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Properties

Label 36.36.2401529537...3125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 5^{27}\cdot 19^{32}$
Root discriminant $79.33$
Ramified primes $3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 69, 222, -16958, -101999, 414379, 2601760, -3815281, -25143775, 19939192, 122941569, -73364329, -340300799, 194732689, 566539690, -346734395, -587870429, 399630129, 387680446, -299264731, -162961903, 148130790, 42405508, -49174374, -6014336, 11009448, 151385, -1651384, 100187, 162163, -18344, -9923, 1498, 341, -61, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 5*x^35 - 61*x^34 + 341*x^33 + 1498*x^32 - 9923*x^31 - 18344*x^30 + 162163*x^29 + 100187*x^28 - 1651384*x^27 + 151385*x^26 + 11009448*x^25 - 6014336*x^24 - 49174374*x^23 + 42405508*x^22 + 148130790*x^21 - 162961903*x^20 - 299264731*x^19 + 387680446*x^18 + 399630129*x^17 - 587870429*x^16 - 346734395*x^15 + 566539690*x^14 + 194732689*x^13 - 340300799*x^12 - 73364329*x^11 + 122941569*x^10 + 19939192*x^9 - 25143775*x^8 - 3815281*x^7 + 2601760*x^6 + 414379*x^5 - 101999*x^4 - 16958*x^3 + 222*x^2 + 69*x + 1)
 
gp: K = bnfinit(x^36 - 5*x^35 - 61*x^34 + 341*x^33 + 1498*x^32 - 9923*x^31 - 18344*x^30 + 162163*x^29 + 100187*x^28 - 1651384*x^27 + 151385*x^26 + 11009448*x^25 - 6014336*x^24 - 49174374*x^23 + 42405508*x^22 + 148130790*x^21 - 162961903*x^20 - 299264731*x^19 + 387680446*x^18 + 399630129*x^17 - 587870429*x^16 - 346734395*x^15 + 566539690*x^14 + 194732689*x^13 - 340300799*x^12 - 73364329*x^11 + 122941569*x^10 + 19939192*x^9 - 25143775*x^8 - 3815281*x^7 + 2601760*x^6 + 414379*x^5 - 101999*x^4 - 16958*x^3 + 222*x^2 + 69*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 5 x^{35} - 61 x^{34} + 341 x^{33} + 1498 x^{32} - 9923 x^{31} - 18344 x^{30} + 162163 x^{29} + 100187 x^{28} - 1651384 x^{27} + 151385 x^{26} + 11009448 x^{25} - 6014336 x^{24} - 49174374 x^{23} + 42405508 x^{22} + 148130790 x^{21} - 162961903 x^{20} - 299264731 x^{19} + 387680446 x^{18} + 399630129 x^{17} - 587870429 x^{16} - 346734395 x^{15} + 566539690 x^{14} + 194732689 x^{13} - 340300799 x^{12} - 73364329 x^{11} + 122941569 x^{10} + 19939192 x^{9} - 25143775 x^{8} - 3815281 x^{7} + 2601760 x^{6} + 414379 x^{5} - 101999 x^{4} - 16958 x^{3} + 222 x^{2} + 69 x + 1 \)

