↑

Properties

Label 36.0.23610692285...6704.4
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{54}\cdot 7^{30}$
Root discriminant $74.38$
Ramified primes $2, 3, 7$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191102976, 0, 286654464, 0, 334430208, 0, 366280704, 0, 394149888, 0, 422019072, 0, 451215360, 0, 178163712, 0, 59222016, 0, 18551808, 0, 5640192, 0, 1651968, 0, 430848, 0, 55296, 0, 7056, 0, 888, 0, 108, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 12*x^34 + 108*x^32 + 888*x^30 + 7056*x^28 + 55296*x^26 + 430848*x^24 + 1651968*x^22 + 5640192*x^20 + 18551808*x^18 + 59222016*x^16 + 178163712*x^14 + 451215360*x^12 + 422019072*x^10 + 394149888*x^8 + 366280704*x^6 + 334430208*x^4 + 286654464*x^2 + 191102976)
 
gp: K = bnfinit(x^36 + 12*x^34 + 108*x^32 + 888*x^30 + 7056*x^28 + 55296*x^26 + 430848*x^24 + 1651968*x^22 + 5640192*x^20 + 18551808*x^18 + 59222016*x^16 + 178163712*x^14 + 451215360*x^12 + 422019072*x^10 + 394149888*x^8 + 366280704*x^6 + 334430208*x^4 + 286654464*x^2 + 191102976, 1)
 

Normalized defining polynomial

\( x^{36} + 12 x^{34} + 108 x^{32} + 888 x^{30} + 7056 x^{28} + 55296 x^{26} + 430848 x^{24} + 1651968 x^{22} + 5640192 x^{20} + 18551808 x^{18} + 59222016 x^{16} + 178163712 x^{14} + 451215360 x^{12} + 422019072 x^{10} + 394149888 x^{8} + 366280704 x^{6} + 334430208 x^{4} + 286654464 x^{2} + 191102976 \)

