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Properties

Label 36.36.4272343582...6117.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $66.54$
Ramified primes $3, 7, 13$
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![1, 24, -216, -2741, 10022, 79647, -220627, -852134, 2090230, 4809468, -10663861, -16533426, 33119347, 37010766, -66860091, -55712580, 90928152, 57204830, -84897695, -40274564, 54933520, 19486869, -24747903, -6496035, 7778369, 1490905, -1701570, -233479, 256293, 24367, -25920, -1613, 1673, 61, -62, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 62*x^34 + 61*x^33 + 1673*x^32 - 1613*x^31 - 25920*x^30 + 24367*x^29 + 256293*x^28 - 233479*x^27 - 1701570*x^26 + 1490905*x^25 + 7778369*x^24 - 6496035*x^23 - 24747903*x^22 + 19486869*x^21 + 54933520*x^20 - 40274564*x^19 - 84897695*x^18 + 57204830*x^17 + 90928152*x^16 - 55712580*x^15 - 66860091*x^14 + 37010766*x^13 + 33119347*x^12 - 16533426*x^11 - 10663861*x^10 + 4809468*x^9 + 2090230*x^8 - 852134*x^7 - 220627*x^6 + 79647*x^5 + 10022*x^4 - 2741*x^3 - 216*x^2 + 24*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 62*x^34 + 61*x^33 + 1673*x^32 - 1613*x^31 - 25920*x^30 + 24367*x^29 + 256293*x^28 - 233479*x^27 - 1701570*x^26 + 1490905*x^25 + 7778369*x^24 - 6496035*x^23 - 24747903*x^22 + 19486869*x^21 + 54933520*x^20 - 40274564*x^19 - 84897695*x^18 + 57204830*x^17 + 90928152*x^16 - 55712580*x^15 - 66860091*x^14 + 37010766*x^13 + 33119347*x^12 - 16533426*x^11 - 10663861*x^10 + 4809468*x^9 + 2090230*x^8 - 852134*x^7 - 220627*x^6 + 79647*x^5 + 10022*x^4 - 2741*x^3 - 216*x^2 + 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 62 x^{34} + 61 x^{33} + 1673 x^{32} - 1613 x^{31} - 25920 x^{30} + 24367 x^{29} + 256293 x^{28} - 233479 x^{27} - 1701570 x^{26} + 1490905 x^{25} + 7778369 x^{24} - 6496035 x^{23} - 24747903 x^{22} + 19486869 x^{21} + 54933520 x^{20} - 40274564 x^{19} - 84897695 x^{18} + 57204830 x^{17} + 90928152 x^{16} - 55712580 x^{15} - 66860091 x^{14} + 37010766 x^{13} + 33119347 x^{12} - 16533426 x^{11} - 10663861 x^{10} + 4809468 x^{9} + 2090230 x^{8} - 852134 x^{7} - 220627 x^{6} + 79647 x^{5} + 10022 x^{4} - 2741 x^{3} - 216 x^{2} + 24 x + 1 \)