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:  \(240152953708250935530977810544721792914847414233751595020294189453125=3^{18}\cdot 5^{27}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.33$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(4,·)$, $\chi_{285}(263,·)$, $\chi_{285}(137,·)$, $\chi_{285}(139,·)$, $\chi_{285}(271,·)$, $\chi_{285}(272,·)$, $\chi_{285}(17,·)$, $\chi_{285}(23,·)$, $\chi_{285}(68,·)$, $\chi_{285}(158,·)$, $\chi_{285}(169,·)$, $\chi_{285}(47,·)$, $\chi_{285}(49,·)$, $\chi_{285}(182,·)$, $\chi_{285}(188,·)$, $\chi_{285}(61,·)$, $\chi_{285}(62,·)$, $\chi_{285}(64,·)$, $\chi_{285}(196,·)$, $\chi_{285}(197,·)$, $\chi_{285}(199,·)$, $\chi_{285}(77,·)$, $\chi_{285}(83,·)$, $\chi_{285}(214,·)$, $\chi_{285}(218,·)$, $\chi_{285}(92,·)$, $\chi_{285}(16,·)$, $\chi_{285}(226,·)$, $\chi_{285}(229,·)$, $\chi_{285}(233,·)$, $\chi_{285}(106,·)$, $\chi_{285}(244,·)$, $\chi_{285}(248,·)$, $\chi_{285}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{151} a^{32} + \frac{40}{151} a^{31} + \frac{74}{151} a^{30} - \frac{2}{151} a^{29} - \frac{25}{151} a^{28} - \frac{10}{151} a^{27} - \frac{56}{151} a^{26} - \frac{1}{151} a^{25} - \frac{50}{151} a^{24} - \frac{3}{151} a^{23} - \frac{70}{151} a^{22} - \frac{53}{151} a^{21} - \frac{75}{151} a^{20} + \frac{44}{151} a^{19} - \frac{48}{151} a^{18} + \frac{75}{151} a^{17} + \frac{15}{151} a^{16} + \frac{72}{151} a^{15} + \frac{28}{151} a^{14} - \frac{64}{151} a^{13} + \frac{69}{151} a^{12} + \frac{51}{151} a^{11} - \frac{52}{151} a^{10} + \frac{14}{151} a^{9} - \frac{10}{151} a^{8} + \frac{48}{151} a^{7} - \frac{1}{151} a^{6} - \frac{55}{151} a^{5} + \frac{29}{151} a^{4} - \frac{33}{151} a^{3} + \frac{15}{151} a^{2} + \frac{38}{151} a + \frac{15}{151}$, $\frac{1}{63269} a^{33} + \frac{107}{63269} a^{32} + \frac{6831}{63269} a^{31} - \frac{28113}{63269} a^{30} + \frac{31249}{63269} a^{29} + \frac{12660}{63269} a^{28} + \frac{22075}{63269} a^{27} - \frac{11605}{63269} a^{26} + \frac{14681}{63269} a^{25} + \frac{13559}{63269} a^{24} - \frac{24280}{63269} a^{23} - \frac{13350}{63269} a^{22} + \frac{16608}{63269} a^{21} - \frac{22648}{63269} a^{20} + \frac{1692}{63269} a^{19} - \frac{5557}{63269} a^{18} + \frac{11684}{63269} a^{17} - \frac{16288}{63269} a^{16} - \frac{17345}{63269} a^{15} + \frac{87}{419} a^{14} + \frac{3633}{63269} a^{13} - \frac{18731}{63269} a^{12} + \frac{25411}{63269} a^{11} - \frac{14191}{63269} a^{10} + \frac{1230}{63269} a^{9} - \frac{6511}{63269} a^{8} + \frac{7594}{63269} a^{7} + \frac{6220}{63269} a^{6} + \frac{30017}{63269} a^{5} + \frac{18520}{63269} a^{4} + \frac{30722}{63269} a^{3} + \frac{24599}{63269} a^{2} + \frac{18114}{63269} a - \frac{22702}{63269}$, $\frac{1}{63269} a^{34} - \frac{9}{63269} a^{32} - \frac{5249}{63269} a^{31} + \frac{27149}{63269} a^{30} + \frac{13056}{63269} a^{29} + \frac{7417}{63269} a^{28} - \frac{15498}{63269} a^{27} - \frac{13992}{63269} a^{26} + \frac{19808}{63269} a^{25} + \frac{2720}{63269} a^{24} - \frac{23246}{63269} a^{23} - \frac{16414}{63269} a^{22} - \frac{19373}{63269} a^{21} - \frac{8524}{63269} a^{20} + \frac{16195}{63269} a^{19} + \frac{5437}{63269} a^{18} + \frac{28234}{63269} a^{17} + \frac{23074}{63269} a^{16} - \frac{13515}{63269} a^{15} - \frac{7594}{63269} a^{14} - \frac{6479}{63269} a^{13} + \frac{6696}{63269} a^{12} - \frac{30618}{63269} a^{11} + \frac{14619}{63269} a^{10} - \frac{10326}{63269} a^{9} + \frac{25491}{63269} a^{8} - \frac{15685}{63269} a^{7} - \frac{7442}{63269} a^{6} - \frac{30268}{63269} a^{5} + \frac{17544}{63269} a^{4} + \frac{1774}{63269} a^{3} - \frac{14084}{63269} a^{2} - \frac{14226}{63269} a + \frac{30758}{63269}$, $\frac{1}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{35} + \frac{3261622492466089923685877849729112961615351669888648917275766618078366202470016158707964}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{34} - \frac{8961131179382633425624569877380126598119994251643759399022395999325699121281125148765888}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{33} - \frac{3727740399308115998749303697840899899681791560772144834381058714727910345043439111708932827}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{32} - \frac{695723327041299766672413919408177460773589906541939711267021931616225446668661214762469893670}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{31} + \frac{259550278154042273956793048441211692161104415243241914512463944793595966378152858656081059767}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{30} - \frac{282467158905066978742429229535639998055581902185710842560435664270710862184592365936022434517}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{29} - \frac{323256252866562588838792285766439452971981892332257335553324053364004882973038596021841690564}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{28} + \frac{95060121229254764471381288143342878094326884914455122099387924235143242415778331926082185416}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