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:  \(23610692285332399309092778573219694177406932269512586359440570056704=2^{54}\cdot 3^{54}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.38$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(131,·)$, $\chi_{504}(265,·)$, $\chi_{504}(11,·)$, $\chi_{504}(145,·)$, $\chi_{504}(275,·)$, $\chi_{504}(409,·)$, $\chi_{504}(25,·)$, $\chi_{504}(155,·)$, $\chi_{504}(289,·)$, $\chi_{504}(419,·)$, $\chi_{504}(169,·)$, $\chi_{504}(299,·)$, $\chi_{504}(433,·)$, $\chi_{504}(179,·)$, $\chi_{504}(73,·)$, $\chi_{504}(313,·)$, $\chi_{504}(59,·)$, $\chi_{504}(193,·)$, $\chi_{504}(323,·)$, $\chi_{504}(97,·)$, $\chi_{504}(457,·)$, $\chi_{504}(337,·)$, $\chi_{504}(83,·)$, $\chi_{504}(347,·)$, $\chi_{504}(395,·)$, $\chi_{504}(107,·)$, $\chi_{504}(481,·)$, $\chi_{504}(227,·)$, $\chi_{504}(361,·)$, $\chi_{504}(491,·)$, $\chi_{504}(241,·)$, $\chi_{504}(467,·)$, $\chi_{504}(121,·)$, $\chi_{504}(251,·)$, $\chi_{504}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{24} a^{6}$, $\frac{1}{24} a^{7}$, $\frac{1}{48} a^{8}$, $\frac{1}{48} a^{9}$, $\frac{1}{96} a^{10}$, $\frac{1}{96} a^{11}$, $\frac{1}{576} a^{12}$, $\frac{1}{576} a^{13}$, $\frac{1}{4608} a^{14} + \frac{1}{4}$, $\frac{1}{4608} a^{15} + \frac{1}{4} a$, $\frac{1}{9216} a^{16} + \frac{1}{8} a^{2}$, $\frac{1}{9216} a^{17} + \frac{1}{8} a^{3}$, $\frac{1}{55296} a^{18} - \frac{1}{16} a^{4}$, $\frac{1}{55296} a^{19} - \frac{1}{16} a^{5}$, $\frac{1}{110592} a^{20} + \frac{1}{96} a^{6}$, $\frac{1}{110592} a^{21} + \frac{1}{96} a^{7}$, $\frac{1}{221184} a^{22} + \frac{1}{192} a^{8}$, $\frac{1}{221184} a^{23} + \frac{1}{192} a^{9}$, $\frac{1}{1327104} a^{24} - \frac{1}{384} a^{10}$, $\frac{1}{1327104} a^{25} - \frac{1}{384} a^{11}$, $\frac{1}{37124407296} a^{26} + \frac{3353}{9281101824} a^{24} + \frac{901}{515616768} a^{22} - \frac{139}{128904192} a^{20} + \frac{499}{96678144} a^{18} + \frac{1469}{42968064} a^{16} - \frac{1189}{64452096} a^{14} - \frac{22139}{32226048} a^{12} + \frac{3643}{2685504} a^{10} - \frac{1777}{447584} a^{8} + \frac{181}{111896} a^{6} - \frac{3083}{55948} a^{4} + \frac{6739}{111896} a^{2} + \frac{27711}{55948}$, $\frac{1}{37124407296} a^{27} + \frac{3353}{9281101824} a^{25} + \frac{901}{515616768} a^{23} - \frac{139}{128904192} a^{21} + \frac{499}{96678144} a^{19} + \frac{1469}{42968064} a^{17} - \frac{1189}{64452096} a^{15} - \frac{22139}{32226048} a^{13} + \frac{3643}{2685504} a^{11} - \frac{1777}{447584} a^{9} + \frac{181}{111896} a^{7} - \frac{3083}{55948} a^{5} + \frac{6739}{111896} a^{3} + \frac{27711}{55948} a$, $\frac{1}{296995258368} a^{28} + \frac{10333}{128904192} a^{14} + \frac{99225}{223792}$, $\frac{1}{296995258368} a^{29} + \frac{10333}{128904192} a^{15} + \frac{99225}{223792} a$, $\frac{1}{1781971550208} a^{30} + \frac{12769}{257808384} a^{16} - \frac{22873}{447584} a^{2}$, $\frac{1}{1781971550208} a^{31} + \frac{12769}{257808384} a^{17} - \frac{22873}{447584} a^{3}$, $\frac{1}{3563943100416} a^{32} + \frac{10333}{1546850304} a^{18} + \frac{33075}{895168} a^{4}$, $\frac{1}{3563943100416} a^{33} + \frac{10333}{1546850304} a^{19} + \frac{33075}{895168} a^{5}$, $\frac{1}{7127886200832} a^{34} + \frac{10333}{3093700608} a^{20} + \frac{33075}{1790336} a^{6}$, $\frac{1}{7127886200832} a^{35} + \frac{10333}{3093700608} a^{21} + \frac{33075}{1790336} a^{7}$

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:  \( -\frac{19}{131997892608} a^{32} + \frac{377893}{1546850304} a^{18} - \frac{1022295}{895168} a^{4} \) (order $14$)
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

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-42}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{6}, \sqrt{-7})\), 6.6.10077696.1, 6.6.24196548096.1, 6.6.24196548096.2, 6.6.33191424.1, 6.0.2250423.1, 6.0.3456649728.1, 6.0.110270727.2, 6.0.169375836672.2, 6.0.110270727.1, 6.0.169375836672.1, \(\Q(\zeta_{7})\), 6.0.232339968.1, 9.9.62523502209.1, 12.0.11948427342082473984.5, 12.0.28688174048340020035584.5, 12.0.28688174048340020035584.7, 12.0.53981860730241024.6, 18.18.14166424145858741301073711988736.1, 18.0.1340851596668237962730583.1, 18.0.4859083482029548266268283212136448.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
7Data not computed