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Properties

Label 36.36.9115316922...8125.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 7^{30}\cdot 13^{24}$
Root discriminant $93.56$
Ramified primes $5, 7, 13$
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![2521, -505872, -1356543, 16680904, 12391295, -212313517, 110647669, 1122925954, -1371167792, -2690155249, 5110286208, 2679130010, -9335764683, -21924592, 9996118628, -2583494750, -6878941470, 2878146054, 3202795146, -1707996749, -1036003947, 651588327, 234314708, -169652114, -36431516, 30780881, 3669506, -3888989, -196141, 335261, -857, -18792, 853, 617, -50, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 50*x^34 + 617*x^33 + 853*x^32 - 18792*x^31 - 857*x^30 + 335261*x^29 - 196141*x^28 - 3888989*x^27 + 3669506*x^26 + 30780881*x^25 - 36431516*x^24 - 169652114*x^23 + 234314708*x^22 + 651588327*x^21 - 1036003947*x^20 - 1707996749*x^19 + 3202795146*x^18 + 2878146054*x^17 - 6878941470*x^16 - 2583494750*x^15 + 9996118628*x^14 - 21924592*x^13 - 9335764683*x^12 + 2679130010*x^11 + 5110286208*x^10 - 2690155249*x^9 - 1371167792*x^8 + 1122925954*x^7 + 110647669*x^6 - 212313517*x^5 + 12391295*x^4 + 16680904*x^3 - 1356543*x^2 - 505872*x + 2521)
 
gp: K = bnfinit(x^36 - 9*x^35 - 50*x^34 + 617*x^33 + 853*x^32 - 18792*x^31 - 857*x^30 + 335261*x^29 - 196141*x^28 - 3888989*x^27 + 3669506*x^26 + 30780881*x^25 - 36431516*x^24 - 169652114*x^23 + 234314708*x^22 + 651588327*x^21 - 1036003947*x^20 - 1707996749*x^19 + 3202795146*x^18 + 2878146054*x^17 - 6878941470*x^16 - 2583494750*x^15 + 9996118628*x^14 - 21924592*x^13 - 9335764683*x^12 + 2679130010*x^11 + 5110286208*x^10 - 2690155249*x^9 - 1371167792*x^8 + 1122925954*x^7 + 110647669*x^6 - 212313517*x^5 + 12391295*x^4 + 16680904*x^3 - 1356543*x^2 - 505872*x + 2521, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 50 x^{34} + 617 x^{33} + 853 x^{32} - 18792 x^{31} - 857 x^{30} + 335261 x^{29} - 196141 x^{28} - 3888989 x^{27} + 3669506 x^{26} + 30780881 x^{25} - 36431516 x^{24} - 169652114 x^{23} + 234314708 x^{22} + 651588327 x^{21} - 1036003947 x^{20} - 1707996749 x^{19} + 3202795146 x^{18} + 2878146054 x^{17} - 6878941470 x^{16} - 2583494750 x^{15} + 9996118628 x^{14} - 21924592 x^{13} - 9335764683 x^{12} + 2679130010 x^{11} + 5110286208 x^{10} - 2690155249 x^{9} - 1371167792 x^{8} + 1122925954 x^{7} + 110647669 x^{6} - 212313517 x^{5} + 12391295 x^{4} + 16680904 x^{3} - 1356543 x^{2} - 505872 x + 2521 \)

