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Properties

Label 36.36.1986567619...6352.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{54}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $108.66$
Ramified primes $2, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-771471, -2159532, 169180380, 652490564, -2116489012, -7464917976, 11547553650, 34634932836, -36970755407, -85597408092, 77381360212, 126071424492, -109205998641, -116627921592, 104690525460, 69170933872, -68587862405, -26088563508, 30988574904, 5836708324, -9735028662, -522495128, 2132039314, -95894188, -323192934, 39516048, 33144748, -6302752, -2182273, 576272, 80914, -31316, -927, 940, -36, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 - 36*x^34 + 940*x^33 - 927*x^32 - 31316*x^31 + 80914*x^30 + 576272*x^29 - 2182273*x^28 - 6302752*x^27 + 33144748*x^26 + 39516048*x^25 - 323192934*x^24 - 95894188*x^23 + 2132039314*x^22 - 522495128*x^21 - 9735028662*x^20 + 5836708324*x^19 + 30988574904*x^18 - 26088563508*x^17 - 68587862405*x^16 + 69170933872*x^15 + 104690525460*x^14 - 116627921592*x^13 - 109205998641*x^12 + 126071424492*x^11 + 77381360212*x^10 - 85597408092*x^9 - 36970755407*x^8 + 34634932836*x^7 + 11547553650*x^6 - 7464917976*x^5 - 2116489012*x^4 + 652490564*x^3 + 169180380*x^2 - 2159532*x - 771471)
 
gp: K = bnfinit(x^36 - 12*x^35 - 36*x^34 + 940*x^33 - 927*x^32 - 31316*x^31 + 80914*x^30 + 576272*x^29 - 2182273*x^28 - 6302752*x^27 + 33144748*x^26 + 39516048*x^25 - 323192934*x^24 - 95894188*x^23 + 2132039314*x^22 - 522495128*x^21 - 9735028662*x^20 + 5836708324*x^19 + 30988574904*x^18 - 26088563508*x^17 - 68587862405*x^16 + 69170933872*x^15 + 104690525460*x^14 - 116627921592*x^13 - 109205998641*x^12 + 126071424492*x^11 + 77381360212*x^10 - 85597408092*x^9 - 36970755407*x^8 + 34634932836*x^7 + 11547553650*x^6 - 7464917976*x^5 - 2116489012*x^4 + 652490564*x^3 + 169180380*x^2 - 2159532*x - 771471, 1)
 

Normalized defining polynomial

\( x^{36} - 12 x^{35} - 36 x^{34} + 940 x^{33} - 927 x^{32} - 31316 x^{31} + 80914 x^{30} + 576272 x^{29} - 2182273 x^{28} - 6302752 x^{27} + 33144748 x^{26} + 39516048 x^{25} - 323192934 x^{24} - 95894188 x^{23} + 2132039314 x^{22} - 522495128 x^{21} - 9735028662 x^{20} + 5836708324 x^{19} + 30988574904 x^{18} - 26088563508 x^{17} - 68587862405 x^{16} + 69170933872 x^{15} + 104690525460 x^{14} - 116627921592 x^{13} - 109205998641 x^{12} + 126071424492 x^{11} + 77381360212 x^{10} - 85597408092 x^{9} - 36970755407 x^{8} + 34634932836 x^{7} + 11547553650 x^{6} - 7464917976 x^{5} - 2116489012 x^{4} + 652490564 x^{3} + 169180380 x^{2} - 2159532 x - 771471 \)

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:  \(19865676195397711398725987827960270735894630080373118478444419023321956352=2^{54}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.66$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(515,·)$, $\chi_{728}(641,·)$, $\chi_{728}(9,·)$, $\chi_{728}(11,·)$, $\chi_{728}(529,·)$, $\chi_{728}(275,·)$, $\chi_{728}(25,·)$, $\chi_{728}(289,·)$, $\chi_{728}(291,·)$, $\chi_{728}(113,·)$, $\chi_{728}(683,·)$, $\chi_{728}(435,·)$, $\chi_{728}(393,·)$, $\chi_{728}(569,·)$, $\chi_{728}(123,·)$, $\chi_{728}(603,·)$, $\chi_{728}(67,·)$, $\chi_{728}(417,·)$, $\chi_{728}(331,·)$, $\chi_{728}(81,·)$, $\chi_{728}(163,·)$, $\chi_{728}(723,·)$, $\chi_{728}(121,·)$, $\chi_{728}(267,·)$, $\chi_{728}(219,·)$, $\chi_{728}(225,·)$, $\chi_{728}(99,·)$, $\chi_{728}(337,·)$, $\chi_{728}(361,·)$, $\chi_{728}(323,·)$, $\chi_{728}(625,·)$, $\chi_{728}(499,·)$, $\chi_{728}(233,·)$, $\chi_{728}(673,·)$, $\chi_{728}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} + \frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{5}{12} a^{5} - \frac{5}{12} a^{4} - \frac{5}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6} a$, $\frac{1}{24} a^{18} - \frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{12} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{8} a^{8} - \frac{5}{24} a^{6} - \frac{1}{2} a^{5} - \frac{7}{24} a^{4} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a + \frac{3}{8}$, $\frac{1}{24} a^{19} - \frac{1}{24} a^{17} - \frac{1}{12} a^{16} - \frac{1}{24} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} + \frac{1}{6} a^{8} + \frac{1}{8} a^{7} + \frac{1}{6} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} + \frac{5}{12} a^{2} - \frac{1}{24} a - \frac{1}{2}$, $\frac{1}{24} a^{20} - \frac{1}{24} a^{14} + \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{8} a^{10} + \frac{1}{6} a^{8} + \frac{1}{12} a^{6} - \frac{1}{3} a^{5} + \frac{1}{24} a^{4} + \frac{1}{6} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{24} a^{21} - \frac{1}{24} a^{15} + \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{8} a^{11} + \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{24} a^{5} - \frac{1}{3} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{24} a^{22} - \frac{1}{24} a^{16} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{8} a^{12} + \frac{1}{6} a^{10} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{24} a^{6} - \frac{1}{3} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{24} a^{23} - \frac{1}{24} a^{17} + \frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{8} a^{13} + \frac{1}{6} a^{11} - \frac{1}{4} a^{10} + \frac{1}{12} a^{9} + \frac{1}{6} a^{8} + \frac{1}{24} a^{7} + \frac{1}{6} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{48} a^{24} + \frac{5}{48} a^{16} + \frac{1}{12} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{24} a^{10} + \frac{1}{6} a^{9} + \frac{5}{24} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{5}{24} a^{4} - \frac{5}{12} a^{3} - \frac{1}{24} a^{2} + \frac{1}{3} a - \frac{7}{16}$, $\frac{1}{48} a^{25} + \frac{1}{48} a^{17} - \frac{1}{12} a^{16} + \frac{1}{24} a^{15} + \frac{1}{12} a^{14} + \frac{1}{24} a^{13} + \frac{1}{12} a^{12} - \frac{1}{24} a^{11} - \frac{1}{12} a^{10} - \frac{5}{24} a^{9} + \frac{1}{6} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{5}{12} a^{2} + \frac{23}{48} a$, $\frac{1}{48} a^{26} - \frac{1}{48} a^{18} + \frac{1}{12} a^{16} - \frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{24} a^{12} - \frac{1}{12} a^{11} + \frac{5}{24} a^{10} - \frac{1}{12} a^{9} + \frac{1}{24} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} + \frac{5}{12} a - \frac{1}{8}$, $\frac{1}{48} a^{27} - \frac{1}{48} a^{19} - \frac{1}{24} a^{13} - \frac{1}{24} a^{11} + \frac{1}{6} a^{10} - \frac{1}{8} a^{9} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} - \frac{1}{3} a^{4} + \frac{11}{48} a^{3} + \frac{1}{6} a^{2} - \frac{11}{24} a - \frac{1}{2}$, $\frac{1}{48} a^{28} - \frac{1}{48} a^{20} - \frac{1}{24} a^{14} - \frac{1}{24} a^{12} + \frac{1}{6} a^{11} - \frac{1}{8} a^{10} - \frac{1}{12} a^{8} + \frac{1}{12} a^{6} - \frac{1}{3} a^{5} + \frac{11}{48} a^{4} + \frac{1}{6} a^{3} - \frac{11}{24} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{29} - \frac{1}{48} a^{21} - \frac{1}{24} a^{15} - \frac{1}{24} a^{13} - \frac{1}{12} a^{12} - \frac{1}{8} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{11}{48} a^{5} - \frac{1}{3} a^{4} - \frac{11}{24} a^{3} + \frac{1}{4}$, $\frac{1}{96} a^{30} - \frac{1}{96} a^{28} - \frac{1}{96} a^{24} + \frac{1}{96} a^{22} + \frac{1}{96} a^{20} + \frac{1}{32} a^{16} + \frac{1}{12} a^{15} - \frac{1}{48} a^{14} - \frac{1}{12} a^{13} - \frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} + \frac{7}{48} a^{8} - \frac{19}{96} a^{6} - \frac{1}{6} a^{5} + \frac{7}{96} a^{4} - \frac{1}{4} a^{3} - \frac{5}{16} a^{2} + \frac{1}{12} a - \frac{5}{32}$, $\frac{1}{288} a^{31} - \frac{1}{96} a^{29} + \frac{1}{144} a^{27} - \frac{1}{144} a^{26} + \frac{1}{288} a^{25} + \frac{5}{288} a^{23} + \frac{1}{96} a^{21} + \frac{1}{72} a^{20} - \frac{1}{144} a^{19} + \frac{1}{144} a^{18} + \frac{1}{32} a^{17} + \frac{1}{36} a^{16} + \frac{5}{48} a^{15} + \frac{5}{72} a^{14} + \frac{1}{72} a^{13} + \frac{7}{72} a^{12} + \frac{5}{72} a^{11} + \frac{1}{18} a^{10} + \frac{11}{144} a^{9} - \frac{1}{24} a^{8} + \frac{1}{32} a^{7} - \frac{1}{9} a^{6} - \frac{13}{96} a^{5} + \frac{1}{8} a^{4} + \frac{5}{24} a^{3} + \frac{67}{144} a^{2} - \frac{13}{32} a - \frac{1}{2}$, $\frac{1}{288} a^{32} - \frac{1}{288} a^{28} - \frac{1}{144} a^{27} + \frac{1}{288} a^{26} + \frac{1}{144} a^{24} - \frac{1}{48} a^{22} + \frac{1}{72} a^{21} + \frac{1}{288} a^{20} + \frac{1}{144} a^{19} - \frac{1}{96} a^{18} + \frac{1}{36} a^{17} + \frac{5}{96} a^{16} - \frac{7}{72} a^{15} + \frac{5}{144} a^{14} + \frac{7}{72} a^{13} + \frac{5}{72} a^{12} + \frac{2}{9} a^{11} + \frac{11}{144} a^{10} - \frac{1}{24} a^{9} + \frac{7}{32} a^{8} + \frac{2}{9} a^{7} - \frac{1}{6} a^{6} - \frac{5}{24} a^{5} + \frac{43}{96} a^{4} + \frac{43}{144} a^{3} + \frac{23}{96} a^{2} + \frac{1}{6} a + \frac{7}{32}$, $\frac{1}{288} a^{33} - \frac{1}{288} a^{29} - \frac{1}{144} a^{28} + \frac{1}{288} a^{27} + \frac{1}{144} a^{25} - \frac{1}{48} a^{23} + \frac{1}{72} a^{22} + \frac{1}{288} a^{21} + \frac{1}{144} a^{20} - \frac{1}{96} a^{19} - \frac{1}{72} a^{18} - \frac{1}{32} a^{17} + \frac{1}{9} a^{16} - \frac{7}{144} a^{15} + \frac{1}{18} a^{14} - \frac{1}{72} a^{13} - \frac{1}{36} a^{12} - \frac{25}{144} a^{11} + \frac{1}{8} a^{10} - \frac{11}{96} a^{9} + \frac{13}{72} a^{8} - \frac{1}{6} a^{6} - \frac{15}{32} a^{5} + \frac{1}{144} a^{4} - \frac{1}{96} a^{3} + \frac{1}{4} a^{2} - \frac{9}{32} a - \frac{1}{8}$, $\frac{1}{4006046524541243907586031427291131110557910553292538656} a^{34} - \frac{351276775446683022522369446332994708196664091100373}{500755815567655488448253928411391388819738819161567332} a^{33} - \frac{2678741341045072480780806469061593384826727539175793}{2003023262270621953793015713645565555278955276646269328} a^{32} + \frac{4278772388377195810617002452090253123482871509431133}{4006046524541243907586031427291131110557910553292538656} a^{31} - \frac{894780171106022682813808951010421135477750897187123}{4006046524541243907586031427291131110557910553292538656} a^{30} + \frac{210817582520687656787364728231820473605903885140247}{445116280504582656398447936365681234506434505921393184} a^{29} + \frac{15436881469548473487567560791398263552077306236800911}{4006046524541243907586031427291131110557910553292538656} a^{28} + \frac{5265276425520246870608245546542796508868035114701831}{1001511631135310976896507856822782777639477638323134664} a^{27} + \frac{11192299438736854834651313932086614571331925325372113}{2003023262270621953793015713645565555278955276646269328} a^{26} - \frac{9312956258902965758070611255917168080591038813691973}{1335348841513747969195343809097043703519303517764179552} a^{25} + \frac{320407665952137917744668538247924074306251589546003}{250377907783827744224126964205695694409869409580783666} a^{24} - \frac{4206986959947682947497159095117631016302983314430841}{1335348841513747969195343809097043703519303517764179552} a^{23} + \frac{50914775715139770838261948869976176493668677397431075}{4006046524541243907586031427291131110557910553292538656} a^{22} - \frac{19388467395497503139355496583115060545458331474410403}{4006046524541243907586031427291131110557910553292538656} a^{21} + \frac{64302870603224215090065070050316962834057928595252575}{4006046524541243907586031427291131110557910553292538656} a^{20} - \frac{232650616995773310300527071431749592830062840249657}{250377907783827744224126964205695694409869409580783666} a^{19} - \frac{13189449174226533092521109654327648905577023074535721}{4006046524541243907586031427291131110557910553292538656} a^{18} + \frac{36606027803266628920629594001620180563935349995317025}{4006046524541243907586031427291131110557910553292538656} a^{17} + \frac{175525003996501651094754492505818206802943738720639