{27} - \frac{180086829838380812500679476144918083530870598007351616820579793358476961140981447370996118717}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{26} + \frac{147154506866074389913309986616327838232507297335341998603020673263162699575636802892694326069}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{25} - \frac{547760950505441303561125810206462962642616680794606877508908482985980365296915820371417411020}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{24} + \frac{379112066503719704131742451561719927100706493391858727030146183048119840190341855120557281861}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{23} + \frac{503499157461340233731882865973449616091014145817476444257304570908855143755731019891182342382}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{22} - \frac{488705528811071397997498938121990120483400813845502254343391051462250868904659758705217722387}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{21} + \frac{415295423501669537934793976502406532537314743306874561522394108378729263777162514879964529573}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{20} - \frac{329403173580951299288685606196630944481917196217751196967400897891858533896605119252151936558}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{19} - \frac{508391476820796953587475338040171752141763407706866676151935640271992419377354996466493096926}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{18} - \frac{602442790263569338680924532668245637945103467003037577436215718869462679164921509179262397880}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{17} - \frac{6209182728129143145533677958223151571112688798712606362325162939895851751596840175673800811}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{16} - \frac{380890937819508301251149704496268345284670212537683604689083712504391694059777380722415305207}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{15} - \frac{150256656408744530036129937091572204435359071791239733205251103526215499931127446795035287931}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{14} - \frac{554290886591868738763953277528471692007460348620244800119085558720650265311808746025143858315}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{13} - \frac{279030256665687335307665671370519602162357580478817939783794976586734606353021258467937951391}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{12} + \frac{207586424703334384326227480975930938710513400525766275450585279105475482539045286582284079289}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{11} - \frac{215844150237611618116930226139256336514681832760126080975079447084379530751253236004188573392}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{10} - \frac{716031164150746629810474864008507385777109944138116065413150334665852737499320781616975457799}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{9} - \frac{554300624270797566290764879597946322259432509042137487333160657460342553475786677814261745284}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{8} - \frac{253682206639636070459257504562889866589008248971192519456903732555495132141146895693298574766}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{7} - \frac{624681591222014819260753569023677026783953311705540923635116284569665956109882853667846048785}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{6} - \frac{283298092185772495710723162623992402796105263668632264815056653264811312745328627482514693109}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{5} - \frac{7611472206142621892646248199377569603338561234536925978658405207077421073344795917052946512}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{4} - \frac{722821451930318996288015414555532251532684630874015022358868064361411039578253794447722829127}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{3} - \frac{434039154860709000890822095956427478425138260946092667973659861678486996659486801516217959234}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559} a^{2} + \frac{2043513162814702739783030939612038585899341105656616008396033907932390008002608287136345130}{9793391636691934353634073821667541320658107344913533826830176503616621922864064159314890209} a - \frac{232178289123032329393564069205808889282159023015973161596655592975406520011949192090824514327}{1478802137140482087398745147071798739419374209081943607851356652046109910352473688056548421559}$

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:  \( 57472811629290726000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\zeta_{15})^+\), 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.24181674720486328125.1, 18.18.563362135874260093126953125.1

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 $36$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ R $36$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ $18^{2}$ $36$ $36$ $36$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
3Data not computed
5Data not computed
19Data not computed