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Properties

Label 36.0.21809974230...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 5^{18}\cdot 19^{32}$
Root discriminant $61.26$
Ramified primes $2, 5, 19$
Class number $81529$ (GRH)
Class group $[81529]$ (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, 135, 0, 4335, 0, 61182, 0, 460449, 0, 2080353, 0, 6090371, 0, 12133089, 0, 16964206, 0, 16953894, 0, 12218191, 0, 6356559, 0, 2373706, 0, 628407, 0, 115592, 0, 14310, 0, 1129, 0, 51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 51*x^34 + 1129*x^32 + 14310*x^30 + 115592*x^28 + 628407*x^26 + 2373706*x^24 + 6356559*x^22 + 12218191*x^20 + 16953894*x^18 + 16964206*x^16 + 12133089*x^14 + 6090371*x^12 + 2080353*x^10 + 460449*x^8 + 61182*x^6 + 4335*x^4 + 135*x^2 + 1)
 
gp: K = bnfinit(x^36 + 51*x^34 + 1129*x^32 + 14310*x^30 + 115592*x^28 + 628407*x^26 + 2373706*x^24 + 6356559*x^22 + 12218191*x^20 + 16953894*x^18 + 16964206*x^16 + 12133089*x^14 + 6090371*x^12 + 2080353*x^10 + 460449*x^8 + 61182*x^6 + 4335*x^4 + 135*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 51 x^{34} + 1129 x^{32} + 14310 x^{30} + 115592 x^{28} + 628407 x^{26} + 2373706 x^{24} + 6356559 x^{22} + 12218191 x^{20} + 16953894 x^{18} + 16964206 x^{16} + 12133089 x^{14} + 6090371 x^{12} + 2080353 x^{10} + 460449 x^{8} + 61182 x^{6} + 4335 x^{4} + 135 x^{2} + 1 \)