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:  \(91153169226034327635845833383424381414975980898647673428058624267578125=5^{27}\cdot 7^{30}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.56$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(3,·)$, $\chi_{455}(261,·)$, $\chi_{455}(9,·)$, $\chi_{455}(16,·)$, $\chi_{455}(274,·)$, $\chi_{455}(152,·)$, $\chi_{455}(27,·)$, $\chi_{455}(412,·)$, $\chi_{455}(29,·)$, $\chi_{455}(289,·)$, $\chi_{455}(157,·)$, $\chi_{455}(48,·)$, $\chi_{455}(178,·)$, $\chi_{455}(432,·)$, $\chi_{455}(313,·)$, $\chi_{455}(191,·)$, $\chi_{455}(68,·)$, $\chi_{455}(326,·)$, $\chi_{455}(328,·)$, $\chi_{455}(74,·)$, $\chi_{455}(204,·)$, $\chi_{455}(79,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(87,·)$, $\chi_{455}(222,·)$, $\chi_{455}(144,·)$, $\chi_{455}(354,·)$, $\chi_{455}(237,·)$, $\chi_{455}(367,·)$, $\chi_{455}(243,·)$, $\chi_{455}(118,·)$, $\chi_{455}(248,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{701} a^{34} - \frac{72}{701} a^{33} + \frac{276}{701} a^{32} + \frac{341}{701} a^{31} - \frac{3}{701} a^{30} - \frac{339}{701} a^{29} + \frac{183}{701} a^{28} - \frac{176}{701} a^{27} - \frac{20}{701} a^{26} + \frac{20}{701} a^{25} - \frac{7}{701} a^{24} + \frac{332}{701} a^{23} + \frac{297}{701} a^{22} - \frac{10}{701} a^{21} - \frac{7}{701} a^{20} + \frac{195}{701} a^{19} + \frac{108}{701} a^{18} - \frac{308}{701} a^{17} - \frac{204}{701} a^{16} - \frac{54}{701} a^{15} + \frac{283}{701} a^{14} + \frac{4}{701} a^{13} + \frac{248}{701} a^{12} - \frac{334}{701} a^{11} + \frac{220}{701} a^{10} - \frac{65}{701} a^{9} + \frac{124}{701} a^{8} - \frac{294}{701} a^{7} - \frac{75}{701} a^{6} - \frac{45}{701} a^{5} + \frac{57}{701} a^{4} - \frac{151}{701} a^{3} - \frac{110}{701} a^{2} - \frac{269}{701} a - \frac{246}{701}$, $\frac{1}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{35} - \frac{97187643016837663461867010721243375863013669539777454887929807884409427569947659774008493837795513501566747290657475356}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{34} - \frac{39317486045953912922026549759773339239138071096942734978989178838342015894194340274799412184689124254287487052536893986102}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{33} + \frac{99303814093569208097473550295970791512239568999028220062375177498948820838564144141668231036289502677934094806962007215764}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{32} + \frac{100149667020893760739812621349287077104691401173100000707812247385922301063948640577862289512355988526723221301499718430726}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{31} - \frac{3965125565371390930575225417085893340507000631925374377429126917829726624552548973925024236342279721020969454635425675233}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{30} + \frac{54733776248685502263118208981274108065380171409479347102685775205709199345133119340144465729019464241787911205237519852011}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{29} + \frac{70971847151548273984389099521238320949807042478144049284657338919495394064280365636371928251946976196009829894198677416545}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{28} - \frac{60111565163370698161660844113733775271879614891931563629077061666713030421648613038230806897389039112925800349496346565556}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{27} + \frac{106726371259578324489047162949181143859979422585823372227017432893307017479434548479546372351724901505264085440858823154397}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{26} - \frac{86376037569780150129453200423054566060526175239277161672650534142774413871438031678323142459353755144592544672713769661307}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{25} + \frac{84964821101770534227143816259240819562211092411446004029570903789514543451072617785242086672283443635677171971584713821603}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{24} + \frac{17879648500347898544791889661959626351196143709816245449665851912359623506850637903248440205406095619991194706177757208236}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{23} - \frac{47563863014164433032148312605837528622349647691499734916696899644814895695730659913024979134292623627312120505558639991955}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{22} + \frac{36922655229041941481314661144778305096638794467243730169005485429565659670004139351281696851047450903310999383231211203863}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{21} - \frac{82728689878964510453723494262600332553478375639529643281550153874846712673892488875450125812071759285909152433257128841941}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{20} - \frac{91528824576940902265672288929147787186801337862299825669788926183140203287010282640795319581807773147630543519741512744168}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{19} + \frac{10775235844008369089199409260464747849693780878205336438625879470164572090005487293619710997290924896942892979919671355791}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{18} - \frac{63005031506675944715499238926939933321520073331121717454851852549517627304873758930401019766244531064237531187815486105616}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{17} - \frac{47687715523706829806422509900884513565908311750472710644511539388588714788628248229421000639881330060132178809872196153772}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{16} + \frac{3986030992159746460289204560492314698515198448431287870572175906769440754915516665099158677691984587460988767580963057059}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{15} + \frac{55678326725947672862383836531191690101578642449512194389554321275623095944969975985815690272021451752628074370566768994264}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{14} - \frac{80638551229860688767639935151323572970124766991447727775809763867472385214060757975252239651969929069766042587574590137084}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{13} + \frac{3602425743765584528703764420712085039905350221362042975161409138474454516992971518574926523030827078292907044449063788837}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{12} + \frac{34506520352170061132504385827659342378573851746912303592146745075563051170078074848113812292187138650733136471738010606603}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{11} + \frac{32353739612500891938173706989594194646191726278823122728196526307676762065807048385358555954059404291165788372673731855757}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{10} - \frac{42815382000373469953654262903989118034521516415674100261755073075498429919611707439051316801594691050379625731848103827188}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{9} + \frac{55371890742139312862505188000032349127981587676691523163658146657271239228687068575183815316003249790207905218025929058080}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{8} + \frac{105008329495666200538558536932334700527064966173486457467150621292195961908349527705065681237796682209329688633093274352624}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{7} - \frac{69973761193342549661469339881443533813899824598937918427520607694347734328433211166589042980981122981667112771075208179775}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{6} + \frac{46012935923041159391048012670136281220235143262906817527331868364733662709669003691126925553215425291443853034555523906800}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{5} + \frac{66427996590030160731342204343096576218123020334763137008718272858167029992704182856619417917027352120407163594232358922193}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{4} - \frac{28825696789029219299295447461999343283573056640503963344149861021616157922091211231134306158854173156784564299932284219260}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{3} + \frac{78417831123228730152157072439790123655767890641866071319951628627901434835714710442305311401158497575796882668769555216387}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a^{2} - \frac{11045734440005822341367478487577581834755529646428584163678897107145573389335922909013750087183721499388542898495974188533}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041} a + \frac{23619217928608726122137374686810727185366362791490389249827028292099734664920649304990018707751730301831624980617059090678}{213523638801757753454390897558702645805357795376571303886997652321427289693036815631511515428013679579518618296905422944041}$

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:  \( 949930772610205500000000 \) (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

\(\Q(\sqrt{5}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 4.4.6125.1, 6.6.3570125.1, 6.6.300125.1, 6.6.8571870125.2, 6.6.8571870125.1, 9.9.567869252041.1, 12.12.187441217958845703125.1, \(\Q(\zeta_{35})^+\), 12.12.450046364319188533203125.1, 12.12.450046364319188533203125.2, 18.18.629834936354696841143908203125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed
$13$13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$