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Properties

Label 36.0.75876531007...6144.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 7^{30}\cdot 19^{24}$
Root discriminant $72.07$
Ramified primes $2, 7, 19$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13841287201, 0, -14123762450, 0, 10739824263, 0, -7494358949, 0, 5086203969, 0, -3420884824, 0, 2294243848, 0, -699981526, 0, 175403900, 0, -40095490, 0, 8663374, 0, -1782270, 0, 338472, 0, -49336, 0, 6985, 0, -946, 0, 119, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201)
 
gp: K = bnfinit(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201, 1)
 

Normalized defining polynomial

\( x^{36} - 13 x^{34} + 119 x^{32} - 946 x^{30} + 6985 x^{28} - 49336 x^{26} + 338472 x^{24} - 1782270 x^{22} + 8663374 x^{20} - 40095490 x^{18} + 175403900 x^{16} - 699981526 x^{14} + 2294243848 x^{12} - 3420884824 x^{10} + 5086203969 x^{8} - 7494358949 x^{6} + 10739824263 x^{4} - 14123762450 x^{2} + 13841287201 \)

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:  \(7587653100749080800127745255338750713360512873262289573077418246144=2^{36}\cdot 7^{30}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.07$
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:  $2, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(532=2^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{532}(1,·)$, $\chi_{532}(387,·)$, $\chi_{532}(391,·)$, $\chi_{532}(11,·)$, $\chi_{532}(45,·)$, $\chi_{532}(121,·)$, $\chi_{532}(277,·)$, $\chi_{532}(153,·)$, $\chi_{532}(201,·)$, $\chi_{532}(239,·)$, $\chi_{532}(159,·)$, $\chi_{532}(163,·)$, $\chi_{532}(39,·)$, $\chi_{532}(425,·)$, $\chi_{532}(429,·)$, $\chi_{532}(125,·)$, $\chi_{532}(305,·)$, $\chi_{532}(115,·)$, $\chi_{532}(311,·)$, $\chi_{532}(191,·)$, $\chi_{532}(267,·)$, $\chi_{532}(197,·)$, $\chi_{532}(457,·)$, $\chi_{532}(463,·)$, $\chi_{532}(83,·)$, $\chi_{532}(87,·)$, $\chi_{532}(349,·)$, $\chi_{532}(353,·)$, $\chi_{532}(419,·)$, $\chi_{532}(229,·)$, $\chi_{532}(235,·)$, $\chi_{532}(495,·)$, $\chi_{532}(467,·)$, $\chi_{532}(501,·)$, $\chi_{532}(505,·)$, $\chi_{532}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8} a^{14} + \frac{1}{8}$, $\frac{1}{8} a^{15} + \frac{1}{8} a$, $\frac{1}{8} a^{16} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{17} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{18} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{5}$, $\frac{1}{8} a^{20} + \frac{1}{8} a^{6}$, $\frac{1}{8} a^{21} + \frac{1}{8} a^{7}$, $\frac{1}{8} a^{22} + \frac{1}{8} a^{8}$, $\frac{1}{8} a^{23} + \frac{1}{8} a^{9}$, $\frac{1}{8} a^{24} + \frac{1}{8} a^{10}$, $\frac{1}{56} a^{25} + \frac{1}{56} a^{23} - \frac{1}{56} a^{19} - \frac{1}{56} a^{17} + \frac{1}{7} a^{13} - \frac{1}{8} a^{11} - \frac{15}{56} a^{9} - \frac{1}{7} a^{7} + \frac{1}{8} a^{5} + \frac{15}{56} a^{3} + \frac{1}{7} a$, $\frac{1}{465319718314472} a^{26} - \frac{3243115003540}{58164964789309} a^{24} - \frac{2505762039743}{66474245473496} a^{22} + \frac{6490631255265}{232659859157236} a^{20} + \frac{976289912167}{232659859157236} a^{18} + \frac{188121821805}{8309280684187} a^{16} + \frac{1810931332925}{465319718314472} a^{14} - \frac{16463970362265}{66474245473496} a^{12} + \frac{5935032075111}{58164964789309} a^{10} + \frac{102997627485423}{465319718314472} a^{8} - \frac{3103496852465}{33237122736748} a^{6} - \frac{24055837713785}{232659859157236} a^{4} - \frac{12110059037173}{58164964789309} a^{2} + \frac{3102595254541}{9496320781928}$, $\frac{1}{3257238028201304} a^{27} - \frac{3243115003540}{407154753525163} a^{25} + \frac{1450879661111}{116329929578618} a^{23} - \frac{45183702278779}{3257238028201304} a^{21} - \frac{172542314543593}{3257238028201304} a^{19} + \frac{9814255258627}{465319718314472} a^{17} + \frac{44076456425213}{814309507050326} a^{15} - \frac{82938215835761}{465319718314472} a^{13} + \frac{180429926443038}{407154753525163} a^{11} + \frac{272950507225919}{814309507050326} a^{9} - \frac{14516274389117}{465319718314472} a^{