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:  \(21809974230617285625493307131308780792777539584000000000000000000=2^{36}\cdot 5^{18}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.26$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(380=2^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(131,·)$, $\chi_{380}(9,·)$, $\chi_{380}(11,·)$, $\chi_{380}(271,·)$, $\chi_{380}(149,·)$, $\chi_{380}(111,·)$, $\chi_{380}(329,·)$, $\chi_{380}(159,·)$, $\chi_{380}(161,·)$, $\chi_{380}(291,·)$, $\chi_{380}(39,·)$, $\chi_{380}(169,·)$, $\chi_{380}(301,·)$, $\chi_{380}(49,·)$, $\chi_{380}(309,·)$, $\chi_{380}(311,·)$, $\chi_{380}(61,·)$, $\chi_{380}(191,·)$, $\chi_{380}(321,·)$, $\chi_{380}(139,·)$, $\chi_{380}(199,·)$, $\chi_{380}(201,·)$, $\chi_{380}(81,·)$, $\chi_{380}(339,·)$, $\chi_{380}(349,·)$, $\chi_{380}(351,·)$, $\chi_{380}(99,·)$, $\chi_{380}(101,·)$, $\chi_{380}(359,·)$, $\chi_{380}(289,·)$, $\chi_{380}(239,·)$, $\chi_{380}(229,·)$, $\chi_{380}(119,·)$, $\chi_{380}(121,·)$, $\chi_{380}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{191} a^{32} - \frac{26}{191} a^{30} + \frac{62}{191} a^{28} - \frac{58}{191} a^{26} + \frac{68}{191} a^{24} - \frac{72}{191} a^{22} + \frac{34}{191} a^{20} - \frac{75}{191} a^{18} - \frac{94}{191} a^{16} - \frac{30}{191} a^{14} + \frac{62}{191} a^{12} + \frac{14}{191} a^{10} - \frac{20}{191} a^{8} + \frac{2}{191} a^{6} + \frac{54}{191} a^{4} + \frac{80}{191} a^{2} - \frac{44}{191}$, $\frac{1}{191} a^{33} - \frac{26}{191} a^{31} + \frac{62}{191} a^{29} - \frac{58}{191} a^{27} + \frac{68}{191} a^{25} - \frac{72}{191} a^{23} + \frac{34}{191} a^{21} - \frac{75}{191} a^{19} - \frac{94}{191} a^{17} - \frac{30}{191} a^{15} + \frac{62}{191} a^{13} + \frac{14}{191} a^{11} - \frac{20}{191} a^{9} + \frac{2}{191} a^{7} + \frac{54}{191} a^{5} + \frac{80}{191} a^{3} - \frac{44}{191} a$, $\frac{1}{338790140617299720099338629} a^{34} - \frac{397072552054221897523927}{338790140617299720099338629} a^{32} + \frac{141362563678141518527336116}{338790140617299720099338629} a^{30} + \frac{96630908446260080571412799}{338790140617299720099338629} a^{28} + \frac{44092964371363913615683696}{338790140617299720099338629} a^{26} - \frac{154604783883035881443092879}{338790140617299720099338629} a^{24} + \frac{59798427387781854796054806}{338790140617299720099338629} a^{22} - \frac{47640908691563914641554989}{338790140617299720099338629} a^{20} + \frac{97908979576965514633910660}{338790140617299720099338629} a^{18} + \frac{30450645831861416557665230}{338790140617299720099338629} a^{16} - \frac{33325894172620589671520700}{338790140617299720099338629} a^{14} + \frac{27983465610529562439295501}{338790140617299720099338629} a^{12} - \frac{162480109388502230759755198}{338790140617299720099338629} a^{10} + \frac{112527384127951252434424045}{338790140617299720099338629} a^{8} + \frac{69738267733371272948418866}{338790140617299720099338629} a^{6} - \frac{84347799751206012809056347}{338790140617299720099338629} a^{4} + \frac{147164494731434794640214441}{338790140617299720099338629} a^{2} - \frac{36440309813201613027660413}{338790140617299720099338629}$, $\frac{1}{338790140617299720099338629} a^{35} - \frac{397072552054221897523927}{338790140617299720099338629} a^{33} + \frac{141362563678141518527336116}{338790140617299720099338629} a^{31} + \frac{96630908446260080571412799}{338790140617299720099338629} a^{29} + \frac{44092964371363913615683696}{338790140617299720099338629} a^{27} - \frac{154604783883035881443092879}{338790140617299720099338629} a^{25} + \frac{59798427387781854796054806}{338790140617299720099338629} a^{23} - \frac{47640908691563914641554989}{338790140617299720099338629} a^{21} + \frac{97908979576965514633910660}{338790140617299720099338629} a^{19} + \frac{30450645831861416557665230}{338790140617299720099338629} a^{17} - \frac{33325894172620589671520700}{338790140617299720099338629} a^{15} + \frac{27983465610529562439295501}{338790140617299720099338629} a^{13} - \frac{162480109388502230759755198}{338790140617299720099338629} a^{11} + \frac{112527384127951252434424045}{338790140617299720099338629} a^{9} + \frac{69738267733371272948418866}{338790140617299720099338629} a^{7} - \frac{84347799751206012809056347}{338790140617299720099338629} a^{5} + \frac{147164494731434794640214441}{338790140617299720099338629} a^{3} - \frac{36440309813201613027660413}{338790140617299720099338629} a$

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

Class group and class number

$C_{81529}$, which has order $81529$ (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:  \( \frac{59561509973630}{15667926926913551} a^{35} + \frac{3112708933605875}{15667926926913551} a^{33} + \frac{69482806818005545}{15667926926913551} a^{31} + \frac{858399173246355855}{15667926926913551} a^{29} + \frac{6284825315838164355}{15667926926913551} a^{27} + \frac{25949709904771875205}{15667926926913551} a^{25} + \frac{190289360008466200}{82031031030961} a^{23} - \frac{200796361645179790130}{15667926926913551} a^{21} - \frac{1346198266545412316466}{15667926926913551} a^{19} - \frac{3964675996476563721921}{15667926926913551} a^{17} - \frac{7093722793120017574377}{15667926926913551} a^{15} - \frac{8245601636152080355870}{15667926926913551} a^{13} - \frac{6287701511979159303664}{15667926926913551} a^{11} - \frac{3080802927178378824528}{15667926926913551} a^{9} - \frac{921573601675037044693}{15667926926913551} a^{7} - \frac{152795559134850489211}{15667926926913551} a^{5} - \frac{11634293724128822660}{15667926926913551} a^{3} - \frac{259134257421887306}{15667926926913551} a \) (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:  \( 3431432369157.3267 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 3.3.361.1, \(\Q(i, \sqrt{5})\), 6.0.8340544.1, 6.6.16290125.1, 6.0.1042568000.1, \(\Q(\zeta_{19})^+\), 12.0.1086948034624000000.1, 18.0.75613185918270483380568064.1, 18.18.563362135874260093126953125.1, 18.0.147682003746622037852672000000000.1

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 $18^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
5Data not computed
19Data not computed