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Properties

Label 36.0.61987675026...8125.1
Degree $36$
Signature $[0, 18]$
Discriminant $5^{27}\cdot 19^{32}$
Root discriminant $45.80$
Ramified primes $5, 19$
Class number $1417$ (GRH)
Class group $[1417]$ (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, -5, 35, -205, 1260, 509, 5542, 4893, 26744, -9196, 59771, -31899, 111183, -26351, 122457, -20386, 114806, -31001, 84375, -24437, 52298, -12149, 24829, -5625, 10339, -2571, 3655, -836, 1100, -206, 257, -49, 54, -10, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 9*x^34 - 10*x^33 + 54*x^32 - 49*x^31 + 257*x^30 - 206*x^29 + 1100*x^28 - 836*x^27 + 3655*x^26 - 2571*x^25 + 10339*x^24 - 5625*x^23 + 24829*x^22 - 12149*x^21 + 52298*x^20 - 24437*x^19 + 84375*x^18 - 31001*x^17 + 114806*x^16 - 20386*x^15 + 122457*x^14 - 26351*x^13 + 111183*x^12 - 31899*x^11 + 59771*x^10 - 9196*x^9 + 26744*x^8 + 4893*x^7 + 5542*x^6 + 509*x^5 + 1260*x^4 - 205*x^3 + 35*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 9*x^34 - 10*x^33 + 54*x^32 - 49*x^31 + 257*x^30 - 206*x^29 + 1100*x^28 - 836*x^27 + 3655*x^26 - 2571*x^25 + 10339*x^24 - 5625*x^23 + 24829*x^22 - 12149*x^21 + 52298*x^20 - 24437*x^19 + 84375*x^18 - 31001*x^17 + 114806*x^16 - 20386*x^15 + 122457*x^14 - 26351*x^13 + 111183*x^12 - 31899*x^11 + 59771*x^10 - 9196*x^9 + 26744*x^8 + 4893*x^7 + 5542*x^6 + 509*x^5 + 1260*x^4 - 205*x^3 + 35*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 9 x^{34} - 10 x^{33} + 54 x^{32} - 49 x^{31} + 257 x^{30} - 206 x^{29} + 1100 x^{28} - 836 x^{27} + 3655 x^{26} - 2571 x^{25} + 10339 x^{24} - 5625 x^{23} + 24829 x^{22} - 12149 x^{21} + 52298 x^{20} - 24437 x^{19} + 84375 x^{18} - 31001 x^{17} + 114806 x^{16} - 20386 x^{15} + 122457 x^{14} - 26351 x^{13} + 111183 x^{12} - 31899 x^{11} + 59771 x^{10} - 9196 x^{9} + 26744 x^{8} + 4893 x^{7} + 5542 x^{6} + 509 x^{5} + 1260 x^{4} - 205 x^{3} + 35 x^{2} - 5 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(619876750267203693326033178758188478035934269428253173828125=5^{27}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.80$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(7,·)$, $\chi_{95}(9,·)$, $\chi_{95}(11,·)$, $\chi_{95}(16,·)$, $\chi_{95}(17,·)$, $\chi_{95}(23,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(28,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(42,·)$, $\chi_{95}(43,·)$, $\chi_{95}(44,·)$, $\chi_{95}(47,·)$, $\chi_{95}(49,·)$, $\chi_{95}(54,·)$, $\chi_{95}(58,·)$, $\chi_{95}(61,·)$, $\chi_{95}(62,·)$, $\chi_{95}(63,·)$, $\chi_{95}(64,·)$, $\chi_{95}(66,·)$, $\chi_{95}(68,·)$, $\chi_{95}(73,·)$, $\chi_{95}(74,·)$, $\chi_{95}(77,·)$, $\chi_{95}(81,·)$, $\chi_{95}(82,·)$, $\chi_{95}(83,·)$, $\chi_{95}(87,·)$, $\chi_{95}(92,·)$, $\chi_{95}(93,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{698211859637920482850137709081} a^{33} + \frac{131012903322039757274846435495}{698211859637920482850137709081} a^{32} - \frac{267695394468470117653961323780}{698211859637920482850137709081} a^{31} - \frac{80625098231052789847542610415}{698211859637920482850137709081} a^{30} + \frac{252575985013411115240052486781}{698211859637920482850137709081} a^{29} + \frac{10468678174382878506170015602}{698211859637920482850137709081} a^{28} + \frac{342353488833241380300367524138}{698211859637920482850137709081} a^{27} - \frac{163566370653597707230342200511}{698211859637920482850137709081} a^{26} - \frac{275340021959194073531288720435}{698211859637920482850137709081} a^{25} + \frac{90611491801064452047583332271}{698211859637920482850137709081} a^{24} - \frac{297278389