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Properties

Label 36.0.45914605021...2733.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{48}\cdot 13^{33}$
Root discriminant $45.42$
Ramified primes $3, 13$
Class number $729$ (GRH)
Class group $[3, 3, 9, 9]$ (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, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 85512, 133425, 45791, 51861, 22158, 20683, 10188, 8406, 4535, 3417, 1845, 1000, -417, 297, -109, 90, -27, 28, -6, 9, -1, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 3 x^{34} - x^{33} + 9 x^{32} - 6 x^{31} + 28 x^{30} - 27 x^{29} + 90 x^{28} - 109 x^{27} + 297 x^{26} - 417 x^{25} + 1000 x^{24} + 1845 x^{23} + 3417 x^{22} + 4535 x^{21} + 8406 x^{20} + 10188 x^{19} + 20683 x^{18} + 22158 x^{17} + 51861 x^{16} + 45791 x^{15} + 133425 x^{14} + 85512 x^{13} + 354484 x^{12} + 123111 x^{11} + 42756 x^{10} + 14849 x^{9} + 5157 x^{8} + 1791 x^{7} + 622 x^{6} + 216 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1 \)

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:  \(459146050215773460843525344476713987772454059613596693862733=3^{48}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.42$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(4,·)$, $\chi_{117}(7,·)$, $\chi_{117}(10,·)$, $\chi_{117}(16,·)$, $\chi_{117}(19,·)$, $\chi_{117}(22,·)$, $\chi_{117}(25,·)$, $\chi_{117}(28,·)$, $\chi_{117}(31,·)$, $\chi_{117}(34,·)$, $\chi_{117}(37,·)$, $\chi_{117}(40,·)$, $\chi_{117}(43,·)$, $\chi_{117}(46,·)$, $\chi_{117}(49,·)$, $\chi_{117}(55,·)$, $\chi_{117}(58,·)$, $\chi_{117}(61,·)$, $\chi_{117}(64,·)$, $\chi_{117}(67,·)$, $\chi_{117}(70,·)$, $\chi_{117}(73,·)$, $\chi_{117}(76,·)$, $\chi_{117}(79,·)$, $\chi_{117}(82,·)$, $\chi_{117}(85,·)$, $\chi_{117}(88,·)$, $\chi_{117}(94,·)$, $\chi_{117}(97,·)$, $\chi_{117}(100,·)$, $\chi_{117}(103,·)$, $\chi_{117}(106,·)$, $\chi_{117}(109,·)$, $\chi_{117}(112,·)$, $\chi_{117}(115,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{405821287} a^{25} + \frac{179067424}{405821287} a^{24} + \frac{3652395}{405821287} a^{23} + \frac{131380984}{405821287} a^{22} - \frac{168110239}{405821287} a^{21} - \frac{15330730}{405821287} a^{20} + \frac{175930873}{405821287} a^{19} + \frac{122118049}{405821287} a^{18} + \frac{137302062}{405821287} a^{17} + \frac{190423274}{405821287} a^{16} - \frac{116033150}{405821287} a^{15} + \frac{28146473}{405821287} a^{14} - \frac{132701437}{405821287} a^{13} + \frac{100237981}{405821287} a^{12} + \frac{20436748}{405821287} a^{11} - \frac{132703546}{405821287} a^{10} - \frac{38927737}{405821287} a^{9} - \frac{12726099}{405821287} a^{8} + \frac{15920335}{405821287} a^{7} + \frac{749440}{405821287} a^{6} + \frac{60487104}{405821287} a^{5} - \frac{13672015}{405821287} a^{4} + \frac{180711872}{405821287} a^{3} - \frac{101503149}{405821287} a^{2} + \frac{149986344}{405821287} a - \frac{79400032}{405821287}$, $\frac{1}{405821287} a^{26} - \frac{100234588}{405821287} a^{13} - \frac{145640894}{405821287}$, $\frac{1}{405821287} a^{27} - \frac{100234588}{405821287} a^{14} - \frac{145640894}{405821287} a$, $\frac{1}{405821287} a^{28} - \frac{100234588}{405821287} a^{15} - \frac{145640894}{405821287} a^{2}$, $\frac{1}{405821287} a^{29} - \frac{100234588}{405821287} a^{16} - \frac{145640894}{405821287} a^{3}$, $\frac{1}{405821287} a^{30} - \frac{100234588}{405821287} a^{17} - \frac{145640894}{405821287} a^{4}$, $\frac{1}{405821287} a^{31} - \frac{100234588}{405821287} a^{18} - \frac{145640894}{405821287} a^{5}$, $\frac{1}{405821287} a^{32} - \frac{100234588}{405821287} a^{19} - \frac{145640894}{405821287} a^{6}$, $\frac{1}{405821287} a^{33} - \frac{100234588}{405821287} a^{20} - \frac{145640894}{405821287} a^{7}$, $\frac{1}{405821287} a^{34} - \frac{100234588}{405821287} a^{21} - \frac{145640894}{405821287} a^{8}$, $\frac{1}{405821287} a^{35} - \frac{100234588}{405821287} a^{22} - \frac{145640894}{405821287} a^{9}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $729$ (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:  \( \frac{10251}{405821287} a^{29} + \frac{34737096}{405821287} a^{16} - \frac{9688000409}{405821287} a^{3} \) (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:  \( 20980577392492.816 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, 4.0.2197.1, 6.6.14414517.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 6.6.2436053373.1, 9.9.2565164201769.1, 12.0.456488925854205933.1, \(\Q(\zeta_{13})\), 12.0.77146628469360802677.1, 12.0.77146628469360802677.2, 18.18.14456408038335708501176406117.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
13Data not computed