7} - \frac{222606569795497}{3257238028201304} a^{5} + \frac{891923929120869}{3257238028201304} a^{3} + \frac{6414089277905}{16618561368374} a$, $\frac{1}{182405329579273024} a^{28} - \frac{5}{5700166549352282} a^{26} + \frac{50243014059881}{814309507050326} a^{24} - \frac{198792879690063}{22800666197409128} a^{22} + \frac{646086344365783}{22800666197409128} a^{20} + \frac{144157829974089}{3257238028201304} a^{18} + \frac{1369253929671671}{22800666197409128} a^{16} - \frac{211887109294689}{13028952112805216} a^{14} - \frac{606052974670761}{5700166549352282} a^{12} + \frac{1665423993674383}{5700166549352282} a^{10} - \frac{18716055839905}{3257238028201304} a^{8} - \frac{7389258841293937}{22800666197409128} a^{6} + \frac{4194543694700143}{22800666197409128} a^{4} - \frac{130736640838241}{465319718314472} a^{2} + \frac{27909818601233}{75970566255424}$, $\frac{1}{1276837307054911168} a^{29} - \frac{5}{39901165845465974} a^{27} + \frac{50243014059881}{5700166549352282} a^{25} + \frac{1043932118042295}{19950582922732987} a^{23} + \frac{646086344365783}{159604663381863896} a^{21} + \frac{137828145874813}{5700166549352282} a^{19} + \frac{7069420479023953}{159604663381863896} a^{17} - \frac{3469125137495993}{91202664789636512} a^{15} + \frac{16494446673386085}{39901165845465974} a^{13} + \frac{13065757092378947}{39901165845465974} a^{11} + \frac{150343525591948}{2850083274676141} a^{9} - \frac{30189925038703065}{159604663381863896} a^{7} - \frac{3939009807008211}{39901165845465974} a^{5} + \frac{916232725369321}{3257238028201304} a^{3} + \frac{84887743292801}{531793963787968} a$, $\frac{1}{8937861149384378176} a^{30} - \frac{13}{8937861149384378176} a^{28} + \frac{45}{159604663381863896} a^{26} + \frac{57469480556058759}{1117232643673047272} a^{24} + \frac{47395416339855947}{1117232643673047272} a^{22} + \frac{248224465313147}{159604663381863896} a^{20} - \frac{41162813595329285}{1117232643673047272} a^{18} + \frac{7090169404570739}{638418653527455584} a^{16} - \frac{15218847034525201}{4468930574692189088} a^{14} - \frac{454568503488813085}{1117232643673047272} a^{12} - \frac{53553374638479055}{159604663381863896} a^{10} - \frac{238146285782543669}{1117232643673047272} a^{8} - \frac{436212156870181859}{1117232643673047272} a^{6} - \frac{4561775703613101}{22800666197409128} a^{4} - \frac{818778848072407}{3722557746515776} a^{2} + \frac{28072101076347}{75970566255424}$, $\frac{1}{62565028045690647232} a^{31} - \frac{13}{62565028045690647232} a^{29} + \frac{45}{1117232643673047272} a^{27} + \frac{57469480556058759}{7820628505711330904} a^{25} - \frac{231912744578405871}{7820628505711330904} a^{23} + \frac{10099403694023067}{558616321836523636} a^{21} + \frac{12311408357975203}{977578563213916363} a^{19} - \frac{232316825668225105}{4468930574692189088} a^{17} + \frac{1102013796638522071}{31282514022845323616} a^{15} + \frac{662664140184234187}{7820628505711330904} a^{13} + \frac{265655952125248737}{1117232643673047272} a^{11} + \frac{1717010840645289057}{7820628505711330904} a^{9} + \frac{410337283630998161}{3910314252855665452} a^{7} + \frac{2636121721059021}{19950582922732987} a^{5} - \frac{5937295749531599}{26057904225610432} a^{3} + \frac{47064742640203}{531793963787968} a$, $\frac{1}{437955196319834530624} a^{32} - \frac{13}{437955196319834530624} a^{30} + \frac{17}{62565028045690647232} a^{28} + \frac{14647}{27372199769989658164} a^{26} + \frac{1921931111283379349}{54744399539979316328} a^{24} + \frac{23715329639203133}{977578563213916363} a^{22} + \frac{2970533480945763517}{54744399539979316328} a^{20} + \frac{970929795739154311}{31282514022845323616} a^{18} - \frac{12105973541660955933}{218977598159917265312} a^{16} + \frac{1749258827229197539}{218977598159917265312} a^{14} - \frac{1516603366277236099}{3910314252855665452} a^{12} + \frac{3045621720157464813}{54744399539979316328} a^{10} - \frac{3331378040955949627}{6843049942497414541} a^{8} - \frac{361346937944844419}{1117232643673047272} a^{6} - \frac{42766734044530495