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Properties

Label 36.0.11027665555...1853.1
Degree $36$
Signature $[0, 18]$
Discriminant $7^{24}\cdot 13^{33}$
Root discriminant $38.42$
Ramified primes $7, 13$
Class number $148$ (GRH)
Class group $[2, 74]$ (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -11, 25, -56, 126, -283, 636, -1429, 3211, -7215, 16212, 793, 7411, 2183, 3838, 1918, 2185, 1386, 1331, 909, 899, 442, 924, -507, 286, -155, 89, -47, 28, -14, 9, -4, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 3*x^34 - 4*x^33 + 9*x^32 - 14*x^31 + 28*x^30 - 47*x^29 + 89*x^28 - 155*x^27 + 286*x^26 - 507*x^25 + 924*x^24 + 442*x^23 + 899*x^22 + 909*x^21 + 1331*x^20 + 1386*x^19 + 2185*x^18 + 1918*x^17 + 3838*x^16 + 2183*x^15 + 7411*x^14 + 793*x^13 + 16212*x^12 - 7215*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 3*x^34 - 4*x^33 + 9*x^32 - 14*x^31 + 28*x^30 - 47*x^29 + 89*x^28 - 155*x^27 + 286*x^26 - 507*x^25 + 924*x^24 + 442*x^23 + 899*x^22 + 909*x^21 + 1331*x^20 + 1386*x^19 + 2185*x^18 + 1918*x^17 + 3838*x^16 + 2183*x^15 + 7411*x^14 + 793*x^13 + 16212*x^12 - 7215*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 3 x^{34} - 4 x^{33} + 9 x^{32} - 14 x^{31} + 28 x^{30} - 47 x^{29} + 89 x^{28} - 155 x^{27} + 286 x^{26} - 507 x^{25} + 924 x^{24} + 442 x^{23} + 899 x^{22} + 909 x^{21} + 1331 x^{20} + 1386 x^{19} + 2185 x^{18} + 1918 x^{17} + 3838 x^{16} + 2183 x^{15} + 7411 x^{14} + 793 x^{13} + 16212 x^{12} - 7215 x^{11} + 3211 x^{10} - 1429 x^{9} + 636 x^{8} - 283 x^{7} + 126 x^{6} - 56 x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1 \)