112878037824160681409}{698211859637920482850137709081} a^{23} - \frac{299451745603226320678618182160}{698211859637920482850137709081} a^{22} - \frac{102416411745038510064777111806}{698211859637920482850137709081} a^{21} - \frac{204963597016465110986870492891}{698211859637920482850137709081} a^{20} + \frac{290562206506801304876123689555}{698211859637920482850137709081} a^{19} - \frac{71294545747504553228287180996}{698211859637920482850137709081} a^{18} - \frac{8482051300718287227115611535}{698211859637920482850137709081} a^{17} - \frac{6033508107422336892850718228}{698211859637920482850137709081} a^{16} - \frac{253183744717209413624751420450}{698211859637920482850137709081} a^{15} + \frac{239301479938985419027621849530}{698211859637920482850137709081} a^{14} - \frac{235976608701145195494808508602}{698211859637920482850137709081} a^{13} - \frac{122106253073649944229224770822}{698211859637920482850137709081} a^{12} + \frac{65364596430874290212909506218}{698211859637920482850137709081} a^{11} + \frac{128065215013971940518813223613}{698211859637920482850137709081} a^{10} + \frac{85060353216947836597696269946}{698211859637920482850137709081} a^{9} + \frac{31520344186828682695857384318}{698211859637920482850137709081} a^{8} + \frac{230137718709906456440193514633}{698211859637920482850137709081} a^{7} + \frac{30722139322867759560906105624}{698211859637920482850137709081} a^{6} + \frac{48799197984348397760324497306}{698211859637920482850137709081} a^{5} - \frac{257703597779627279724704527085}{698211859637920482850137709081} a^{4} + \frac{137838274359802381509428694098}{698211859637920482850137709081} a^{3} + \frac{218895832888364970075200049153}{698211859637920482850137709081} a^{2} - \frac{23951982212856891938289287853}{698211859637920482850137709081} a + \frac{317966013919781976560964289098}{698211859637920482850137709081}$, $\frac{1}{698211859637920482850137709081} a^{34} - \frac{119407484671173865137392703427}{698211859637920482850137709081} a^{32} - \frac{248478702421176319232058468573}{698211859637920482850137709081} a^{31} - \frac{8569315310294119016945449783}{698211859637920482850137709081} a^{30} - \frac{74775955230673383325754261556}{698211859637920482850137709081} a^{29} + \frac{23705137231795057675856571112}{698211859637920482850137709081} a^{28} + \frac{320137855501273434092156694307}{698211859637920482850137709081} a^{27} - \frac{67755538570947584212024516752}{698211859637920482850137709081} a^{26} - \frac{273596043157170774539525679156}{698211859637920482850137709081} a^{25} - \frac{13744090401501551926067257654}{698211859637920482850137709081} a^{24} - \frac{24974768257345850717527415104}{698211859637920482850137709081} a^{23} + \frac{34659141261623362958766506530}{698211859637920482850137709081} a^{22} - \frac{178298341401245840060711570311}{698211859637920482850137709081} a^{21} + \frac{191925955406102403224757947194}{698211859637920482850137709081} a^{20} - \frac{66144256395533447641506598055}{698211859637920482850137709081} a^{19} - \frac{224860596596469675660441377765}{698211859637920482850137709081} a^{18} + \frac{81853383864793960085740966511}{698211859637920482850137709081} a^{17} - \frac{155402877409807787378493738522}{698211859637920482850137709081} a^{16} + \frac{251283218750449587229840842402}{698211859637920482850137709081} a^{15} + \frac{157417298890673208176222777358}{698211859637920482850137709081} a^{14} + \frac{122569346825627569812118589382}{698211859637920482850137709081} a^{13} + \frac{283725587182929717561354173053}{698211859637920482850137709081} a^{12} - \frac{12086234340061547058