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Properties

Label 36.36.5273328194...1568.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{99}\cdot 19^{32}$
Root discriminant $92.15$
Ramified primes $2, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![512, 0, -46080, 0, 1025280, 0, -9968640, 0, 51473664, 0, -159622144, 0, 321513984, 0, -442226304, 0, 428794592, 0, -298819648, 0, 151170256, 0, -55646624, 0, 14836664, 0, -2832128, 0, 379376, 0, -34520, 0, 2018, 0, -68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 68*x^34 + 2018*x^32 - 34520*x^30 + 379376*x^28 - 2832128*x^26 + 14836664*x^24 - 55646624*x^22 + 151170256*x^20 - 298819648*x^18 + 428794592*x^16 - 442226304*x^14 + 321513984*x^12 - 159622144*x^10 + 51473664*x^8 - 9968640*x^6 + 1025280*x^4 - 46080*x^2 + 512)
 
gp: K = bnfinit(x^36 - 68*x^34 + 2018*x^32 - 34520*x^30 + 379376*x^28 - 2832128*x^26 + 14836664*x^24 - 55646624*x^22 + 151170256*x^20 - 298819648*x^18 + 428794592*x^16 - 442226304*x^14 + 321513984*x^12 - 159622144*x^10 + 51473664*x^8 - 9968640*x^6 + 1025280*x^4 - 46080*x^2 + 512, 1)
 

Normalized defining polynomial

\( x^{36} - 68 x^{34} + 2018 x^{32} - 34520 x^{30} + 379376 x^{28} - 2832128 x^{26} + 14836664 x^{24} - 55646624 x^{22} + 151170256 x^{20} - 298819648 x^{18} + 428794592 x^{16} - 442226304 x^{14} + 321513984 x^{12} - 159622144 x^{10} + 51473664 x^{8} - 9968640 x^{6} + 1025280 x^{4} - 46080 x^{2} + 512 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(52733281945045886724167383478270850720626086921526306402773390818541568=2^{99}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.15$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(304=2^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{304}(1,·)$, $\chi_{304}(277,·)$, $\chi_{304}(5,·)$, $\chi_{304}(9,·)$, $\chi_{304}(45,·)$, $\chi_{304}(17,·)$, $\chi_{304}(149,·)$, $\chi_{304}(73,·)$, $\chi_{304}(25,·)$, $\chi_{304}(153,·)$, $\chi_{304}(157,·)$, $\chi_{304}(161,·)$, $\chi_{304}(49,·)$, $\chi_{304}(169,·)$, $\chi_{304}(301,·)$, $\chi_{304}(177,·)$, $\chi_{304}(137,·)$, $\chi_{304}(61,·)$, $\chi_{304}(197,·)$, $\chi_{304}(289,·)$, $\chi_{304}(201,·)$, $\chi_{304}(77,·)$, $\chi_{304}(81,·)$, $\chi_{304}(213,·)$, $\chi_{304}(93,·)$, $\chi_{304}(101,·)$, $\chi_{304}(225,·)$, $\chi_{304}(229,·)$, $\chi_{304}(273,·)$, $\chi_{304}(233,·)$, $\chi_{304}(237,·)$, $\chi_{304}(125,·)$, $\chi_{304}(245,·)$, $\chi_{304}(121,·)$, $\chi_{304}(253,·)$, $\chi_{304}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{32} a^{22}$, $\frac{1}{32} a^{23}$, $\frac{1}{64} a^{24}$, $\frac{1}{64} a^{25}$, $\frac{1}{64} a^{26}$, $\frac{1}{64} a^{27}$, $\frac{1}{128} a^{28}$, $\frac{1}{128} a^{29}$, $\frac{1}{128} a^{30}$, $\frac{1}{128} a^{31}$, $\frac{1}{256} a^{32}$, $\frac{1}{256} a^{33}$, $\frac{1}{9838194480600500022779146496} a^{34} + \frac{3047648600927921434859255}{9838194480600500022779146496} a^{32} - \frac{7821866586152340547902955}{4919097240300250011389573248} a^{30} + \frac{556737791481845673381583}{307443577518765625711848328} a^{28} - \frac{3059496375300335479604393}{1229774310075062502847393312} a^{26} + \frac{16734253114894943123475575}{2459548620150125005694786624} a^{24} + \frac{13690648011774797429922561}{1229774310075062502847393312} a^{22} + \frac{4823664717118115182357807}{1229774310075062502847393312} a^{20} - \frac{2424414434422235086932437}{153721788759382812855924164} a^{18} - \frac{899495020033158764632468}{38430447189845703213981041} a^{16} + \frac{10555225049744598637305425}{307443577518765625711848328} a^{14} - \frac{435896290189265978280496}{38430447189845703213981041} a^{12} - \frac{1088674020444146639094483}{153721788759382812855924164} a^{10} + \frac{16454158736600611455034041}{153721788759382812855924164} a^{8} + \frac{11521181360004005359991297}{76860894379691406427962082} a^{6} + \frac{7410693796314200244070708}{38430447189845703213981041} a^{4} + \frac{1259630643487964574080605}{38430447189845703213981041} a^{2} + \frac{10831374811041636665356826}{38430447189845703213981041}$, $\frac{1}{9838194480600500022779146496} a^{35} + \frac{3047648600927921434859255}{9838194480600500022779146496} a^{33} - \frac{7821866586152340547902955}{4919097240300250011389573248} a^{31} + \frac{556737791481845673381583}{307443577518765625711848328} a^{29} - \frac{3059496375300335479604393}{1229774310075062502847393312} a^{27} + \frac{16734253114894943123475575}{2459548620150125005694786624} a^{25} + \frac{13690648011774797429922561}{1229774310075062502847393312} a^{23} + \frac{4823664717118115182357807}{1229774310075062502847393312} a^{21} - \frac{2424414434422235086932437}{153721788759382812855924164} a^{19} - \frac{899495020033158764632468}{38430447189845703213981041} a^{17} + \frac{10555225049744598637305425}{307443577518765625711848328} a^{15} - \frac{435896290189265978280496}{38430447189845703213981041} a^{13} - \frac{1088674020444146639094483}{153721788759382812855924164} a^{11} + \frac{16454158736600611455034041}{153721788759382812855924164} a^{9} + \frac{11521181360004005359991297}{76860894379691406427962082} a^{7} + \frac{7410693796314200244070708}{38430447189845703213981041} a^{5} + \frac{1259630643487964574080605}{38430447189845703213981041} a^{3} + \frac{10831374811041636665356826}{38430447189845703213981041} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:  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:  \( 990111643439917000000000 \) (assuming GRH)
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

\(\Q(\sqrt{2}) \), 3.3.361.1, \(\Q(\zeta_{16})^+\), 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.12.145887695661298614272.1, 18.18.38713951190154487490850848768.1

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 $36$ $36$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ $36$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$ R $18^{2}$ $36$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ $18^{2}$ $36$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $36$ $36$

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
19Data not computed