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Properties

Label 36.0.14361527681...3104.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{48}\cdot 13^{30}$
Root discriminant $73.36$
Ramified primes $2, 3, 13$
Class number $876096$ (GRH)
Class group $[936, 936]$ (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![1, 0, 189, 0, 7476, 0, 132111, 0, 1293075, 0, 7688325, 0, 28999138, 0, 70684362, 0, 111976572, 0, 115641708, 0, 78637230, 0, 35728458, 0, 10994901, 0, 2308518, 0, 329559, 0, 31364, 0, 1899, 0, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 1)
 
gp: K = bnfinit(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 66 x^{34} + 1899 x^{32} + 31364 x^{30} + 329559 x^{28} + 2308518 x^{26} + 10994901 x^{24} + 35728458 x^{22} + 78637230 x^{20} + 115641708 x^{18} + 111976572 x^{16} + 70684362 x^{14} + 28999138 x^{12} + 7688325 x^{10} + 1293075 x^{8} + 132111 x^{6} + 7476 x^{4} + 189 x^{2} + 1 \)

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:  \(14361527681487998235176534750111030030248040605937785686475463983104=2^{36}\cdot 3^{48}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.36$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(259,·)$, $\chi_{468}(133,·)$, $\chi_{468}(391,·)$, $\chi_{468}(139,·)$, $\chi_{468}(367,·)$, $\chi_{468}(277,·)$, $\chi_{468}(25,·)$, $\chi_{468}(283,·)$, $\chi_{468}(157,·)$, $\chi_{468}(415,·)$, $\chi_{468}(289,·)$, $\chi_{468}(295,·)$, $\chi_{468}(43,·)$, $\chi_{468}(433,·)$, $\chi_{468}(181,·)$, $\chi_{468}(55,·)$, $\chi_{468}(313,·)$, $\chi_{468}(445,·)$, $\chi_{468}(451,·)$, $\chi_{468}(49,·)$, $\chi_{468}(199,·)$, $\chi_{468}(439,·)$, $\chi_{468}(205,·)$, $\chi_{468}(79,·)$, $\chi_{468}(337,·)$, $\chi_{468}(211,·)$, $\chi_{468}(217,·)$, $\chi_{468}(355,·)$, $\chi_{468}(103,·)$, $\chi_{468}(361,·)$, $\chi_{468}(235,·)$, $\chi_{468}(61,·)$, $\chi_{468}(373,·)$, $\chi_{468}(121,·)$, $\chi_{468}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{53} a^{28} + \frac{13}{53} a^{26} - \frac{4}{53} a^{24} + \frac{2}{53} a^{20} + \frac{6}{53} a^{18} - \frac{7}{53} a^{16} + \frac{22}{53} a^{14} + \frac{16}{53} a^{12} + \frac{1}{53} a^{10} - \frac{12}{53} a^{8} + \frac{19}{53} a^{6} - \frac{24}{53} a^{4} - \frac{8}{53} a^{2} - \frac{25}{53}$, $\frac{1}{53} a^{29} + \frac{13}{53} a^{27} - \frac{4}{53} a^{25} + \frac{2}{53} a^{21} + \frac{6}{53} a^{19} - \frac{7}{53} a^{17} + \frac{22}{53} a^{15} + \frac{16}{53} a^{13} + \frac{1}{53} a^{11} - \frac{12}{53} a^{9} + \frac{19}{53} a^{7} - \frac{24}{53} a^{5} - \frac{8}{53} a^{3} - \frac{25}{53} a$, $\frac{1}{53} a^{30} - \frac{14}{53} a^{26} - \frac{1}{53} a^{24} + \frac{2}{53} a^{22} - \frac{20}{53} a^{20} + \frac{21}{53} a^{18} + \frac{7}{53} a^{16} - \frac{5}{53} a^{14} + \frac{5}{53} a^{12} - \frac{25}{53} a^{10} + \frac{16}{53} a^{8} - \frac{6}{53} a^{6} - \frac{14}{53} a^{4} + \frac{26}{53} a^{2} + \frac{7}{53}$, $\frac{1}{53} a^{31} - \frac{14}{53} a^{27} - \frac{1}{53} a^{25} + \frac{2}{53} a^{23} - \frac{20}{53} a^{21} + \frac{21}{53} a^{19} + \frac{7}{53} a^{17} - \frac{5}{53} a^{15} + \frac{5}{53} a^{13} - \frac{25}{53} a^{11} + \frac{16}{53} a^{9} - \frac{6}{53} a^{7} - \frac{14}{53} a^{5} + \frac{26}{53} a^{3} + \frac{7}{53} a$, $\frac{1}{53} a^{32} + \frac{22}{53} a^{26} - \frac{1}{53} a^{24} - \frac{20}{53} a^{22} - \frac{4}{53} a^{20} - \frac{15}{53} a^{18} + \frac{3}{53} a^{16} - \frac{5}{53} a^{14} - \frac{13}{53} a^{12} - \frac{23}{53} a^{10} - \frac{15}{53} a^{8} - \frac{13}{53} a^{6} + \frac{8}{53} a^{4} + \frac{1}{53} a^{2} + \frac{21}{53}$, $\frac{1}{53} a^{33} + \frac{22}{53} a^{27} - \frac{1}{53} a^{25} - \frac{20}{53} a^{23} - \frac{4}{53} a^{21} - \frac{15}{53} a^{19} + \frac{3}{53} a^{17} - \frac{5}{53} a^{15} - \frac{13}{53} a^{13} - \frac{23}{53} a^{11} - \frac{15}{53} a^{9} - \frac{13}{53} a^{7} + \frac{8}{53} a^{5} + \frac{1}{53} a^{3} + \frac{21}{53} a$, $\frac{1}{47627645614351126040333609718698858653} a^{34} + \frac{385060092666894419836841445863780923}{47627645614351126040333609718698858653} a^{32} - \frac{335709082852599056362848897687665427}{47627645614351126040333609718698858653} a^{30} - \frac{203065593312347055274868365374327878}{47627645614351126040333609718698858653} a^{28} + \frac{22912909058595372891628011995173016551}{47627645614351126040333609718698858653} a^{26} - \frac{16802694538891608488662426890554913190}{47627645614351126040333609718698858653} a^{24} + \frac{8995802178764411942677860534634091995}{47627645614351126040333609718698858653} a^{22} + \frac{15420555374884449955338128387815741098}{47627645614351126040333609718698858653} a^{20} + \frac{10496070157960533169269683215393045591}{47627645614351126040333609718698858653} a^{18} - \frac{16951053928953742057984193722544352405}{47627645614351126040333609718698858653} a^{16} - \frac{12171041619921106962173993947598768442}{47627645614351126040333609718698858653} a^{14} - \frac{8015630782060113203341517721488470248}{47627645614351126040333609718698858653} a^{12} + \frac{19645959090516967005218126728335596670}{47627645614351126040333609718698858653} a^{10} - \frac{22887315403573206972867513232949860555}{47627645614351126040333609718698858653} a^{8} - \frac{14787769669364402439812449341016545717}{47627645614351126040333609718698858653} a^{6} + \frac{13709226887332515280267930868756019085}{47627645614351126040333609718698858653} a^{4} + \frac{4746715704729199305974218564182002332}{47627645614351126040333609718698858653} a^{2} - \frac{5570328814094086494429682492342169639}{47627645614351126040333609718698858653}$, $\frac{1}{47627645614351126040333609718698858653} a^{35} + \frac{385060092666894419836841445863780923}{47627645614351126040333609718698858653} a^{33} - \frac{335709082852599056362848897687665427}{47627645614351126040333609718698858653} a^{31} - \frac{203065593312347055274868365374327878}{47627645614351126040333609718698858653} a^{29} + \frac{22912909058595372891628011995173016551}{47627645614351126040333609718698858653} a^{27} - \frac{16802694538891608488662426890554913190}{47627645614351126040333609718698858653} a^{25} + \frac{8995802178764411942677860534634091995}{47627645614351126040333609718698858653} a^{23} + \frac{15420555374884449955338128387815741098}{47627645614351126040333609718698858653} a^{21} + \frac{10496070157960533169269683215393045591}{47627645614351126040333609718698858653} a^{19} - \frac{16951053928953742057984193722544352405}{47627645614351126040333609718698858653} a^{17} - \frac{12171041619921106962173993947598768442}{47627645614351126040333609718698858653} a^{15} - \frac{8015630782060113203341517721488470248}{47627645614351126040333609718698858653} a^{13} + \frac{19645959090516967005218126728335596670}{47627645614351126040333609718698858653} a^{11} - \frac{22887315403573206972867513232949860555}{47627645614351126040333609718698858653} a^{9} - \frac{14787769669364402439812449341016545717}{47627645614351126040333609718698858653} a^{7} + \frac{13709226887332515280267930868756019085}{47627645614351126040333609718698858653} a^{5} + \frac{4746715704729199305974218564182002332}{47627645614351126040333609718698858653} a^{3} - \frac{5570328814094086494429682492342169639}{47627645614351126040333609718698858653} a$