013946272}{698211859637920482850137709081} a^{11} - \frac{146728651082879844732401159061}{698211859637920482850137709081} a^{10} - \frac{297329812812391587984115494286}{698211859637920482850137709081} a^{9} - \frac{294155173343552407748933608526}{698211859637920482850137709081} a^{8} - \frac{325542153364207924942809080904}{698211859637920482850137709081} a^{7} + \frac{161579280411242806168067745582}{698211859637920482850137709081} a^{6} - \frac{140574610520460726885287478401}{698211859637920482850137709081} a^{5} + \frac{16109256241553530922050027146}{698211859637920482850137709081} a^{4} - \frac{180569528629678416722613252599}{698211859637920482850137709081} a^{3} + \frac{51628200782795446143575802529}{698211859637920482850137709081} a^{2} - \frac{15668085255167470105861214489}{698211859637920482850137709081} a - \frac{113545916594682554626066724331}{698211859637920482850137709081}$, $\frac{1}{698211859637920482850137709081} a^{35} - \frac{222372925298093539824950833128}{698211859637920482850137709081} a^{32} + \frac{189128887271399901391655969424}{698211859637920482850137709081} a^{31} + \frac{74012592232591404634116617705}{698211859637920482850137709081} a^{30} - \frac{170102668173068796200558569264}{698211859637920482850137709081} a^{29} + \frac{193904028014533442237944702360}{698211859637920482850137709081} a^{28} - \frac{266265884981512768510481450329}{698211859637920482850137709081} a^{27} - \frac{310265248669234771947901422475}{698211859637920482850137709081} a^{26} + \frac{214478213929187085277414217532}{698211859637920482850137709081} a^{25} - \frac{53821394123402713283686047688}{698211859637920482850137709081} a^{24} - \frac{187793896872875896953077586091}{698211859637920482850137709081} a^{23} + \frac{277254377149902113430296490468}{698211859637920482850137709081} a^{22} + \frac{33002305598496792659229865626}{698211859637920482850137709081} a^{21} + \frac{261811260149939480702228212157}{698211859637920482850137709081} a^{20} - \frac{84185115741363942570211305282}{698211859637920482850137709081} a^{19} - \frac{296653583511777372941478044507}{698211859637920482850137709081} a^{18} + \frac{75651975046443749923506122841}{698211859637920482850137709081} a^{17} - \frac{190789140554927301487609426307}{698211859637920482850137709081} a^{16} + \frac{28722743072033092594284929289}{698211859637920482850137709081} a^{15} + \frac{103027779309382459161443547929}{698211859637920482850137709081} a^{14} + \frac{23972033068915069247214997232}{698211859637920482850137709081} a^{13} - \frac{122769122419943756419787349815}{698211859637920482850137709081} a^{12} + \frac{8076239671350273586762336435}{698211859637920482850137709081} a^{11} - \frac{8425860296940468485236962620}{698211859637920482850137709081} a^{10} - \frac{48198492663849875615496504013}{698211859637920482850137709081} a^{9} - \frac{205726959274294077746349417959}{698211859637920482850137709081} a^{8} - \frac{44403857525766507210283348038}{698211859637920482850137709081} a^{7} + \frac{27745913724128503447124440645}{698211859637920482850137709081} a^{6} - \frac{170322726123929789859026284150}{698211859637920482850137709081} a^{5} - \frac{250264230499287400316110079242}{698211859637920482850137709081} a^{4} + \frac{276335053969016920603316021554}{698211859637920482850137709081} a^{3} - \frac{137074070689471358527594115962}{698211859637920482850137709081} a^{2} + \frac{171871334791934800232113520}{1222787845250298568914426811} a + \frac{224147551739256680263320867998}{698211859637920482850137709081}$

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