885}{2003023262270621953793015713645565555278955276646269328} a^{16} - \frac{181960142754008642291885103916460213714704395030491539}{2003023262270621953793015713645565555278955276646269328} a^{15} - \frac{34016655594425738852232084645066528238801568879197243}{333837210378436992298835952274260925879825879441044888} a^{14} - \frac{14491571669233527375721117927995367907137968330829721}{125188953891913872112063482102847847204934704790391833} a^{13} + \frac{65860456953137330648178021959331042439954314267656347}{2003023262270621953793015713645565555278955276646269328} a^{12} - \frac{23882518001014530389203748973282427406513915241599830}{125188953891913872112063482102847847204934704790391833} a^{11} + \frac{976775482176013308791071402311010791164800491685156715}{4006046524541243907586031427291131110557910553292538656} a^{10} + \frac{378731195153557828597475227912886863291533384575874535}{2003023262270621953793015713645565555278955276646269328} a^{9} - \frac{63723460450191670314218165098561992630976533044599285}{2003023262270621953793015713645565555278955276646269328} a^{8} + \frac{311775883935384060403549720688266168681526517185105}{2447187858607968178122193907935938369308436501705888} a^{7} + \frac{213194522041838196505697192998066633238833776987500395}{4006046524541243907586031427291131110557910553292538656} a^{6} - \frac{794045805323446770407554596880512075915442596428451061}{4006046524541243907586031427291131110557910553292538656} a^{5} - \frac{1860336002615780003748923260695382711434292444706687593}{4006046524541243907586031427291131110557910553292538656} a^{4} + \frac{822885668048371864289865151806410609966391788632286395}{2003023262270621953793015713645565555278955276646269328} a^{3} + \frac{57556733108301158242282704662908247746342875930977123}{4006046524541243907586031427291131110557910553292538656} a^{2} + \frac{462586048761103490095020856103968149701118131486297221}{1335348841513747969195343809097043703519303517764179552} a + \frac{16016682001642632861256263725763480880915894683514839}{55639535063072832049805992045710154313304313240174148}$, $\frac{1}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{35} + \frac{520275544679359}{4476656913678613618906031821896665908854550607191149676863319758515472} a^{34} + \frac{12767164099821666828770110293331799153784365904436272676728289275029}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{33} + \frac{13876100096815164707501302264730333287346903681541566403450171614279}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{32} - \frac{686183546640193180435110182136409068113328174319823079406195222903}{746109485613102269817671970316110984809091767865191612810553293085912} a^{31} - \frac{8495657047570594236092306648378714453540171558957325452544313064227}{2238328456839306809453015910948332954427275303595574838431659879257736} a^{30} - \frac{66231276874803897622194651081445103651409789818347652785101734370095}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{29} + \frac{14737313576614429604203394451062932301456976713459010773749274962661}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{28} - \frac{14351324304474511059897770919965798228500519991369953261332494467201}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{27} + \frac{53215527123932348952137237992760974479850541337808086303862573431877}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{26} + \frac{82095754407982147324346721076705510810706844357113789601696668364605}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{25} + \frac{2419123443432625739511690591296089566017370103354074822487713413}{683667824324773002276425140790572069159216647402435808928423909364} a^{24} + \frac{20071492641101738004933952533213908516497705476885471994198650327961}{1119164228419653404726507955474166477213637651797787419215829939628868} a^{23} - \frac{2404235174301332831741794764949064087068533974487