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Properties

Label 36.36.3155227631...9488.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 3^{48}\cdot 13^{33}$
Root discriminant $90.85$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (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![1, -120, 1335, 8268, -162048, 371922, 3090085, -15365958, 3573618, 92545112, -116564358, -223846182, 440313718, 261081888, -806374134, -135354640, 884137635, -8009280, -635559496, 51886254, 314149779, -31916038, -109335657, 10295208, 26995916, -2006406, -4707315, 242208, 571011, -17640, -46837, 708, 2466, -12, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 75*x^34 - 12*x^33 + 2466*x^32 + 708*x^31 - 46837*x^30 - 17640*x^29 + 571011*x^28 + 242208*x^27 - 4707315*x^26 - 2006406*x^25 + 26995916*x^24 + 10295208*x^23 - 109335657*x^22 - 31916038*x^21 + 314149779*x^20 + 51886254*x^19 - 635559496*x^18 - 8009280*x^17 + 884137635*x^16 - 135354640*x^15 - 806374134*x^14 + 261081888*x^13 + 440313718*x^12 - 223846182*x^11 - 116564358*x^10 + 92545112*x^9 + 3573618*x^8 - 15365958*x^7 + 3090085*x^6 + 371922*x^5 - 162048*x^4 + 8268*x^3 + 1335*x^2 - 120*x + 1)
 
gp: K = bnfinit(x^36 - 75*x^34 - 12*x^33 + 2466*x^32 + 708*x^31 - 46837*x^30 - 17640*x^29 + 571011*x^28 + 242208*x^27 - 4707315*x^26 - 2006406*x^25 + 26995916*x^24 + 10295208*x^23 - 109335657*x^22 - 31916038*x^21 + 314149779*x^20 + 51886254*x^19 - 635559496*x^18 - 8009280*x^17 + 884137635*x^16 - 135354640*x^15 - 806374134*x^14 + 261081888*x^13 + 440313718*x^12 - 223846182*x^11 - 116564358*x^10 + 92545112*x^9 + 3573618*x^8 - 15365958*x^7 + 3090085*x^6 + 371922*x^5 - 162048*x^4 + 8268*x^3 + 1335*x^2 - 120*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 75 x^{34} - 12 x^{33} + 2466 x^{32} + 708 x^{31} - 46837 x^{30} - 17640 x^{29} + 571011 x^{28} + 242208 x^{27} - 4707315 x^{26} - 2006406 x^{25} + 26995916 x^{24} + 10295208 x^{23} - 109335657 x^{22} - 31916038 x^{21} + 314149779 x^{20} + 51886254 x^{19} - 635559496 x^{18} - 8009280 x^{17} + 884137635 x^{16} - 135354640 x^{15} - 806374134 x^{14} + 261081888 x^{13} + 440313718 x^{12} - 223846182 x^{11} - 116564358 x^{10} + 92545112 x^{9} + 3573618 x^{8} - 15365958 x^{7} + 3090085 x^{6} + 371922 x^{5} - 162048 x^{4} + 8268 x^{3} + 1335 x^{2} - 120 x + 1 \)

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:  \(31552276316229132122682846845993932976454945211245315153186594370879488=2^{36}\cdot 3^{48}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.85$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(133,·)$, $\chi_{468}(7,·)$, $\chi_{468}(271,·)$, $\chi_{468}(19,·)$, $\chi_{468}(277,·)$, $\chi_{468}(151,·)$, $\chi_{468}(25,·)$, $\chi_{468}(157,·)$, $\chi_{468}(31,·)$, $\chi_{468}(289,·)$, $\chi_{468}(163,·)$, $\chi_{468}(49,·)$, $\chi_{468}(427,·)$, $\chi_{468}(175,·)$, $\chi_{468}(433,·)$, $\chi_{468}(307,·)$, $\chi_{468}(181,·)$, $\chi_{468}(313,·)$, $\chi_{468}(187,·)$, $\chi_{468}(61,·)$, $\chi_{468}(319,·)$, $\chi_{468}(67,·)$, $\chi_{468}(331,·)$, $\chi_{468}(205,·)$, $\chi_{468}(463,·)$, $\chi_{468}(337,·)$, $\chi_{468}(343,·)$, $\chi_{468}(217,·)$, $\chi_{468}(223,·)$, $\chi_{468}(361,·)$, $\chi_{468}(445,·)$, $\chi_{468}(115,·)$, $\chi_{468}(373,·)$, $\chi_{468}(121,·)$, $\chi_{468}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{53} a^{31} - \frac{2}{53} a^{30} - \frac{17}{53} a^{29} - \frac{14}{53} a^{28} - \frac{12}{53} a^{27} + \frac{3}{53} a^{26} - \frac{7}{53} a^{25} - \frac{16}{53} a^{24} - \frac{16}{53} a^{23} - \frac{18}{53} a^{22} + \frac{23}{53} a^{21} + \frac{25}{53} a^{20} + \frac{3}{53} a^{19} + \frac{22}{53} a^{18} - \frac{22}{53} a^{17} + \frac{21}{53} a^{16} + \frac{5}{53} a^{15} + \frac{10}{53} a^{14} - \frac{25}{53} a^{13} - \frac{11}{53} a^{12} - \frac{8}{53} a^{11} + \frac{6}{53} a^{10} - \frac{19}{53} a^{9} + \frac{10}{53} a^{8} + \frac{7}{53} a^{7} - \frac{7}{53} a^{6} - \frac{10}{53} a^{5} + \frac{18}{53} a^{4} - \frac{11}{53} a^{3} - \frac{3}{53} a^{2} - \frac{26}{53} a - \frac{9}{53}$, $\frac{1}{53} a^{32} - \frac{21}{53} a^{30} + \frac{5}{53} a^{29} + \frac{13}{53} a^{28} - \frac{21}{53} a^{27} - \frac{1}{53} a^{26} + \frac{23}{53} a^{25} + \frac{5}{53} a^{24} + \frac{3}{53} a^{23} - \frac{13}{53} a^{22} + \frac{18}{53} a^{21} - \frac{25}{53} a^{19} + \frac{22}{53} a^{18} - \frac{23}{53} a^{17} - \frac{6}{53} a^{16} + \frac{20}{53} a^{15} - \frac{5}{53} a^{14} - \frac{8}{53} a^{13} + \frac{23}{53} a^{12} - \frac{10}{53} a^{11} - \frac{7}{53} a^{10} + \frac{25}{53} a^{9} - \frac{26}{53} a^{8} + \frac{7}{53} a^{7} - \frac{24}{53} a^{6} - \frac{2}{53} a^{5} + \frac{25}{53} a^{4} - \frac{25}{53} a^{3} + \frac{21}{53} a^{2} - \frac{8}{53} a - \frac{18}{53}$, $\frac{1}{53} a^{33} + \frac{16}{53} a^{30} - \frac{26}{53} a^{29} + \frac{3}{53} a^{28} + \frac{12}{53} a^{27} - \frac{20}{53} a^{26} + \frac{17}{53} a^{25} - \frac{15}{53} a^{24} + \frac{22}{53} a^{23} + \frac{11}{53} a^{22} + \frac{6}{53} a^{21} + \frac{23}{53} a^{20} - \frac{21}{53} a^{19} + \frac{15}{53} a^{18} + \frac{9}{53} a^{17} - \frac{16}{53} a^{16} - \frac{6}{53} a^{15} - \frac{10}{53} a^{14} - \frac{25}{53} a^{13} + \frac{24}{53} a^{12} - \frac{16}{53} a^{11} - \frac{8}{53} a^{10} - \frac{1}{53} a^{9} + \frac{5}{53} a^{8} + \frac{17}{53} a^{7} + \frac{10}{53} a^{6} - \frac{26}{53} a^{5} - \frac{18}{53} a^{4} + \frac{2}{53} a^{3} - \frac{18}{53} a^{2} + \frac{19}{53} a + \frac{23}{53}$, $\frac{1}{1572344661415527453963040669527414766291418597} a^{34} + \frac{48526588163216646687150207015592202643586}{29666880404066555735151710745800278609272049} a^{33} + \frac{10370876944257505228732102302922501800309425}{1572344661415527453963040669527414766291418597} a^{32} - \frac{11519645569641369526653958831556021671544543}{1572344661415527453963040669527414766291418597} a^{31} + \frac{527664966263306576286914426522169796805209174}{1572344661415527453963040669527414766291418597} a^{30} - \frac{5934436308647469916427725434261950155560990}{1572344661415527453963040669527414766291418597} a^{29} - \frac{706775477687853739435248051027422924659231414}{1572344661415527453963040669527414766291418597} a^{28} - \frac{540653641343079904198191494999757435092152979}{1572344661415527453963040669527414766291418597} a^{27} + \frac{632236446658626799338044762365933514900897786}{1572344661415527453963040669527414766291418597} a^{26} - \frac{193640681031448816135669897750237527753858770}{1572344661415527453963040669527414766291418597} a^{25} - \frac{402595926814146175457167879721404305918153145}{1572344661415527453963040669527414766291418597} a^{24} + \frac{308409185507351090157436210644764693200691167}{1572344661415527453963040669527414766291418597} a^{23} - \frac{526303309136790372416451452952404428880295758}{1572344661415527453963040669527414766291418597} a^{22} + \frac{545664153989779548190667834234092476203506242}{1572344661415527453963040669527414766291418597} a^{21} - \frac{679828706904219589160580041116597667792406348}{1572344661415527453963040669527414766291418597} a^{20} + \frac{526939590720470205866653991294620189224081898}{1572344661415527453963040669527414