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

Class group and class number

$C_{936}\times C_{936}$, which has order $876096$ (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:  \( \frac{749195149491219199743}{630534726674038762237} a^{35} + \frac{49264890649408878897431}{630534726674038762237} a^{33} + \frac{1410742182733092748400454}{630534726674038762237} a^{31} + \frac{23154259172514622529873625}{630534726674038762237} a^{29} + \frac{241256361838291759281330911}{630534726674038762237} a^{27} + \frac{1670546293159551855975576174}{630534726674038762237} a^{25} + \frac{7827576734875697768556950247}{630534726674038762237} a^{23} + \frac{24838854752462209026830995479}{630534726674038762237} a^{21} + \frac{52753326546701508835173786339}{630534726674038762237} a^{19} + \frac{73420312961266711721355136572}{630534726674038762237} a^{17} + \frac{65205404853817539475253905471}{630534726674038762237} a^{15} + \frac{35940153314839751341488004896}{630534726674038762237} a^{13} + \frac{11960879065473313770190126920}{630534726674038762237} a^{11} + \frac{2290566807458779505722607146}{630534726674038762237} a^{9} + \frac{225799277096236679342538870}{630534726674038762237} a^{7} + \frac{7930801479464632737507834}{630534726674038762237} a^{5} - \frac{164570800343422927644771}{630534726674038762237} a^{3} - \frac{9001957169534022418083}{630534726674038762237} a \) (order $4$)
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:  \( 20980577392492.816 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, \(\Q(i, \sqrt{13})\), 6.6.14414517.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 6.6.2436053373.1, 6.0.922529088.2, 6.0.419904.1, 6.0.23762752.1, 6.0.1827904.1, 6.0.155907415872.2, 6.0.11992878144.4, 6.0.155907415872.1, 6.0.11992878144.3, 9.9.2565164201769.1, 12.0.851059918206111744.1, 12.0.564668382613504.1, 12.0.24307122323884757520384.2, 12.0.24307122323884757520384.1, 18.18.14456408038335708501176406117.1, 18.0.3789660628801475969332387805134848.1, 18.0.1724925183796757382490845609984.3

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed
13Data not computed