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:  \(427234358221042548307849539075210285658672906173431042163517056117=3^{18}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.54$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(2,·)$, $\chi_{273}(4,·)$, $\chi_{273}(8,·)$, $\chi_{273}(137,·)$, $\chi_{273}(128,·)$, $\chi_{273}(11,·)$, $\chi_{273}(142,·)$, $\chi_{273}(16,·)$, $\chi_{273}(149,·)$, $\chi_{273}(22,·)$, $\chi_{273}(25,·)$, $\chi_{273}(158,·)$, $\chi_{273}(32,·)$, $\chi_{273}(43,·)$, $\chi_{273}(44,·)$, $\chi_{273}(176,·)$, $\chi_{273}(50,·)$, $\chi_{273}(172,·)$, $\chi_{273}(64,·)$, $\chi_{273}(197,·)$, $\chi_{273}(71,·)$, $\chi_{273}(200,·)$, $\chi_{273}(205,·)$, $\chi_{273}(79,·)$, $\chi_{273}(211,·)$, $\chi_{273}(86,·)$, $\chi_{273}(88,·)$, $\chi_{273}(100,·)$, $\chi_{273}(235,·)$, $\chi_{273}(239,·)$, $\chi_{273}(242,·)$, $\chi_{273}(121,·)$, $\chi_{273}(254,·)$, $\chi_{273}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{3} a^{32} + \frac{1}{3} a^{30} + \frac{1}{3} a^{28} - \frac{1}{3} a^{26} + \frac{1}{3} a^{25} + \frac{1}{3} a^{23} - \frac{1}{3} a^{19} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{33} + \frac{1}{3} a^{31} + \frac{1}{3} a^{29} - \frac{1}{3} a^{27} + \frac{1}{3} a^{26} + \frac{1}{3} a^{24} - \frac{1}{3} a^{20} - \frac{1}{3} a^{19} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{34} + \frac{1}{9} a^{33} - \frac{1}{9} a^{32} + \frac{1}{9} a^{31} - \frac{4}{9} a^{30} + \frac{4}{9} a^{29} - \frac{1}{3} a^{28} + \frac{1}{3} a^{26} - \frac{4}{9} a^{25} + \frac{4}{9} a^{24} - \frac{2}{9} a^{23} + \frac{2}{9} a^{21} + \frac{4}{9} a^{20} - \frac{1}{3} a^{19} - \frac{2}{9} a^{17} + \frac{1}{3} a^{16} - \frac{2}{9} a^{15} + \frac{2}{9} a^{14} - \frac{1}{3} a^{13} + \frac{4}{9} a^{12} - \frac{4}{9} a^{11} - \frac{2}{9} a^{10} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{35} - \frac{2344547413816685980340047792014972486268055712057174740463770403814659694308251}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{34} + \frac{10273333258048115534959968536361150489270813476005728031220623846195391953154537}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{33} - \frac{3824820536516350762811468667620104894556985817093258733791624692986954449644215}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{32} - \frac{5333083084573566528495511087286363791067679122747258703890321568210400181937934}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{31} + \frac{4736216478176351505501614904748928910274877941300135196416497698408852494383475}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{30} - \frac{6637503002698340239069668565518524220442813991793445301084695516415217005460334}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{29} - \frac{2404866869614325466749659235985856547680153409526602838879220980542363758489798}{7246543291003768889392956102835587231021051114899173003468095180976252556880277} a^{28} + \frac{5580979283643912813361554941342337656067498021922052024741258232333400863266227}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{27} - \frac{6014002644561326297457221702393424046094015545545422622703940143099906573876534}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{26} + \frac{18694284069558587761734555824455555592023111100149237644650995269244011845926354}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{25} - \frac{8923864111196950190203595902181529592037067946527366300427178794184240383070687}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{24} + \frac{2954826304923347113120562917769772127618404083158475851027260521700073556514157}{7246543291003768889392956102835587231021051114899173003468095180976252556880277} a^{23} + \frac{4701843131303471126818913221092939332813670328885137847769204253479037928563866}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{22} - \frac{21568166055119328773524451363655983532935851352424637348548401037312156548237696}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{21} - \frac{7666737231839119405851060417490068382778360855588669896227223963637282241540940}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{20} - \frac{1668489464058546934084112000394043011583140941268167366918679921866717572971139}{7246543291003768889392956102835587231021051114899173003468095180976252556880277} a^{19} - \frac{21238378559886351052454597287838269106246287237709595896678141058524693811851525}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{18} + \frac{1710537684411284439219049335167232205549502476510613910945338196360896072135587}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{17} + \frac{2627785523774207787932264971162873293173651749081009133662960786114753259531698}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{16} - \frac{30347651140502773633943413965343431671600378108220719239673871122819584033931539}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{15} + \frac{10678322872390065091638190446110094430817990695646143821634770243146363672470190}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{14} - \frac{19411256803216550067712971150129854295091401159146405646562572878595729190550132}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{13} - \frac{5227127210662717316455320270569315812655690230970420482638840523844452994427262}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{12} - \frac{16159610912493231435633394372302808056572579546990744255500999990146635205104537}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{11} - \frac{1053656005881909604785104757316124381579183883869712808527775029988118259968850}{7246543291003768889392956102835587231021051114899173003468095180976252556880277} a^{10} - \frac{4635212183986521948712132037160048216769822311367331987650949805065291923399547}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{9} + \frac{7350243227201402290595773253845321206783488058802643061319736831777909950874846}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{8} - \frac{20252409871286943338477113930634020204559000328376576460683635921265687336000481}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{7} - \frac{11353248543117521259575424423249561503689803415698342957605150453916477326072141}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{6} + \frac{952448825667729637898166828487683897657907559740883642809342573801273921172507}{7246543291003768889392956102835587231021051114899173003468095180976252556880277} a^{5} + \frac{23785384671135491292970464532367968643052478398772566300132039633084974974812389}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{4} - \frac{851074541283446813747155584729434546408437308511268517358069780286117992252211}{21739629873011306668178868308506761693063153344697519010404285542928757670640831} a^{3} - \frac{17624808805749622685797567294959135536535252189375214661574388579073524053713110}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a^{2} - \frac{2752672236043256762074110505997982271672455437030075681994246448785157938050954}{65218889619033920004536604925520285079189460034092557031212856628786273011922493} a - \frac{798950920977298789542078715640956712883495754739350266038750409746519752921256}{7246543291003768889392956102835587231021051114899173003468095180976252556880277}$

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:  \( 2899632352845428700000 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 4.4.19773.1, \(\Q(\zeta_{13})^+\), 6.6.5274997.1, 6.6.891474493.1, 6.6.891474493.2, 9.9.567869252041.1, \(\Q(\zeta_{39})^+\), 12.12.44565832042087964517.1, 12.12.7531625615112866003373.1, 12.12.7531625615112866003373.2, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$