↑

Properties

Label 36.36.5744476530...5625.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 37^{35}$
Root discriminant $111.91$
Ramified primes $5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{36}$ (as 36T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -112, 3701, -37038, 40590, 983052, -2202202, -11623218, 20891644, 68588675, -97607738, -225724125, 273263612, 444437006, -488912671, -548111526, 572080385, 440605249, -447142705, -238127228, 238235194, 88382528, -87888652, -22805506, 22653732, 4104557, -4083615, -511341, 509935, 43067, -43030, -2332, 2332, 73, -73, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 73*x^34 + 73*x^33 + 2332*x^32 - 2332*x^31 - 43030*x^30 + 43067*x^29 + 509935*x^28 - 511341*x^27 - 4083615*x^26 + 4104557*x^25 + 22653732*x^24 - 22805506*x^23 - 87888652*x^22 + 88382528*x^21 + 238235194*x^20 - 238127228*x^19 - 447142705*x^18 + 440605249*x^17 + 572080385*x^16 - 548111526*x^15 - 488912671*x^14 + 444437006*x^13 + 273263612*x^12 - 225724125*x^11 - 97607738*x^10 + 68588675*x^9 + 20891644*x^8 - 11623218*x^7 - 2202202*x^6 + 983052*x^5 + 40590*x^4 - 37038*x^3 + 3701*x^2 - 112*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 73*x^34 + 73*x^33 + 2332*x^32 - 2332*x^31 - 43030*x^30 + 43067*x^29 + 509935*x^28 - 511341*x^27 - 4083615*x^26 + 4104557*x^25 + 22653732*x^24 - 22805506*x^23 - 87888652*x^22 + 88382528*x^21 + 238235194*x^20 - 238127228*x^19 - 447142705*x^18 + 440605249*x^17 + 572080385*x^16 - 548111526*x^15 - 488912671*x^14 + 444437006*x^13 + 273263612*x^12 - 225724125*x^11 - 97607738*x^10 + 68588675*x^9 + 20891644*x^8 - 11623218*x^7 - 2202202*x^6 + 983052*x^5 + 40590*x^4 - 37038*x^3 + 3701*x^2 - 112*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 73 x^{34} + 73 x^{33} + 2332 x^{32} - 2332 x^{31} - 43030 x^{30} + 43067 x^{29} + 509935 x^{28} - 511341 x^{27} - 4083615 x^{26} + 4104557 x^{25} + 22653732 x^{24} - 22805506 x^{23} - 87888652 x^{22} + 88382528 x^{21} + 238235194 x^{20} - 238127228 x^{19} - 447142705 x^{18} + 440605249 x^{17} + 572080385 x^{16} - 548111526 x^{15} - 488912671 x^{14} + 444437006 x^{13} + 273263612 x^{12} - 225724125 x^{11} - 97607738 x^{10} + 68588675 x^{9} + 20891644 x^{8} - 11623218 x^{7} - 2202202 x^{6} + 983052 x^{5} + 40590 x^{4} - 37038 x^{3} + 3701 x^{2} - 112 x + 1 \)