}{182405329579273024} a^{4} + \frac{1206087150896419}{3722557746515776} a^{2} - \frac{10699939378865}{75970566255424}$, $\frac{1}{3065686374238841714368} a^{33} - \frac{13}{3065686374238841714368} a^{31} + \frac{17}{437955196319834530624} a^{29} + \frac{14647}{191605398389927607148} a^{27} + \frac{1921931111283379349}{383210796779855214296} a^{25} + \frac{1072439881770728895}{27372199769989658164} a^{23} - \frac{484064557693956378}{47901349597481901787} a^{21} + \frac{12701872554306150667}{218977598159917265312} a^{19} - \frac{66850373081640272261}{1532843187119420857184} a^{17} + \frac{56493658367208513867}{1532843187119420857184} a^{15} + \frac{6304025139434094805}{27372199769989658164} a^{13} + \frac{167278820340095413797}{383210796779855214296} a^{11} + \frac{75634137088642590525}{191605398389927607148} a^{9} - \frac{481587368677889643}{977578563213916363} a^{7} - \frac{156770065031576135}{1276837307054911168} a^{5} + \frac{275447714267475}{26057904225610432} a^{3} + \frac{1184671740713}{75970566255424} a$, $\frac{1}{21459804619671892000576} a^{34} - \frac{13}{21459804619671892000576} a^{32} + \frac{17}{3065686374238841714368} a^{30} - \frac{473}{10729902309835946000288} a^{28} - \frac{719727}{2682475577458986500072} a^{26} + \frac{16124887576979796959}{383210796779855214296} a^{24} - \frac{116585102829424402067}{2682475577458986500072} a^{22} + \frac{23861863614638170223}{1532843187119420857184} a^{20} + \frac{363483010066018745395}{10729902309835946000288} a^{18} + \frac{385084237479126786695}{10729902309835946000288} a^{16} - \frac{36885404969438080889}{766421593559710428592} a^{14} - \frac{6765554154083058655}{2682475577458986500072} a^{12} + \frac{10916550041436124793}{2682475577458986500072} a^{10} + \frac{21660977151667525125}{54744399539979316328} a^{8} + \frac{3554350852308769073}{8937861149384378176} a^{6} - \frac{5089731938085117}{182405329579273024} a^{4} + \frac{16409972761735}{75970566255424} a^{2} - \frac{907430289301}{37985283127712}$, $\frac{1}{150218632337703244004032} a^{35} - \frac{13}{150218632337703244004032} a^{33} + \frac{17}{21459804619671892000576} a^{31} - \frac{473}{75109316168851622002016} a^{29} - \frac{719727}{18777329042212905500504} a^{27} + \frac{16124887576979796959}{2682475577458986500072} a^{25} - \frac{116585102829424402067}{18777329042212905500504} a^{23} - \frac{167743534775289436925}{10729902309835946000288} a^{21} - \frac{2318992567392967754677}{75109316168851622002016} a^{19} - \frac{956153551250366463341}{75109316168851622002016} a^{17} - \frac{132688104164401884463}{5364951154917973000144} a^{15} + \frac{5358185600763889941489}{18777329042212905500504} a^{13} - \frac{5354034604876536875351}{18777329042212905500504} a^{11} - \frac{87827821928291107531}{383210796779855214296} a^{9} + \frac{29250701656788856329}{62565028045690647232} a^{7} + \frac{131714265246369651}{1276837307054911168} a^{5} + \frac{6913651979807}{531793963787968} a^{3} + \frac{10044996510737}{37985283127712} a$

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

Class group and class number

Not computed

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:  \( -\frac{10625579}{1276837307054911168} a^{33} - \frac{2680158372175}{638418653527455584} a^{19} - \frac{468901578337858483}{1276837307054911168} a^{5} \) (order $28$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), 3.3.17689.1, \(\Q(\zeta_{7})^+\), 3.3.361.1, 3.3.17689.2, \(\Q(i, \sqrt{7})\), 6.0.20025646144.2, 6.0.153664.1, 6.0.8340544.1, 6.0.20025646144.1, 6.6.140179523008.1, 6.0.2190305047.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.2860806592.1, 6.0.44700103.1, 6.6.140179523008.2, 6.0.2190305047.2, 9.9.5534900853769.1, 12.0.19650298670750401368064.1, \(\Q(\zeta_{28})\), 12.0.8184214356830654464.1, 12.0.19650298670750401368064.2, 18.0.8030814853150226545771901353984.1, 18.18.2754569494630527705199762164416512.1, 18.0.10507848719141112156676338823.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
19Data not computed