529482045284104753}{497406323742068179878447980210740656539394511910127741873702195390608} a^{22} + \frac{126506294672400427309576396743800829699037386345935587955942557572677}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{21} - \frac{15338393213086318196101908512282273643996343566965216989985797403085}{994812647484136359756895960421481313078789023820255483747404390781216} a^{20} - \frac{6710812781140142876705273537000204059211299706774897741911863788867}{373054742806551134908835985158055492404545883932595806405276646542956} a^{19} + \frac{103174730571687306607860325756865793952186768488365316617373037869683}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{18} + \frac{17759335760124399863317705323026146132823790127785926657102258992519}{1492218971226204539635343940632221969618183535730383225621106586171824} a^{17} - \frac{86888569873841347324179208937368810470253020020900074979692284360693}{994812647484136359756895960421481313078789023820255483747404390781216} a^{16} - \frac{146300550670122236927531526016254064623293241536309966158914869315279}{2238328456839306809453015910948332954427275303595574838431659879257736} a^{15} + \frac{419343843753427214300405712088547649176429275294007364441249267906079}{4476656913678613618906031821896665908854550607191149676863319758515472} a^{14} + \frac{57922707443685216364117182954516020093818356521767657040285706263591}{4476656913678613618906031821896665908854550607191149676863319758515472} a^{13} + \frac{5251782909599121766391863635429217383447170211553260849986049109877}{746109485613102269817671970316110984809091767865191612810553293085912} a^{12} + \frac{483724082855897199269468564556541108789045942733030282054322121765991}{2984437942452409079270687881264443939236367071460766451242213172343648} a^{11} - \frac{399493177005308749440612815923823325377647859299116943112341405947977}{2238328456839306809453015910948332954427275303595574838431659879257736} a^{10} + \frac{739203346500528916148573331928633469150953404674103772225196804257367}{2984437942452409079270687881264443939236367071460766451242213172343648} a^{9} - \frac{1047872324359550764416369876682890820673029374945881701895039264047327}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{8} - \frac{925570581063812966813714081550309818960039351051896678874730845878175}{4476656913678613618906031821896665908854550607191149676863319758515472} a^{7} + \frac{63382370369153753782853027758400650708217836290581603044333174995539}{373054742806551134908835985158055492404545883932595806405276646542956} a^{6} - \frac{376396117029588586903600326089258668769202792509885642291564144989577}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{5} - \frac{1734272929247532417255159510059157679474153751130025998180088295737865}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{4} - \frac{1724608950155464534012962879939438418419663427648790839068660054292915}{4476656913678613618906031821896665908854550607191149676863319758515472} a^{3} - \frac{4088388158670090708247901630330363414649564666188276153483160473350405}{8953313827357227237812063643793331817709101214382299353726639517030944} a^{2} + \frac{246095772871464139107812892906151495992276740315434994340245106474683}{1492218971226204539635343940632221969618183535730383225621106586171824} a + \frac{184947444198002372684930169363232811294589574084589413549859761875315}{994812647484136359756895960421481313078789023820255483747404390781216}$

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, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 4.4.140608.1, \(\Q(\zeta_{13})^+\), 6.6.891474493.2, 6.6.5274997.1, 6.6.891474493.1, 9.9.567869252041.1, 12.12.469804094334435328.1, 12.12.2708327112823247113289728.1, 12.12.16025604217889036173312.1, 12.12.2708327112823247113289728.2, 18.18.708478645847689707516501157.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$