766291418597} a^{19} + \frac{7811560497949064051075338976557833803214155}{1572344661415527453963040669527414766291418597} a^{18} - \frac{772850678578635055567069092687295177053936916}{1572344661415527453963040669527414766291418597} a^{17} - \frac{408347955240202906770239905076696161249784130}{1572344661415527453963040669527414766291418597} a^{16} - \frac{540730394287286415582866463306069125207681613}{1572344661415527453963040669527414766291418597} a^{15} + \frac{627956585889555071206775761556959379564320358}{1572344661415527453963040669527414766291418597} a^{14} - \frac{322388198442545317764441523981948572500468283}{1572344661415527453963040669527414766291418597} a^{13} - \frac{254607744285628490742108131549212655347136205}{1572344661415527453963040669527414766291418597} a^{12} - \frac{545734718056494384396767635025845254238459207}{1572344661415527453963040669527414766291418597} a^{11} + \frac{556334124356600046475738436599349262918634172}{1572344661415527453963040669527414766291418597} a^{10} + \frac{469386707943630060685093332312077640797010331}{1572344661415527453963040669527414766291418597} a^{9} - \frac{10911882919870276586868048036621270679250907}{29666880404066555735151710745800278609272049} a^{8} - \frac{401157004821151396266384399640226930178335674}{1572344661415527453963040669527414766291418597} a^{7} - \frac{468316111538723570286426013734551409627829477}{1572344661415527453963040669527414766291418597} a^{6} + \frac{666124979900749495501960093085150288633586334}{1572344661415527453963040669527414766291418597} a^{5} - \frac{177022748224187841386166224529316504910006396}{1572344661415527453963040669527414766291418597} a^{4} + \frac{353651873503062139651923844649494702485116322}{1572344661415527453963040669527414766291418597} a^{3} - \frac{658339126232551370460795702393708985624200722}{1572344661415527453963040669527414766291418597} a^{2} - \frac{583483684527710448688155042898263257586507590}{1572344661415527453963040669527414766291418597} a + \frac{218833456073220786900103041319793263119151752}{1572344661415527453963040669527414766291418597}$, $\frac{1}{6476379368276691911251427721283389060717154412169968819} a^{35} - \frac{1809165661}{6476379368276691911251427721283389060717154412169968819} a^{34} - \frac{39977038999566678634661936553837070026633062466122631}{6476379368276691911251427721283389060717154412169968819} a^{33} - \frac{31481171751892824010120024528944116373400697548496209}{6476379368276691911251427721283389060717154412169968819} a^{32} - \frac{60511213946866087820343205725103708648536566438760904}{6476379368276691911251427721283389060717154412169968819} a^{31} + \frac{1631999655531895741808874637199666058456675484004076568}{6476379368276691911251427721283389060717154412169968819} a^{30} + \frac{1840038033923101165519047442635063348391651179420830082}{6476379368276691911251427721283389060717154412169968819} a^{29} - \frac{2428484332566052057375195165035550801171574678703229026}{6476379368276691911251427721283389060717154412169968819} a^{28} + \frac{1708524392758907435813202311922216431139149483367271581}{6476379368276691911251427721283389060717154412169968819} a^{27} - \frac{209061503079558095768417270280008251037728961089138311}{6476379368276691911251427721283389060717154412169968819} a^{26} - \frac{1834537834983710433336377863880175020490722451289442796}{6476379368276691911251427721283389060717154412169968819} a^{25} + \frac{2678083107784487867713541344897919808600224643639550928}{6476379368276691911251427721283389060717154412169968819} a^{24} - \frac{