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Properties

Label 36.0.22150200378...8944.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{18}\cdot 19^{32}$
Root discriminant $47.45$
Ramified primes $2, 3, 19$
Class number $171$ (GRH)
Class group $[171]$ (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, -45, 0, 1695, 0, -13002, 0, 66033, 0, -190091, 0, 384931, 0, -549123, 0, 592078, 0, -487810, 0, 316711, 0, -162613, 0, 66874, 0, -21853, 0, 5664, 0, -1130, 0, 169, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 17*x^34 + 169*x^32 - 1130*x^30 + 5664*x^28 - 21853*x^26 + 66874*x^24 - 162613*x^22 + 316711*x^20 - 487810*x^18 + 592078*x^16 - 549123*x^14 + 384931*x^12 - 190091*x^10 + 66033*x^8 - 13002*x^6 + 1695*x^4 - 45*x^2 + 1)
 
gp: K = bnfinit(x^36 - 17*x^34 + 169*x^32 - 1130*x^30 + 5664*x^28 - 21853*x^26 + 66874*x^24 - 162613*x^22 + 316711*x^20 - 487810*x^18 + 592078*x^16 - 549123*x^14 + 384931*x^12 - 190091*x^10 + 66033*x^8 - 13002*x^6 + 1695*x^4 - 45*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 17 x^{34} + 169 x^{32} - 1130 x^{30} + 5664 x^{28} - 21853 x^{26} + 66874 x^{24} - 162613 x^{22} + 316711 x^{20} - 487810 x^{18} + 592078 x^{16} - 549123 x^{14} + 384931 x^{12} - 190091 x^{10} + 66033 x^{8} - 13002 x^{6} + 1695 x^{4} - 45 x^{2} + 1 \)

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:  \(2215020037800761116296816339199940379209022060324490202578944=2^{36}\cdot 3^{18}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.45$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(131,·)$, $\chi_{228}(5,·)$, $\chi_{228}(7,·)$, $\chi_{228}(137,·)$, $\chi_{228}(11,·)$, $\chi_{228}(17,·)$, $\chi_{228}(149,·)$, $\chi_{228}(23,·)$, $\chi_{228}(25,·)$, $\chi_{228}(47,·)$, $\chi_{228}(157,·)$, $\chi_{228}(161,·)$, $\chi_{228}(35,·)$, $\chi_{228}(163,·)$, $\chi_{228}(169,·)$, $\chi_{228}(43,·)$, $\chi_{228}(175,·)$, $\chi_{228}(49,·)$, $\chi_{228}(55,·)$, $\chi_{228}(187,·)$, $\chi_{228}(61,·)$, $\chi_{228}(191,·)$, $\chi_{228}(139,·)$, $\chi_{228}(197,·)$, $\chi_{228}(199,·)$, $\chi_{228}(73,·)$, $\chi_{228}(77,·)$, $\chi_{228}(83,·)$, $\chi_{228}(85,·)$, $\chi_{228}(215,·)$, $\chi_{228}(101,·)$, $\chi_{228}(115,·)$, $\chi_{228}(119,·)$, $\chi_{228}(121,·)$, $\chi_{228}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{37} a^{28} + \frac{4}{37} a^{26} + \frac{16}{37} a^{24} - \frac{11}{37} a^{16} - \frac{7}{37} a^{14} + \frac{9}{37} a^{12} + \frac{10}{37} a^{4} + \frac{3}{37} a^{2} + \frac{12}{37}$, $\frac{1}{37} a^{29} + \frac{4}{37} a^{27} + \frac{16}{37} a^{25} - \frac{11}{37} a^{17} - \frac{7}{37} a^{15} + \frac{9}{37} a^{13} + \frac{10}{37} a^{5} + \frac{3}{37} a^{3} + \frac{12}{37} a$, $\frac{1}{37} a^{30} + \frac{10}{37} a^{24} - \frac{11}{37} a^{18} + \frac{1}{37} a^{12} + \frac{10}{37} a^{6} - \frac{11}{37}$, $\frac{1}{37} a^{31} + \frac{10}{37} a^{25} - \frac{11}{37} a^{19} + \frac{1}{37} a^{13} + \frac{10}{37} a^{7} - \frac{11}{37} a$, $\frac{1}{37} a^{32} + \frac{10}{37} a^{26} - \frac{11}{37} a^{20} + \frac{1}{37} a^{14} + \frac{10}{37} a^{8} - \frac{11}{37} a^{2}$, $\frac{1}{37} a^{33} + \frac{10}{37} a^{27} - \frac{11}{37} a^{21} + \frac{1}{37} a^{15} + \frac{10}{37} a^{9} - \frac{11}{37} a^{3}$, $\frac{1}{57118061071237599258033288313} a^{34} - \frac{562600824958963995812081486}{57118061071237599258033288313} a^{32} - \frac{448807443347415556458904948}{57118061071237599258033288313} a^{30} - \frac{533817949968534897231750798}{57118061071237599258033288313} a^{28} + \frac{33962486031347116744561722}{249423847472653271869140997} a^{26} - \frac{9930900060489045750964940968}{57118061071237599258033288313} a^{24} + \frac{15691695729946070417197777447}{57118061071237599258033288313} a^{22} + \frac{25774056983632964157176648148}{57118061071237599258033288313} a^{20} - \frac{1616026988670696881567624688}{57118061071237599258033288313} a^{18} - \frac{22156432587936718189242127121}{57118061071237599258033288313} a^{16} - \frac{13201468499411361331930015822}{57118061071237599258033288313} a^{14} - \frac{15882856580042423864710821167}{57118061071237599258033288313} a^{12} + \frac{4471241508449024007978869677}{57118061071237599258033288313} a^{10} + \frac{27263888700260645096868472200}{57118061071237599258033288313} a^{8} - \frac{20560851795679057364022555702}{57118061071237599258033288313} a^{6} + \frac{20976725452887937630990242929}{57118061071237599258033288313} a^{4} + \frac{23706273027634887149215856239}{57118061071237599258033288313} a^{2} - \frac{9961109638644629283251629416}{57118061071237599258033288313}$, $\frac{1}{57118061071237599258033288313} a^{35} - \frac{562600824958963995812081486}{57118061071237599258033288313} a^{33} - \frac{448807443347415556458904948}{57118061071237599258033288313} a^{31} - \frac{533817949968534897231750798}{57118061071237599258033288313} a^{29} + \frac{33962486031347116744561722}{249423847472653271869140997} a^{27} - \frac{9930900060489045750964940968}{57118061071237599258033288313} a^{25} + \frac{15691695729946070417197777447}{57118061071237599258033288313} a^{23} + \frac{25774056983632964157176648148}{57118061071237599258033288313} a^{21} - \frac{1616026988670696881567624688}{57118061071237599258033288313} a^{19} - \frac{22156432587936718189242127121}{57118061071237599258033288313} a^{17} - \frac{13201468499411361331930015822}{57118061071237599258033288313} a^{15} - \frac{15882856580042423864710821167}{57118061071237599258033288313} a^{13} + \frac{4471241508449024007978869677}{57118061071237599258033288313} a^{11} + \frac{27263888700260645096868472200}{57118061071237599258033288313} a^{9} - \frac{20560851795679057364022555702}{57118061071237599258033288313} a^{7} + \frac{20976725452887937630990242929}{57118061071237599258033288313} a^{5} + \frac{23706273027634887149215856239}{57118061071237599258033288313} a^{3} - \frac{9961109638644629283251629416}{57118061071237599258033288313} a$

