↑

Properties

Label 36.0.79962223364...8784.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{18}\cdot 19^{34}$
Root discriminant $55.89$
Ramified primes $2, 3, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, 0, 5415, 0, 57399, 0, 262086, 0, 806113, 0, 1629193, 0, 2415451, 0, 2670317, 0, 2298126, 0, 1558038, 0, 848787, 0, 372723, 0, 132810, 0, 38095, 0, 8740, 0, 1558, 0, 209, 0, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361)
 
gp: K = bnfinit(x^36 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361, 1)
 

Normalized defining polynomial

\( x^{36} + 19 x^{34} + 209 x^{32} + 1558 x^{30} + 8740 x^{28} + 38095 x^{26} + 132810 x^{24} + 372723 x^{22} + 848787 x^{20} + 1558038 x^{18} + 2298126 x^{16} + 2670317 x^{14} + 2415451 x^{12} + 1629193 x^{10} + 806113 x^{8} + 262086 x^{6} + 57399 x^{4} + 5415 x^{2} + 361 \)

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:  \(799622233646074762983150698451178476894456963777140963130998784=2^{36}\cdot 3^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.89$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(5,·)$, $\chi_{228}(137,·)$, $\chi_{228}(143,·)$, $\chi_{228}(17,·)$, $\chi_{228}(149,·)$, $\chi_{228}(151,·)$, $\chi_{228}(25,·)$, $\chi_{228}(155,·)$, $\chi_{228}(157,·)$, $\chi_{228}(31,·)$, $\chi_{228}(161,·)$, $\chi_{228}(167,·)$, $\chi_{228}(169,·)$, $\chi_{228}(49,·)$, $\chi_{228}(179,·)$, $\chi_{228}(59,·)$, $\chi_{228}(61,·)$, $\chi_{228}(67,·)$, $\chi_{228}(197,·)$, $\chi_{228}(71,·)$, $\chi_{228}(73,·)$, $\chi_{228}(203,·)$, $\chi_{228}(77,·)$, $\chi_{228}(79,·)$, $\chi_{228}(211,·)$, $\chi_{228}(85,·)$, $\chi_{228}(91,·)$, $\chi_{228}(223,·)$, $\chi_{228}(227,·)$, $\chi_{228}(101,·)$, $\chi_{228}(103,·)$, $\chi_{228}(107,·)$, $\chi_{228}(121,·)$, $\chi_{228}(125,·)$, $\chi_{228}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18}$, $\frac{1}{19} a^{19}$, $\frac{1}{19} a^{20}$, $\frac{1}{19} a^{21}$, $\frac{1}{19} a^{22}$, $\frac{1}{19} a^{23}$, $\frac{1}{19} a^{24}$, $\frac{1}{19} a^{25}$, $\frac{1}{703} a^{26} + \frac{3}{703} a^{24} + \frac{12}{703} a^{22} - \frac{1}{703} a^{18} - \frac{6}{37} a^{16} - \frac{7}{37} a^{14} - \frac{2}{37} a^{12} + \frac{10}{37} a^{10} - \frac{1}{37} a^{8} - \frac{10}{37} a^{6} + \frac{4}{37} a^{4} - \frac{3}{37} a^{2} + \frac{3}{37}$, $\frac{1}{703} a^{27} + \frac{3}{703} a^{25} + \frac{12}{703} a^{23} - \frac{1}{703} a^{19} - \frac{6}{37} a^{17} - \frac{7}{37} a^{15} - \frac{2}{37} a^{13} + \frac{10}{37} a^{11} - \frac{1}{37} a^{9} - \frac{10}{37} a^{7} + \frac{4}{37} a^{5} - \frac{3}{37} a^{3} + \frac{3}{37} a$, $\frac{1}{703} a^{28} + \frac{3}{703} a^{24} + \frac{1}{703} a^{22} - \frac{1}{703} a^{20} + \frac{11}{37} a^{16} - \frac{18}{37} a^{14} + \frac{16}{37} a^{12} + \frac{6}{37} a^{10} - \frac{7}{37} a^{8} - \frac{3}{37} a^{6} - \frac{15}{37} a^{4} + \frac{12}{37} a^{2} - \frac{9}{37}$, $\frac{1}{703} a^{29} + \frac{3}{703} a^{25} + \frac{1}{703} a^{23} - \frac{1}{703} a^{21} + \frac{11}{37} a^{17} - \frac{18}{37} a^{15} + \frac{16}{37} a^{13} + \frac{6}{37} a^{11} - \frac{7}{37} a^{9} - \frac{3}{37} a^{7} - \frac{15}{37} a^{5} + \frac{12}{37} a^{3} - \frac{9}{37} a$, $\frac{1}{703} a^{30} - \frac{8}{703} a^{24} - \frac{10}{703} a^{18} + \frac{12}{37} a^{12} + \frac{15}{37} a^{6} - \frac{9}{37}$, $\frac{1}{703} a^{31} - \frac{8}{703} a^{25} - \frac{10}{703} a^{19} + \frac{12}{37} a^{13} + \frac{15}{37} a^{7} - \frac{9}{37} a$, $\frac{1}{703} a^{32} - \frac{13}{703} a^{24} - \frac{15}{703} a^{22} - \frac{10}{703} a^{20} - \frac{8}{703} a^{18} - \frac{11}{37} a^{16} - \frac{7}{37} a^{14} - \frac{16}{37} a^{12} + \frac{6}{37} a^{10} + \frac{7}{37} a^{8} - \frac{6}{37} a^{6} - \frac{5}{37} a^{4} + \frac{4}{37} a^{2} - \frac{13}{37}$, $\frac{1}{703} a^{33} - \frac{13}{703} a^{25} - \frac{15}{703} a^{23} - \frac{10}{703} a^{21} - \frac{8}{703} a^{19} - \frac{11}{37} a^{17} - \frac{7}{37} a^{15} - \frac{16}{37} a^{13} + \frac{6}{37} a^{11} + \frac{7}{37} a^{9} - \frac{6}{37} a^{7} - \frac{5}{37} a^{5} + \frac{4}{37} a^{3} - \frac{13}{37} a$, $\frac{1}{66495926797178861491219} a^{34} + \frac{95473288288552618}{184199243205481610779} a^{32} + \frac{1807951731514635866}{3499785620904150604801} a^{30} + \frac{171309968879160652}{3499785620904150604801} a^{28} + \frac{440090776031058913}{3499785620904150604801} a^{26} - \frac{53585764503177685364}{3499785620904150604801} a^{24} + \frac{26683600630186601723}{3499785620904150604801} a^{22} + \frac{32781662307927217072}{3499785620904150604801} a^{20} + \frac{91988776569193916607}{3499785620904150604801} a^{18} - \frac{1307974449556231788942}{3499785620904150604801} a^{16} - \frac{30943815699806536848}{184199243205481610779} a^{14} - \frac{43424296143470230158}{184199243205481610779} a^{12} - \frac{71260072058689014988}{184199243205481610779} a^{10} - \frac{55846283841363957147}{184199243205481610779} a^{8} + \frac{26505569839203403346}{184199243205481610779} a^{6} - \frac{34104452947850036366}{184199243205481610779} a^{4} - \frac{38520759123752461115}{184199243205481610779} a^{2} + \frac{65912108193554012131}{184199243205481610779}$, $\frac{1}{66495926797178861491219} a^{35} + \frac{95473288288552618}{184199243205481610779} a^{33} + \frac{1807951731514635866}{3499785620904150604801} a^{31} + \frac{171309968879160652}{3499785620904150604801} a^{29} + \frac{440090776031058913}{3499785620904150604801} a^{27} - \frac{53585764503177685364}{3499785620904150604801} a^{25} + \frac{26683600630186601723}{3499785620904150604801} a^{23} + \frac{32781662307927217072}{3499785620904150604801} a^{21} + \frac{91988776569193916607}{3499785620904150604801} a^{19} - \frac{1307974449556231788942}{3499785620904150604801} a^{17} - \frac{30943815699806536848}{184199243205481610779} a^{15} - \frac{43424296143470230158}{184199243205481610779} a^{13} - \frac{71260072058689014988}{184199243205481610779} a^{11} - \frac{55846283841363957147}{184199243205481610779} a^{9} + \frac{26505569839203403346}{184199243205481610779} a^{7} - \frac{34104452947850036366}{184199243205481610779} a^{5} - \frac{38520759123752461115}{184199243205481610779} a^{3} + \frac{65912108193554012131}{184199243205481610779} a$

