718052354266693338573}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{12} - \frac{520127058285458866277434830291638133151241362376605746567378146904301146755282437950620433913840171799}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{11} - \frac{172721072418744497639046830109463768698249433951117037561874787123270907330880646885665147475107079019}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{10} + \frac{432800531576746587618664378971735237859339099995777086973484643890581350068798592792597329835880623669}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{9} + \frac{47683485437751054142022730275772858838770069856787993816393645069987943097916027710532357902413531467}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{8} + \frac{824686540301142228855792369365952761013493874342188165682779319498767628275347192772937911134135433913}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{7} + \frac{1929452978064545865382942560895105897755509559476333871818433711313803544521632963266656584716751641}{16660966578208693410498653845170302205095030254739255421876028074773505192497293274666033055982399960} a^{6} - \frac{218205088148569822151202075881938622114712292818834697843761626227461122300765477307066019874370267947}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{5} - \frac{523256513444783007555007868651500507301436120768944823100583325632040282120393407379118875813402650431}{1066301861005356378271913846090899341126081936303312347000065796785504332319826769578626115582873597440} a^{4} - \frac{393851551530626993147001043264773170520560187746978780565794980459937071982232811114973452872993988247}{2132603722010712756543827692181798682252163872606624694000131593571008664639653539157252231165747194880} a^{3} + \frac{50462558344335638371541083589338210112277049979597753652669401639423581437575869758060704220752845985}{106630186100535637827191384609089934112608193630331234700006579678550433231982676957862611558287359744} a^{2} - \frac{64489574752535281482806322603407128420610002362837203690832707750901119990638261956478386892024517557}{533150930502678189135956923045449670563040968151656173500032898392752166159913384789313057791436798720} a + \frac{21859545646277317801045491937013312449615988371897453914366180975505499761213640981842607878163049671}{53315093050267818913595692304544967056304096815165617350003289839275216615991338478931305779143679872}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $35$
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:  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:  \( 837390418804361600000000000 \) (assuming GRH)
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

\(\Q(\sqrt{65}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 4.4.274625.1, 6.6.46411625.1, 6.6.111434311625.2, 6.6.111434311625.1, 6.6.659374625.1, 9.9.567869252041.1, 12.12.3500313269603515625.2, 12.12.20178609436923616478515625.3, 12.12.20178609436923616478515625.2, 12.12.119400055839784712890625.2, 18.18.1383747355171268959993166322265625.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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$5.12.9.3$x^{12} - 25 x^{4} + 250$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.3$x^{12} - 25 x^{4} + 250$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.3$x^{12} - 25 x^{4} + 250$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed
$13$13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$