1991250937806381879781014473451407667362517596059989774}{6476379368276691911251427721283389060717154412169968819} a^{23} - \frac{2184466374768514187713006260615002825357328807535809805}{6476379368276691911251427721283389060717154412169968819} a^{22} + \frac{1204452036527078914957412545636062107138822877974949745}{6476379368276691911251427721283389060717154412169968819} a^{21} - \frac{695897157051872645580539019495852638356678449156208308}{6476379368276691911251427721283389060717154412169968819} a^{20} + \frac{2337005225350339546301650902532678310836493555739970788}{6476379368276691911251427721283389060717154412169968819} a^{19} + \frac{1791203095604290610166124216588949394023805953823261739}{6476379368276691911251427721283389060717154412169968819} a^{18} - \frac{613966846866516249021199094519254659072807599449686691}{6476379368276691911251427721283389060717154412169968819} a^{17} - \frac{1418147565853124765875789904969246553551301077549350676}{6476379368276691911251427721283389060717154412169968819} a^{16} - \frac{810551544157051517918842308664214456319029976740743473}{6476379368276691911251427721283389060717154412169968819} a^{15} + \frac{162305391234235732861434726091609264853124030836150296}{6476379368276691911251427721283389060717154412169968819} a^{14} + \frac{1882451372818504524478677922467463963817177825527264225}{6476379368276691911251427721283389060717154412169968819} a^{13} - \frac{1183216017190393401737140941114951457984169681405590628}{6476379368276691911251427721283389060717154412169968819} a^{12} + \frac{191154025821524028289104650839396345273205516363842471}{6476379368276691911251427721283389060717154412169968819} a^{11} - \frac{1323784966685351384391825313570838226258930238696149851}{6476379368276691911251427721283389060717154412169968819} a^{10} + \frac{810832098179016691479804988246135120249251169913937935}{6476379368276691911251427721283389060717154412169968819} a^{9} - \frac{2197433781602022758203230646912749699315066320443403491}{6476379368276691911251427721283389060717154412169968819} a^{8} + \frac{800857124021047389979030564772934325507693534937014033}{6476379368276691911251427721283389060717154412169968819} a^{7} - \frac{1379371749807232259932825116311550837888414608368019529}{6476379368276691911251427721283389060717154412169968819} a^{6} + \frac{158942331765576394219541859494075816155936606378569690}{6476379368276691911251427721283389060717154412169968819} a^{5} - \frac{3099667862953061587372858430966960157469148262025848789}{6476379368276691911251427721283389060717154412169968819} a^{4} - \frac{2442435937050264197454037743767991310943191534921551790}{6476379368276691911251427721283389060717154412169968819} a^{3} - \frac{1760744013814125502343008406311748742413511999423352302}{6476379368276691911251427721283389060717154412169968819} a^{2} + \frac{1885472856462125164283531874181211290747360899189769618}{6476379368276691911251427721283389060717154412169968819} a - \frac{1247027361762428041706840999889361748406434850640869009}{6476379368276691911251427721283389060717154412169968819}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1056313367466610800000000 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 3.3.13689.2, 4.4.35152.1, 6.6.14414517.1, 6.6.2436053373.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 9.9.2565164201769.1, 12.12.1869778640298827501568.1, 12.12.315992590210501847764992.1, \(\Q(\zeta_{52})^+\), 12.12.315992590210501847764992.2, 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
3Data not computed
13Data not computed