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

Class group and class number

$C_{171}$, which has order $171$ (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:  \( \frac{3119768034844994278395666934}{57118061071237599258033288313} a^{35} - \frac{52686989737560164127211774886}{57118061071237599258033288313} a^{33} + \frac{521262559950391673415837953047}{57118061071237599258033288313} a^{31} - \frac{3465598390908035834055204859481}{57118061071237599258033288313} a^{29} + \frac{75408414660554749807210242843}{249423847472653271869140997} a^{27} - \frac{66150057942679514451058229463121}{57118061071237599258033288313} a^{25} + \frac{200759166176499311751503439400129}{57118061071237599258033288313} a^{23} - \frac{483041260331622135857931022469482}{57118061071237599258033288313} a^{21} + \frac{928503978511893230542710915712580}{57118061071237599258033288313} a^{19} - \frac{1404660975853884939458076936570698}{57118061071237599258033288313} a^{17} + \frac{1664382056444629854639293416812681}{57118061071237599258033288313} a^{15} - \frac{1488216613926939599168448112804054}{57118061071237599258033288313} a^{13} + \frac{988768418221129181284238929520978}{57118061071237599258033288313} a^{11} - \frac{442023473554713121530554571519222}{57118061071237599258033288313} a^{9} + \frac{130253680987610904059613358788156}{57118061071237599258033288313} a^{7} - \frac{14528014841503868895874615064517}{57118061071237599258033288313} a^{5} + \frac{387861834973960968728909210295}{57118061071237599258033288313} a^{3} + \frac{337657092403592416021801765204}{57118061071237599258033288313} a \) (order $12$)
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:  \( 119587984215961.08 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), 3.3.361.1, \(\Q(\zeta_{12})\), 6.0.3518667.1, 6.6.225194688.1, 6.0.8340544.1, \(\Q(\zeta_{19})^+\), 12.0.50712647503417344.1, 18.0.5677392343251487443465123.1, 18.18.1488294338429317924379721203712.1, 18.0.75613185918270483380568064.1

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 R $18^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{36}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
3Data not computed
19Data not computed