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:  \( \frac{26829997518280183021}{66495926797178861491219} a^{34} + \frac{26598244133079038772}{3499785620904150604801} a^{32} + \frac{290816878133015963118}{3499785620904150604801} a^{30} + \frac{2153288311705603483287}{3499785620904150604801} a^{28} + \frac{11998687105984961447667}{3499785620904150604801} a^{26} + \frac{51899600479398829656543}{3499785620904150604801} a^{24} + \frac{4849522942561243997067}{94588800564977043373} a^{22} + \frac{498605177959280803164861}{3499785620904150604801} a^{20} + \frac{1122593033140817945162022}{3499785620904150604801} a^{18} + \frac{2031702862437939905122865}{3499785620904150604801} a^{16} + \frac{155072516601073134700314}{184199243205481610779} a^{14} + \frac{176194500030210489789879}{184199243205481610779} a^{12} + \frac{155039966107495776853800}{184199243205481610779} a^{10} + \frac{100425776080331302112142}{184199243205481610779} a^{8} + \frac{47425533042550358380716}{184199243205481610779} a^{6} + \frac{14082041180781203080581}{184199243205481610779} a^{4} + \frac{3119154647880984572367}{184199243205481610779} a^{2} + \frac{291248148377115816489}{184199243205481610779} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-57}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{19})\), 6.0.3518667.1, 6.6.158470336.1, 6.0.4278699072.1, \(\Q(\zeta_{19})^+\), 12.0.18307265748733661184.2, 18.0.5677392343251487443465123.1, \(\Q(\zeta_{76})^+\), 18.0.28277592430157040563214702870528.1

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 R $18^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
3Data not computed
19Data not computed