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Properties

Label 36.0.90881971984...6609.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 19^{18}$
Root discriminant $67.95$
Ramified primes $3, 19$
Class number Not computed
Class group Not computed
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![3814697265625, 0, 0, 0, 0, 0, 0, 0, 0, -4894531250, 0, 0, 0, 0, 0, 0, 0, 0, 4326911, 0, 0, 0, 0, 0, 0, 0, 0, -2506, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 2506*x^27 + 4326911*x^18 - 4894531250*x^9 + 3814697265625)
 
gp: K = bnfinit(x^36 - 2506*x^27 + 4326911*x^18 - 4894531250*x^9 + 3814697265625, 1)
 

Normalized defining polynomial

\( x^{36} - 2506 x^{27} + 4326911 x^{18} - 4894531250 x^{9} + 3814697265625 \)

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:  \(908819719841676525092557848566442535650646971382385935623234726609=3^{90}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.95$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(513=3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{513}(512,·)$, $\chi_{513}(1,·)$, $\chi_{513}(134,·)$, $\chi_{513}(265,·)$, $\chi_{513}(398,·)$, $\chi_{513}(400,·)$, $\chi_{513}(20,·)$, $\chi_{513}(151,·)$, $\chi_{513}(284,·)$, $\chi_{513}(286,·)$, $\chi_{513}(419,·)$, $\chi_{513}(37,·)$, $\chi_{513}(170,·)$, $\chi_{513}(172,·)$, $\chi_{513}(305,·)$, $\chi_{513}(436,·)$, $\chi_{513}(56,·)$, $\chi_{513}(58,·)$, $\chi_{513}(191,·)$, $\chi_{513}(322,·)$, $\chi_{513}(455,·)$, $\chi_{513}(457,·)$, $\chi_{513}(77,·)$, $\chi_{513}(208,·)$, $\chi_{513}(341,·)$, $\chi_{513}(343,·)$, $\chi_{513}(476,·)$, $\chi_{513}(94,·)$, $\chi_{513}(227,·)$, $\chi_{513}(229,·)$, $\chi_{513}(362,·)$, $\chi_{513}(493,·)$, $\chi_{513}(113,·)$, $\chi_{513}(115,·)$, $\chi_{513}(248,·)$, $\chi_{513}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{284} a^{18} + \frac{25}{284} a^{9} + \frac{57}{284}$, $\frac{1}{1420} a^{19} + \frac{309}{1420} a^{10} + \frac{341}{1420} a$, $\frac{1}{7100} a^{20} - \frac{2531}{7100} a^{11} + \frac{1761}{7100} a^{2}$, $\frac{1}{35500} a^{21} - \frac{9631}{35500} a^{12} - \frac{5339}{35500} a^{3}$, $\frac{1}{177500} a^{22} - \frac{80631}{177500} a^{13} + \frac{65661}{177500} a^{4}$, $\frac{1}{887500} a^{23} - \frac{80631}{887500} a^{14} + \frac{420661}{887500} a^{5}$, $\frac{1}{4437500} a^{24} - \frac{80631}{4437500} a^{15} + \frac{420661}{4437500} a^{6}$, $\frac{1}{22187500} a^{25} - \frac{80631}{22187500} a^{16} + \frac{420661}{22187500} a^{7}$, $\frac{1}{110937500} a^{26} - \frac{22268131}{110937500} a^{17} + \frac{420661}{110937500} a^{8}$, $\frac{1}{2400083445312500} a^{27} + \frac{151979}{554687500} a^{18} + \frac{82031251}{554687500} a^{9} - \frac{93029213}{307210681}$, $\frac{1}{12000417226562500} a^{28} + \frac{151979}{2773437500} a^{19} - \frac{1027343749}{2773437500} a^{10} - \frac{141490115}{307210681} a$, $\frac{1}{60002086132812500} a^{29} + \frac{151979}{13867187500} a^{20} + \frac{4519531251}{13867187500} a^{11} + \frac{165720566}{1536053405} a^{2}$, $\frac{1}{300010430664062500} a^{30} + \frac{151979}{69335937500} a^{21} - \frac{23214843749}{69335937500} a^{12} - \frac{2906386244}{7680267025} a^{3}$, $\frac{1}{1500052153320312500} a^{31} + \frac{151979}{346679687500} a^{22} - \frac{161886718749}{346679687500} a^{13} - \frac{10586653269}{38401335125} a^{4}$, $\frac{1}{7500260766601562500} a^{32} + \frac{151979}{1733398437500} a^{23} - \frac{508566406249}{1733398437500} a^{14} - \frac{87389323519}{192006675625} a^{5}$, $\frac{1}{37501303833007812500} a^{33} + \frac{151979}{8666992187500} a^{24} - \frac{2241964843749}{8666992187500} a^{15} - \frac{471402674769}{960033378125} a^{6}$, $\frac{1}{187506519165039062500} a^{34} + \frac{151979}{43334960937500} a^{25} - \frac{10908957031249}{43334960937500} a^{16} - \frac{1431436052894}{4800166890625} a^{7}$, $\frac{1}{937532595825195312500} a^{35} + \frac{151979}{216674804687500} a^{26} - \frac{97578878906249}{216674804687500} a^{17} + \frac{3368730837731}{24000834453125} a^{8}$

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

Class group and class number

Not computed

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:  \( \frac{19645787}{750026076660156250} a^{31} - \frac{15679}{346679687500} a^{22} + \frac{21713449}{346679687500} a^{13} - \frac{48996875}{1228842724} a^{4} \) (order $54$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\zeta_{9})\), 6.6.135005697.1, 6.0.45001899.1, \(\Q(\zeta_{27})^+\), 12.0.18226538222455809.1, \(\Q(\zeta_{27})\), 18.18.953320365796134938048201146413897.1, 18.0.317773455265378312682733715471299.2

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 $18^{2}$ R $18^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
3Data not computed
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$