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Properties

Label 36.0.12040604416...0625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 5^{18}\cdot 13^{24}$
Root discriminant $64.24$
Ramified primes $3, 5, 13$
Class number $9604$ (GRH)
Class group $[7, 14, 98]$ (GRH)
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![15625, 0, 0, -359375, 0, 0, 8883000, 0, 0, 12785625, 0, 0, 40530596, 0, 0, -22224984, 0, 0, 36900263, 0, 0, 5639039, 0, 0, 969798, 0, 0, 13298, 0, 0, 997, 0, 0, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 2*x^33 + 997*x^30 + 13298*x^27 + 969798*x^24 + 5639039*x^21 + 36900263*x^18 - 22224984*x^15 + 40530596*x^12 + 12785625*x^9 + 8883000*x^6 - 359375*x^3 + 15625)
 
gp: K = bnfinit(x^36 - 2*x^33 + 997*x^30 + 13298*x^27 + 969798*x^24 + 5639039*x^21 + 36900263*x^18 - 22224984*x^15 + 40530596*x^12 + 12785625*x^9 + 8883000*x^6 - 359375*x^3 + 15625, 1)
 

Normalized defining polynomial

\( x^{36} - 2 x^{33} + 997 x^{30} + 13298 x^{27} + 969798 x^{24} + 5639039 x^{21} + 36900263 x^{18} - 22224984 x^{15} + 40530596 x^{12} + 12785625 x^{9} + 8883000 x^{6} - 359375 x^{3} + 15625 \)