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:  \(57444765302724909954814307473256133361395843470561362005770206451416015625=5^{27}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.91$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(128,·)$, $\chi_{185}(1,·)$, $\chi_{185}(2,·)$, $\chi_{185}(4,·)$, $\chi_{185}(133,·)$, $\chi_{185}(8,·)$, $\chi_{185}(139,·)$, $\chi_{185}(13,·)$, $\chi_{185}(142,·)$, $\chi_{185}(16,·)$, $\chi_{185}(23,·)$, $\chi_{185}(153,·)$, $\chi_{185}(26,·)$, $\chi_{185}(159,·)$, $\chi_{185}(32,·)$, $\chi_{185}(162,·)$, $\chi_{185}(169,·)$, $\chi_{185}(43,·)$, $\chi_{185}(172,·)$, $\chi_{185}(46,·)$, $\chi_{185}(177,·)$, $\chi_{185}(52,·)$, $\chi_{185}(181,·)$, $\chi_{185}(183,·)$, $\chi_{185}(184,·)$, $\chi_{185}(57,·)$, $\chi_{185}(64,·)$, $\chi_{185}(71,·)$, $\chi_{185}(81,·)$, $\chi_{185}(86,·)$, $\chi_{185}(92,·)$, $\chi_{185}(93,·)$, $\chi_{185}(99,·)$, $\chi_{185}(104,·)$, $\chi_{185}(114,·)$, $\chi_{185}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{487} a^{33} - \frac{145}{487} a^{32} + \frac{165}{487} a^{31} + \frac{208}{487} a^{30} + \frac{238}{487} a^{29} + \frac{219}{487} a^{28} - \frac{96}{487} a^{27} - \frac{107}{487} a^{26} - \frac{205}{487} a^{25} + \frac{130}{487} a^{24} - \frac{105}{487} a^{23} - \frac{200}{487} a^{22} + \frac{156}{487} a^{21} + \frac{19}{487} a^{20} - \frac{205}{487} a^{19} + \frac{3}{487} a^{18} + \frac{135}{487} a^{17} - \frac{103}{487} a^{16} + \frac{52}{487} a^{15} + \frac{75}{487} a^{14} + \frac{240}{487} a^{13} + \frac{49}{487} a^{12} + \frac{89}{487} a^{11} + \frac{52}{487} a^{10} - \frac{29}{487} a^{9} + \frac{152}{487} a^{8} + \frac{186}{487} a^{7} - \frac{62}{487} a^{6} - \frac{41}{487} a^{5} - \frac{56}{487} a^{4} - \frac{93}{487} a^{3} + \frac{92}{487} a + \frac{166}{487}$, $\frac{1}{72713152327} a^{34} + \frac{34330185}{72713152327} a^{33} + \frac{1472329655}{72713152327} a^{32} + \frac{8518273036}{72713152327} a^{31} - \frac{4423559032}{72713152327} a^{30} - \frac{8619842899}{72713152327} a^{29} - \frac{26040729452}{72713152327} a^{28} + \frac{3294187153}{72713152327} a^{27} + \frac{20037938888}{72713152327} a^{26} + \frac{36015884678}{72713152327} a^{25} + \frac{968108071}{72713152327} a^{24} - \frac{19016271266}{72713152327} a^{23} - \frac{36181569953}{72713152327} a^{22} - \frac{8675837990}{72713152327} a^{21} + \frac{33694709358}{72713152327} a^{20} - \frac{17254994692}{72713152327} a^{19} - \frac{1051356663}{72713152327} a^{18} - \frac{26996248405}{72713152327} a^{17} - \frac{28780484176}{72713152327} a^{16} - \frac{8224283274}{72713152327} a^{15} + \frac{30967978045}{72713152327} a^{14} - \frac{32137937990}{72713152327} a^{13} - \frac{15038512162}{72713152327} a^{12} + \frac{32172705245}{72713152327} a^{11} + \frac{22648070123}{72713152327} a^{10} - \frac{5937500065}{72713152327} a^{9} - \frac{29335652284}{72713152327} a^{8} + \frac{24185359011}{72713152327} a^{7} - \frac{26485628200}{72713152327} a^{6} + \frac{3321401247}{72713152327} a^{5} - \frac{6263525504}{72713152327} a^{4} - \frac{7142596526}{72713152327} a^{3} - \frac{2910058668}{72713152327} a^{2} + \frac{30974856039}{72713152327} a + \frac{35331927660}{72713152327}$, $\frac{1}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{35} - \frac{10248285015864411327756390799779596458295926529875081669212013098223819}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{34} + \frac{9032017366993452727113480808370255967006771898948773136581128937602656905478952}{16833472527601308912127237156438174199229624800552445767036007557623413816131854799} a^{33} + \frac{297541072949466421582952187161516656710682197563657354451451511568111168811012641139}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{32} + \frac{138144154643505427255429603618617540417462377270821293007194394817101726588411877965}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{31} - \frac{159314069381762154028820210405407378123570598561277279106291490022762804254897738915}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{30} - \frac{288906340316313902783132748470158029707204333256311118521763419758852350943310047790}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{29} - \frac{43847736711316190581905153572769928598937964934814003826246650770182138066732106926}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{28} + \frac{194890835229065449079117946665898014819254362654673165588654292682359445185160260260}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{27} - \frac{191124437227333211767437447407167573499795162329263995478201530351838866230432553249}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{26} + \frac{333553735259766459185041359399266896969797183463037648694432235448873305822169295143}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{25} + \frac{221699669863712666766510323747982126607702888602820499433461273916783144888347057690}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{24} + \frac{115902503915054525097849926176624470438972746241726311890883594066951312497219138073}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{23} + \frac{331377329938017221683931495952614389961957265328831446940731325885837532734297302226}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{22} + \frac{251248228151482107996284268658399832985408098796131169318381289753844289338164982417}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{21} - \frac{136340766426326575266891001603789952567414270329813899477330524133722253134499961497}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{20} - \frac{149731478661410210886735097187605134138568811183446009906367498446460329321097925557}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{19} + \frac{194106764790463976875964211100477119116726053220715789002132663733237800181129884543}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{18} + \frac{197772837806390194381356161205741400356929965106516200732674818950318878454160454957}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{17} + \frac{124803963438419814596532600775480881420603494608731268372399456868669275939951896329}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{16} - \frac{103541314284982458995742227090907606832650072333213236549613223728778905710452213225}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{15} - \frac{7909082349024423259609570579196828211594278706574154616866009888593544558528105199}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{14} - \frac{153880389851713213660420461733892493045061435909912163459549679553510021742002767642}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{13} + \frac{65882338710130628445745628734110111435131849081856391826086219612290196923499182500}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{12} + \frac{219848271157747479164679769475022792783930068937987007504089808256918191082010611923}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{11} - \frac{218543057692960971558985886177594839616641573209305554084426657869500209545055759583}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{10} + \frac{278910768698778873230284919003984900714142778402228267060269059581733285559957623257}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{9} + \frac{145679967836062498459829311582369264326022895566271948457669083456731844553370418634}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{8} + \frac{9127374055384628068428576437586117947702284702026968351719334023379878097845936756}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{7} + \frac{202480411443192128805361429505162687908675880563013903730936347713052513555971348838}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{6} - \frac{319633257184367197409472355054529339764094367596593068582400600746309938516571983288}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{5} + \frac{171034682279986660040559784951189281420316569290263274504074520563458092338801116072}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{4} + \frac{335658546416366178714758862933501661591734170624570373373650593600789544399097627743}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{3} - \frac{160586908964774616361922442164971351446158896995988015279410882980388827460344059626}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a^{2} - \frac{148343674902720598116484785992907933313802611440042490099790223526476019859591568074}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357} a + \frac{195599921496135052880044027714833087121992149005206911086347601365559339677326396684}{723839318686856283221471197726841490566873866423755167982548324977806794093669756357}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 11027752026555245000000000 \) (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

\(\Q(\sqrt{185}) \), 3.3.1369.1, 4.4.6331625.1, 6.6.8667994625.1, 9.9.3512479453921.1, 12.12.347495355038008619140625.2, 18.18.891578009425849912898724447265625.1

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 $18^{2}$ $36$ R $36$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R $18^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ $36$ $36$

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
37Data not computed