Class group and class number

$C_{1417}$, which has order $1417$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{98018614603376248755638426887}{698211859637920482850137709081} a^{35} - \frac{77957495843925714929975481484}{698211859637920482850137709081} a^{34} + \frac{862411637926344248365493007986}{698211859637920482850137709081} a^{33} - \frac{800350758196794064491879104033}{698211859637920482850137709081} a^{32} + \frac{5095354822265598531637471164998}{698211859637920482850137709081} a^{31} - \frac{3726043831537884794738635959811}{698211859637920482850137709081} a^{30} + \frac{24226493356547394782218122554116}{698211859637920482850137709081} a^{29} - \frac{15070265556555738508634675258685}{698211859637920482850137709081} a^{28} + \frac{103774749969720191272627890150758}{698211859637920482850137709081} a^{27} - \frac{60029068712046560540659652121352}{698211859637920482850137709081} a^{26} + \frac{341851612176577391061422317593523}{698211859637920482850137709081} a^{25} - \frac{179320454596751802972625993660648}{698211859637920482850137709081} a^{24} + \frac{963058691935447486156170852747226}{698211859637920482850137709081} a^{23} - \frac{345944131698702167858601037752092}{698211859637920482850137709081} a^{22} + \frac{2324267885866910053028056340284198}{698211859637920482850137709081} a^{21} - \frac{697794779012632487408312034781440}{698211859637920482850137709081} a^{20} + \frac{4890058057468305769685686441449209}{698211859637920482850137709081} a^{19} - \frac{1357823927921002608813853637536163}{698211859637920482850137709081} a^{18} + \frac{7795633891756236022988070847970870}{698211859637920482850137709081} a^{17} - \frac{1369960091523807512821314650702732}{698211859637920482850137709081} a^{16} + \frac{10655621153439239851104224729587305}{698211859637920482850137709081} a^{15} + \frac{271057445897344586221999627297660}{698211859637920482850137709081} a^{14} + \frac{11623160278901377609099406021338138}{698211859637920482850137709081} a^{13} - \frac{164947979001725573592594889658194}{698211859637920482850137709081} a^{12} + \frac{10390321101751810372790451390091558}{698211859637920482850137709081} a^{11} - \frac{939070558154886093810413548555968}{698211859637920482850137709081} a^{10} + \frac{5238814762249825672048685538080554}{698211859637920482850137709081} a^{9} + \frac{259458766280017392596327653538553}{698211859637920482850137709081} a^{8} + \frac{2445369531531812359813664767472238}{698211859637920482850137709081} a^{7} + \frac{1006220166043971741659192684974359}{698211859637920482850137709081} a^{6} + \frac{640133813668034625377434461857485}{698211859637920482850137709081} a^{5} + \frac{158975791703915596210541900674723}{698211859637920482850137709081} a^{4} + \frac{128864394822851049184824277119801}{698211859637920482850137709081} a^{3} + \frac{4619392013553739456590457599211}{698211859637920482850137709081} a^{2} - \frac{589806502558576625488581748024}{698211859637920482850137709081} a + \frac{196486457262692508655909698442}{698211859637920482850137709081} \) (order $10$)
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:  \( 3431432369157.3267 \) (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, \(\Q(\zeta_{5})\), 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.33171021564453125.1, 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 $36$ $36$ R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $36$ $36$ R $36$ $18^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $36$ $36$ $18^{2}$

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
5Data not computed
19Data not computed