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:  \(120406044167774900818448125272170303354510035249351535797119140625=3^{54}\cdot 5^{18}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.24$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(386,·)$, $\chi_{585}(131,·)$, $\chi_{585}(391,·)$, $\chi_{585}(521,·)$, $\chi_{585}(139,·)$, $\chi_{585}(269,·)$, $\chi_{585}(14,·)$, $\chi_{585}(16,·)$, $\chi_{585}(529,·)$, $\chi_{585}(274,·)$, $\chi_{585}(404,·)$, $\chi_{585}(406,·)$, $\chi_{585}(536,·)$, $\chi_{585}(29,·)$, $\chi_{585}(289,·)$, $\chi_{585}(419,·)$, $\chi_{585}(61,·)$, $\chi_{585}(191,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(581,·)$, $\chi_{585}(326,·)$, $\chi_{585}(74,·)$, $\chi_{585}(334,·)$, $\chi_{585}(79,·)$, $\chi_{585}(464,·)$, $\chi_{585}(209,·)$, $\chi_{585}(211,·)$, $\chi_{585}(341,·)$, $\chi_{585}(94,·)$, $\chi_{585}(224,·)$, $\chi_{585}(484,·)$, $\chi_{585}(146,·)$, $\chi_{585}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{5} a^{26} - \frac{2}{5} a^{23} + \frac{2}{5} a^{20} - \frac{2}{5} a^{17} - \frac{2}{5} a^{14} - \frac{1}{5} a^{11} - \frac{2}{5} a^{8} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{10} a^{27} - \frac{1}{5} a^{24} + \frac{1}{5} a^{21} + \frac{3}{10} a^{18} - \frac{1}{5} a^{15} + \frac{2}{5} a^{12} - \frac{1}{5} a^{9} - \frac{2}{5} a^{6} - \frac{2}{5} a^{3} - \frac{1}{2}$, $\frac{1}{50} a^{28} - \frac{1}{25} a^{25} + \frac{11}{25} a^{22} + \frac{23}{50} a^{19} - \frac{1}{25} a^{16} + \frac{7}{25} a^{13} - \frac{6}{25} a^{10} + \frac{8}{25} a^{7} - \frac{2}{25} a^{4} - \frac{1}{2} a$, $\frac{1}{50} a^{29} - \frac{1}{25} a^{26} + \frac{11}{25} a^{23} + \frac{23}{50} a^{20} - \frac{1}{25} a^{17} + \frac{7}{25} a^{14} - \frac{6}{25} a^{11} + \frac{8}{25} a^{8} - \frac{2}{25} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{302915431422250} a^{30} - \frac{10159252435777}{302915431422250} a^{27} - \frac{601556013021}{1389520327625} a^{24} + \frac{99005872569623}{302915431422250} a^{21} + \frac{141813090777}{607044952750} a^{18} - \frac{22884476969393}{151457715711125} a^{15} + \frac{37335672162019}{151457715711125} a^{12} + \frac{52524973469283}{151457715711125} a^{9} + \frac{65204940794348}{151457715711125} a^{6} + \frac{1132372870229}{12116617256890} a^{3} + \frac{368233796145}{2423323451378}$, $\frac{1}{302915431422250} a^{31} + \frac{1957364821113}{302915431422250} a^{28} + \frac{676802688394}{1389520327625} a^{25} + \frac{62656020798953}{302915431422250} a^{22} + \frac{93249494557}{607044952750} a^{19} - \frac{35001094226283}{151457715711125} a^{16} - \frac{29305722750876}{151457715711125} a^{13} - \frac{20174730072057}{151457715711125} a^{10} + \frac{10680163138343}{151457715711125} a^{7} - \frac{4031429454367}{60583086284450} a^{4} + \frac{368233796145}{2423323451378} a$, $\frac{1}{1514577157111250} a^{32} - \frac{10159252435777}{1514577157111250} a^{29} - \frac{601556013021}{6947601638125} a^{26} + \frac{704836735414123}{1514577157111250} a^{23} + \frac{1355902996277}{3035224763750} a^{20} + \frac{128573238741732}{757288578555625} a^{17} + \frac{340251103584269}{757288578555625} a^{14} + \frac{52524973469283}{757288578555625} a^{11} - \frac{237710490627902}{757288578555625} a^{8} + \frac{1132372870229}{60583086284450} a^{5} + \frac{73646759229}{2423323451378} a^{2}$, $\frac{1}{18352159863030835502450294648497811234083750} a^{33} - \frac{7772817840406240680486954451}{9176079931515417751225147324248905617041875} a^{30} + \frac{313113420239723657123482722864129242213461}{9176079931515417751225147324248905617041875} a^{27} + \frac{3029552507171838736156561631160895475034373}{18352159863030835502450294648497811234083750} a^{24} - \frac{3000351753935378235612642746265355886764951}{9176079931515417751225147324248905617041875} a^{21} + \frac{4246589334219287254402881889301304022926232}{9176079931515417751225147324248905617041875} a^{18} - \frac{1562930351606987004239468361556996154541856}{9176079931515417751225147324248905617041875} a^{15} - \frac{2656321486250024435217616753425418012607217}{9176079931515417751225147324248905617041875} a^{12} - \frac{4422617019950786898746493740747180942766527}{9176079931515417751225147324248905617041875} a^{9} - \frac{67033210235780768465950531136790347120601}{734086394521233420098011785939912449363350} a^{6} + \frac{30944797045055611500613278285743096944452}{73408639452123342009801178593991244936335} a^{3} + \frac{21749427485466580989302184787244150493}{473604125497569948450330184477362870557}$, $\frac{1}{91760799315154177512251473242489056170418750} a^{34} - \frac{7772817840406240680486954451}{45880399657577088756125736621244528085209375} a^{31} + \frac{313113420239723657123482722864129242213461}{45880399657577088756125736621244528085209375} a^{28} + \frac{39733872233233509741057150928156517943201873}{91760799315154177512251473242489056170418750} a^{25} - \frac{21352511616966213738062937394763167120848701}{45880399657577088756125736621244528085209375} a^{22} + \frac{13422669265734705005628029213550209639968107}{45880399657577088756125736621244528085209375} a^{19} + \frac{16789229511423848498210826286940815079541894}{45880399657577088756125736621244528085209375} a^{16} - \frac{2656321486250024435217616753425418012607217}{45880399657577088756125736621244528085209375} a^{13} + \frac{13929542843080048603703800907750630291317223}{45880399657577088756125736621244528085209375} a^{10} - \frac{1535205999278247608661974103016615245847301}{3670431972606167100490058929699562246816750} a^{7} + \frac{30944797045055611500613278285743096944452}{367043197260616710049005892969956224681675} a^{4} + \frac{99070710596607305887926473852921404210}{473604125497569948450330184477362870557} a$, $\frac{1}{91760799315154177512251473242489056170418750} a^{35} - \frac{7772817840406240680486954451}{45880399657577088756125736621244528085209375} a^{32} + \frac{313113420239723657123482722864129242213461}{45880399657577088756125736621244528085209375} a^{29} + \frac{3029552507171838736156561631160895475034373}{91760799315154177512251473242489056170418750} a^{26} + \frac{15351808109095457266837651902232455347318799}{45880399657577088756125736621244528085209375} a^{23} + \frac{22598749197250122756853176537799115257009982}{45880399657577088756125736621244528085209375} a^{20} + \frac{7613149579908430746985678962691909462500019}{45880399657577088756125736621244528085209375} a^{17} - \frac{11832401417765442186442764077674323629649092}{45880399657577088756125736621244528085209375} a^{14} - \frac{13598696951466204649971641064996086559808402}{45880399657577088756125736621244528085209375} a^{11} + \frac{1401139578806686071730073040743034551606099}{3670431972606167100490058929699562246816750} a^{8} - \frac{115872481859191072518989078902239392928218}{367043197260616710049005892969956224681675} a^{5} - \frac{451854698012103367461027999690118720064}{2368020627487849742251650922386814352785} a^{2}$

