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Properties

Label 36.0.25422036462...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{88}\cdot 5^{18}$
Root discriminant $65.59$
Ramified primes $2, 3, 5$
Class number $237177$ (GRH)
Class group $[19, 12483]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 243, 0, 10341, 0, 156168, 0, 1161216, 0, 5001777, 0, 13732377, 0, 25472799, 0, 33080274, 0, 30703482, 0, 20569950, 0, 9962190, 0, 3468195, 0, 857304, 0, 147510, 0, 17118, 0, 1269, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 54*x^34 + 1269*x^32 + 17118*x^30 + 147510*x^28 + 857304*x^26 + 3468195*x^24 + 9962190*x^22 + 20569950*x^20 + 30703482*x^18 + 33080274*x^16 + 25472799*x^14 + 13732377*x^12 + 5001777*x^10 + 1161216*x^8 + 156168*x^6 + 10341*x^4 + 243*x^2 + 1)
 
gp: K = bnfinit(x^36 + 54*x^34 + 1269*x^32 + 17118*x^30 + 147510*x^28 + 857304*x^26 + 3468195*x^24 + 9962190*x^22 + 20569950*x^20 + 30703482*x^18 + 33080274*x^16 + 25472799*x^14 + 13732377*x^12 + 5001777*x^10 + 1161216*x^8 + 156168*x^6 + 10341*x^4 + 243*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 54 x^{34} + 1269 x^{32} + 17118 x^{30} + 147510 x^{28} + 857304 x^{26} + 3468195 x^{24} + 9962190 x^{22} + 20569950 x^{20} + 30703482 x^{18} + 33080274 x^{16} + 25472799 x^{14} + 13732377 x^{12} + 5001777 x^{10} + 1161216 x^{8} + 156168 x^{6} + 10341 x^{4} + 243 x^{2} + 1 \)

