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Properties

Label 36.0.15377735736...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 5^{27}\cdot 19^{34}$
Root discriminant $107.89$
Ramified primes $2, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![705078125, 0, 10576171875, 0, 59649609375, 0, 176833593750, 0, 318610703125, 0, 379529453125, 0, 315440671875, 0, 189620890625, 0, 84453693750, 0, 28304418750, 0, 7200394375, 0, 1393946875, 0, 204644250, 0, 22545875, 0, 1827800, 0, 105450, 0, 4085, 0, 95, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 95*x^34 + 4085*x^32 + 105450*x^30 + 1827800*x^28 + 22545875*x^26 + 204644250*x^24 + 1393946875*x^22 + 7200394375*x^20 + 28304418750*x^18 + 84453693750*x^16 + 189620890625*x^14 + 315440671875*x^12 + 379529453125*x^10 + 318610703125*x^8 + 176833593750*x^6 + 59649609375*x^4 + 10576171875*x^2 + 705078125)
 
gp: K = bnfinit(x^36 + 95*x^34 + 4085*x^32 + 105450*x^30 + 1827800*x^28 + 22545875*x^26 + 204644250*x^24 + 1393946875*x^22 + 7200394375*x^20 + 28304418750*x^18 + 84453693750*x^16 + 189620890625*x^14 + 315440671875*x^12 + 379529453125*x^10 + 318610703125*x^8 + 176833593750*x^6 + 59649609375*x^4 + 10576171875*x^2 + 705078125, 1)
 

Normalized defining polynomial

\( x^{36} + 95 x^{34} + 4085 x^{32} + 105450 x^{30} + 1827800 x^{28} + 22545875 x^{26} + 204644250 x^{24} + 1393946875 x^{22} + 7200394375 x^{20} + 28304418750 x^{18} + 84453693750 x^{16} + 189620890625 x^{14} + 315440671875 x^{12} + 379529453125 x^{10} + 318610703125 x^{8} + 176833593750 x^{6} + 59649609375 x^{4} + 10576171875 x^{2} + 705078125 \)

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:  \(15377735736821953341412273192192323957407601152000000000000000000000000000=2^{36}\cdot 5^{27}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.89$
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$$\chi_{380}(1,·)$, $\chi_{380}(3,·)$, $\chi_{380}(9,·)$, $\chi_{380}(143,·)$, $\chi_{380}(147,·)$, $\chi_{380}(149,·)$, $\chi_{380}(27,·)$, $\chi_{380}(329,·)$, $\chi_{380}(287,·)$, $\chi_{380}(289,·)$, $\chi_{380}(167,·)$, $\chi_{380}(169,·)$, $\chi_{380}(301,·)$, $\chi_{380}(303,·)$, $\chi_{380}(49,·)$, $\chi_{380}(307,·)$, $\chi_{380}(309,·)$, $\chi_{380}(183,·)$, $\chi_{380}(61,·)$, $\chi_{380}(321,·)$, $\chi_{380}(67,·)$, $\chi_{380}(161,·)$, $\chi_{380}(201,·)$, $\chi_{380}(203,·)$, $\chi_{380}(81,·)$, $\chi_{380}(349,·)$, $\chi_{380}(223,·)$, $\chi_{380}(107,·)$, $\chi_{380}(227,·)$, $\chi_{380}(101,·)$, $\chi_{380}(103,·)$, $\chi_{380}(363,·)$, $\chi_{380}(229,·)$, $\chi_{380}(243,·)$, $\chi_{380}(121,·)$, $\chi_{380}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{11875} a^{18}$, $\frac{1}{11875} a^{19}$, $\frac{1}{59375} a^{20}$, $\frac{1}{59375} a^{21}$, $\frac{1}{59375} a^{22}$, $\frac{1}{59375} a^{23}$, $\frac{1}{296875} a^{24}$, $\frac{1}{296875} a^{25}$, $\frac{1}{296875} a^{26}$, $\frac{1}{296875} a^{27}$, $\frac{1}{1484375} a^{28}$, $\frac{1}{1484375} a^{29}$, $\frac{1}{1484375} a^{30}$, $\frac{1}{1484375} a^{31}$, $\frac{1}{7421875} a^{32}$, $\frac{1}{7421875} a^{33}$, $\frac{1}{24574651953125} a^{34} - \frac{63628}{1293402734375} a^{32} - \frac{11663}{51736109375} a^{30} + \frac{52747}{258680546875} a^{28} - \frac{14232}{10347221875} a^{26} + \frac{68982}{51736109375} a^{24} + \frac{58252}{10347221875} a^{22} + \frac{25629}{10347221875} a^{20} - \frac{71707}{2069444375} a^{18} - \frac{1616226}{2069444375} a^{16} - \frac{77373}{21783625} a^{14} + \frac{9108}{21783625} a^{12} + \frac{69108}{4356725} a^{10} + \frac{75688}{4356725} a^{8} - \frac{60292}{871345} a^{6} + \frac{30682}{871345} a^{4} - \frac{58138}{174269} a^{2} + \frac{56265}{174269}$, $\frac{1}{24574651953125} a^{35} - \frac{63628}{1293402734375} a^{33} - \frac{11663}{51736109375} a^{31} + \frac{52747}{258680546875} a^{29} - \frac{14232}{10347221875} a^{27} + \frac{68982}{51736109375} a^{25} + \frac{58252}{10347221875} a^{23} + \frac{25629}{10347221875} a^{21} - \frac{71707}{2069444375} a^{19} - \frac{1616226}{2069444375} a^{17} - \frac{77373}{21783625} a^{15} + \frac{9108}{21783625} a^{13} + \frac{69108}{4356725} a^{11} + \frac{75688}{4356725} a^{9} - \frac{60292}{871345} a^{7} + \frac{30682}{871345} a^{5} - \frac{58138}{174269} a^{3} + \frac{56265}{174269} a$

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:  \( -1 \) (order $2$)
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

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, 4.0.722000.3, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.49048530062408000000000.1, 18.18.563362135874260093126953125.1

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 $36$ R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ R $36$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ $18^{2}$ $36$ $36$ $36$ $18^{2}$

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