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

Class group and class number

$C_{7}\times C_{14}\times C_{98}$, which has order $9604$ (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:  \( \frac{70234324305228434953808927601019626}{60768741268314024842550644531449706073125} a^{35} - \frac{181219211472720127626919497364975002}{60768741268314024842550644531449706073125} a^{32} + \frac{140205173144562333481843384156663813869}{121537482536628049685101289062899412146250} a^{29} + \frac{893352142819922892919504981518900696798}{60768741268314024842550644531449706073125} a^{26} + \frac{67568679854157333224200478844806972253798}{60768741268314024842550644531449706073125} a^{23} + \frac{712999789831863228735665272437686392403953}{121537482536628049685101289062899412146250} a^{20} + \frac{2359403469088100305249440189904429426523388}{60768741268314024842550644531449706073125} a^{17} - \frac{28253203359376402484622059730920744813376}{557511387782697475619730683774767945625} a^{14} + \frac{3655530029568538385363200255108371758143221}{60768741268314024842550644531449706073125} a^{11} - \frac{5756313417307213147633343950389469625406}{486149930146512198740405156251597648585} a^{8} + \frac{46600553649166691188562635272705338850}{97229986029302439748081031250319529717} a^{5} - \frac{8224755265625072224103500533884292642361}{972299860293024397480810312503195297170} a^{2} \) (order $18$)
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:  \( 478341710831896.44 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 3.3.13689.2, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{9})\), 6.0.562166163.2, 6.0.771147.1, 6.0.562166163.1, 6.0.2460375.1, 6.6.820125.1, 6.0.70270770375.5, 6.6.23423590125.1, 6.0.96393375.1, 6.6.3570125.1, 6.0.70270770375.7, 6.6.23423590125.2, 9.9.2565164201769.1, 12.0.6053445140625.1, 12.0.4937981169095977640625.1, 12.0.9291682743890625.1, 12.0.4937981169095977640625.2, 18.0.177661819315004155453692747.1, 18.0.346995740849617491120493646484375.1, 18.18.12851694105541388560018283203125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
13Data not computed