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:  \(1102766555593920971763188134004988790509406708671598011853=7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.42$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(2,·)$, $\chi_{91}(4,·)$, $\chi_{91}(8,·)$, $\chi_{91}(9,·)$, $\chi_{91}(11,·)$, $\chi_{91}(15,·)$, $\chi_{91}(16,·)$, $\chi_{91}(18,·)$, $\chi_{91}(22,·)$, $\chi_{91}(23,·)$, $\chi_{91}(25,·)$, $\chi_{91}(29,·)$, $\chi_{91}(30,·)$, $\chi_{91}(32,·)$, $\chi_{91}(36,·)$, $\chi_{91}(37,·)$, $\chi_{91}(43,·)$, $\chi_{91}(44,·)$, $\chi_{91}(46,·)$, $\chi_{91}(50,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(57,·)$, $\chi_{91}(58,·)$, $\chi_{91}(60,·)$, $\chi_{91}(64,·)$, $\chi_{91}(67,·)$, $\chi_{91}(71,·)$, $\chi_{91}(72,·)$, $\chi_{91}(74,·)$, $\chi_{91}(79,·)$, $\chi_{91}(81,·)$, $\chi_{91}(85,·)$, $\chi_{91}(86,·)$, $\chi_{91}(88,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{8}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{9}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{10}$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{11}$, $\frac{1}{1257987} a^{25} - \frac{35565}{419329} a^{24} - \frac{120667}{1257987} a^{23} - \frac{92722}{1257987} a^{22} + \frac{54674}{419329} a^{21} - \frac{50804}{1257987} a^{20} - \frac{44401}{419329} a^{19} + \frac{195617}{1257987} a^{18} - \frac{31166}{419329} a^{17} - \frac{22600}{419329} a^{16} + \frac{76421}{1257987} a^{15} + \frac{113810}{1257987} a^{14} - \frac{28768}{1257987} a^{13} - \frac{461542}{1257987} a^{12} + \frac{107525}{419329} a^{11} + \frac{546037}{1257987} a^{10} - \frac{362429}{1257987} a^{9} + \frac{33254}{419329} a^{8} + \frac{560075}{1257987} a^{7} - \frac{101217}{419329} a^{6} + \frac{265576}{1257987} a^{5} + \frac{175285}{419329} a^{4} + \frac{40325}{419329} a^{3} - \frac{61676}{1257987} a^{2} + \frac{410152}{1257987} a - \frac{412529}{1257987}$, $\frac{1}{3773961} a^{26} + \frac{463636}{3773961} a^{13} + \frac{86830}{3773961}$, $\frac{1}{3773961} a^{27} + \frac{463636}{3773961} a^{14} + \frac{86830}{3773961} a$, $\frac{1}{3773961} a^{28} + \frac{463636}{3773961} a^{15} + \frac{86830}{3773961} a^{2}$, $\frac{1}{3773961} a^{29} + \frac{463636}{3773961} a^{16} + \frac{86830}{3773961} a^{3}$, $\frac{1}{3773961} a^{30} + \frac{463636}{3773961} a^{17} + \frac{86830}{3773961} a^{4}$, $\frac{1}{3773961} a^{31} + \frac{463636}{3773961} a^{18} + \frac{86830}{3773961} a^{5}$, $\frac{1}{3773961} a^{32} + \frac{463636}{3773961} a^{19} + \frac{86830}{3773961} a^{6}$, $\frac{1}{3773961} a^{33} + \frac{463636}{3773961} a^{20} + \frac{86830}{3773961} a^{7}$, $\frac{1}{3773961} a^{34} + \frac{463636}{3773961} a^{21} + \frac{86830}{3773961} a^{8}$, $\frac{1}{3773961} a^{35} + \frac{463636}{3773961} a^{22} + \frac{86830}{3773961} a^{9}$

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

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (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:  \( \frac{332648}{3773961} a^{35} - \frac{332648}{1257987} a^{34} + \frac{1330592}{3773961} a^{33} - \frac{332648}{419329} a^{32} + \frac{4657072}{3773961} a^{31} - \frac{9314144}{3773961} a^{30} + \frac{15634456}{3773961} a^{29} - \frac{29605672}{3773961} a^{28} + \frac{51560440}{3773961} a^{27} - \frac{95137328}{3773961} a^{26} + \frac{56217512}{1257987} a^{25} - \frac{102455584}{1257987} a^{24} + \frac{183178099}{1257987} a^{23} - \frac{299050552}{3773961} a^{22} - \frac{33597448}{419329} a^{21} - \frac{442754488}{3773961} a^{20} - \frac{51227792}{419329} a^{19} - \frac{726835880}{3773961} a^{18} - \frac{638018864}{3773961} a^{17} - \frac{1276703024}{3773961} a^{16} - \frac{726170584}{3773961} a^{15} - \frac{2465254328}{3773961} a^{14} - \frac{263789864}{3773961} a^{13} - \frac{1797629792}{1257987} a^{12} + \frac{800018440}{1257987} a^{11} - \frac{4483351483}{1257987} a^{10} + \frac{475353992}{3773961} a^{9} - \frac{70521376}{1257987} a^{8} + \frac{94139384}{3773961} a^{7} - \frac{4657072}{419329} a^{6} + \frac{18628288}{3773961} a^{5} - \frac{8316200}{3773961} a^{4} + \frac{3659128}{3773961} a^{3} - \frac{1663240}{3773961} a^{2} + \frac{665296}{3773961} a - \frac{332648}{3773961} \) (order $26$)
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:  \( 4866030378143.887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 4.0.2197.1, \(\Q(\zeta_{13})^+\), 6.6.5274997.1, 6.6.891474493.1, 6.6.891474493.2, 9.9.567869252041.1, \(\Q(\zeta_{13})\), 12.0.61132828589969773.1, 12.0.10331448031704891637.1, 12.0.10331448031704891637.2, 18.18.708478645847689707516501157.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
7Data not computed
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$