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:  \(254220364621420587352541611206660801314202845184000000000000000000=2^{36}\cdot 3^{88}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.59$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(540=2^{2}\cdot 3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{540}(1,·)$, $\chi_{540}(259,·)$, $\chi_{540}(391,·)$, $\chi_{540}(139,·)$, $\chi_{540}(271,·)$, $\chi_{540}(529,·)$, $\chi_{540}(19,·)$, $\chi_{540}(151,·)$, $\chi_{540}(409,·)$, $\chi_{540}(31,·)$, $\chi_{540}(289,·)$, $\chi_{540}(421,·)$, $\chi_{540}(169,·)$, $\chi_{540}(301,·)$, $\chi_{540}(49,·)$, $\chi_{540}(181,·)$, $\chi_{540}(439,·)$, $\chi_{540}(61,·)$, $\chi_{540}(319,·)$, $\chi_{540}(451,·)$, $\chi_{540}(199,·)$, $\chi_{540}(331,·)$, $\chi_{540}(79,·)$, $\chi_{540}(211,·)$, $\chi_{540}(469,·)$, $\chi_{540}(91,·)$, $\chi_{540}(349,·)$, $\chi_{540}(481,·)$, $\chi_{540}(229,·)$, $\chi_{540}(361,·)$, $\chi_{540}(109,·)$, $\chi_{540}(241,·)$, $\chi_{540}(499,·)$, $\chi_{540}(121,·)$, $\chi_{540}(379,·)$, $\chi_{540}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{109} a^{32} + \frac{27}{109} a^{30} + \frac{23}{109} a^{28} + \frac{31}{109} a^{26} - \frac{51}{109} a^{24} - \frac{25}{109} a^{22} + \frac{43}{109} a^{20} + \frac{18}{109} a^{18} - \frac{30}{109} a^{16} + \frac{41}{109} a^{14} - \frac{12}{109} a^{12} - \frac{10}{109} a^{10} - \frac{54}{109} a^{8} - \frac{36}{109} a^{6} + \frac{44}{109} a^{4} - \frac{45}{109} a^{2} + \frac{35}{109}$, $\frac{1}{109} a^{33} + \frac{27}{109} a^{31} + \frac{23}{109} a^{29} + \frac{31}{109} a^{27} - \frac{51}{109} a^{25} - \frac{25}{109} a^{23} + \frac{43}{109} a^{21} + \frac{18}{109} a^{19} - \frac{30}{109} a^{17} + \frac{41}{109} a^{15} - \frac{12}{109} a^{13} - \frac{10}{109} a^{11} - \frac{54}{109} a^{9} - \frac{36}{109} a^{7} + \frac{44}{109} a^{5} - \frac{45}{109} a^{3} + \frac{35}{109} a$, $\frac{1}{102140268639359960440241042941} a^{34} - \frac{239583062672104440671165527}{102140268639359960440241042941} a^{32} + \frac{13335369090428041609783965327}{102140268639359960440241042941} a^{30} - \frac{39684868003465336383669622279}{102140268639359960440241042941} a^{28} + \frac{37576589533067046675033133932}{102140268639359960440241042941} a^{26} - \frac{12204273708471496285781340761}{102140268639359960440241042941} a^{24} - \frac{17003731897076903252373894695}{102140268639359960440241042941} a^{22} + \frac{17917587013035285879934198850}{102140268639359960440241042941} a^{20} + \frac{6148817521222229056183149323}{102140268639359960440241042941} a^{18} - \frac{10945077120088504974479525518}{102140268639359960440241042941} a^{16} - \frac{28252491669675871003333986781}{102140268639359960440241042941} a^{14} + \frac{28032487541945859397735128895}{102140268639359960440241042941} a^{12} + \frac{15967866237577143095154266299}{102140268639359960440241042941} a^{10} - \frac{28651765768666262006343582494}{102140268639359960440241042941} a^{8} - \frac{40619983613778289672513054049}{102140268639359960440241042941} a^{6} - \frac{349980046154614650239449453}{937066684764770279268266449} a^{4} - \frac{41913816524167683171399855437}{102140268639359960440241042941} a^{2} - \frac{633939339880283672343386737}{102140268639359960440241042941}$, $\frac{1}{102140268639359960440241042941} a^{35} - \frac{239583062672104440671165527}{102140268639359960440241042941} a^{33} + \frac{13335369090428041609783965327}{102140268639359960440241042941} a^{31} - \frac{39684868003465336383669622279}{102140268639359960440241042941} a^{29} + \frac{37576589533067046675033133932}{102140268639359960440241042941} a^{27} - \frac{12204273708471496285781340761}{102140268639359960440241042941} a^{25} - \frac{17003731897076903252373894695}{102140268639359960440241042941} a^{23} + \frac{17917587013035285879934198850}{102140268639359960440241042941} a^{21} + \frac{6148817521222229056183149323}{102140268639359960440241042941} a^{19} - \frac{10945077120088504974479525518}{102140268639359960440241042941} a^{17} - \frac{28252491669675871003333986781}{102140268639359960440241042941} a^{15} + \frac{28032487541945859397735128895}{102140268639359960440241042941} a^{13} + \frac{15967866237577143095154266299}{102140268639359960440241042941} a^{11} - \frac{28651765768666262006343582494}{102140268639359960440241042941} a^{9} - \frac{40619983613778289672513054049}{102140268639359960440241042941} a^{7} - \frac{349980046154614650239449453}{937066684764770279268266449} a^{5} - \frac{41913816524167683171399855437}{102140268639359960440241042941} a^{3} - \frac{633939339880283672343386737}{102140268639359960440241042941} a$

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

Class group and class number

$C_{19}\times C_{12483}$, which has order $237177$ (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:  \( -\frac{9156161337255}{404563972627459} a^{35} - \frac{553753748529990}{404563972627459} a^{33} - \frac{14776692695059680}{404563972627459} a^{31} - \frac{229566483260619435}{404563972627459} a^{29} - \frac{2309390501936937694}{404563972627459} a^{27} - \frac{15849660139449985557}{404563972627459} a^{25} - \frac{76307421122774899101}{404563972627459} a^{23} - \frac{261550565854389792942}{404563972627459} a^{21} - \frac{642586339561222429380}{404563972627459} a^{19} - \frac{1132379039636458057134}{404563972627459} a^{17} - \frac{1423142833573441910592}{404563972627459} a^{15} - \frac{1257865456133169734787}{404563972627459} a^{13} - \frac{762157772162799663924}{404563972627459} a^{11} - \frac{303227678870839233290}{404563972627459} a^{9} - \frac{73736314364009817522}{404563972627459} a^{7} - \frac{9701599181189978691}{404563972627459} a^{5} - \frac{557393906101162491}{404563972627459} a^{3} - \frac{8259308572854447}{404563972627459} a \) (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:  \( 6550249244897.024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), \(\Q(i, \sqrt{5})\), 6.0.419904.1, 6.6.820125.1, 6.0.52488000.1, \(\Q(\zeta_{27})^+\), 12.0.2754990144000000.1, 18.0.258151783382020583032356864.7, 18.18.1923380668327365689220703125.1, 18.0.504202701918008951235072000000000.2

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 R $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
2Data not computed
3Data not computed
5Data not computed