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Properties

Label 36.0.72166039996...8656.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{72}\cdot 3^{48}\cdot 7^{24}$
Root discriminant $63.33$
Ramified primes $2, 3, 7$
Class number $9072$ (GRH)
Class group $[2, 2, 18, 126]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 1794, 0, 0, 0, 73593, 0, 0, 0, 761144, 0, 0, 0, 2847312, 0, 0, 0, 3439800, 0, 0, 0, 423088, 0, 0, 0, 16497, 0, 0, 0, 234, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 1)
 
gp: K = bnfinit(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 234 x^{32} + 16497 x^{28} + 423088 x^{24} + 3439800 x^{20} + 2847312 x^{16} + 761144 x^{12} + 73593 x^{8} + 1794 x^{4} + 1 \)

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:  \(72166039996680932841568261508505246616460162082637588781812678656=2^{72}\cdot 3^{48}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.33$
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$$\chi_{504}(1,·)$, $\chi_{504}(403,·)$, $\chi_{504}(277,·)$, $\chi_{504}(151,·)$, $\chi_{504}(25,·)$, $\chi_{504}(415,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(37,·)$, $\chi_{504}(295,·)$, $\chi_{504}(169,·)$, $\chi_{504}(43,·)$, $\chi_{504}(445,·)$, $\chi_{504}(319,·)$, $\chi_{504}(193,·)$, $\chi_{504}(67,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(205,·)$, $\chi_{504}(79,·)$, $\chi_{504}(337,·)$, $\chi_{504}(211,·)$, $\chi_{504}(85,·)$, $\chi_{504}(463,·)$, $\chi_{504}(421,·)$, $\chi_{504}(487,·)$, $\chi_{504}(361,·)$, $\chi_{504}(235,·)$, $\chi_{504}(109,·)$, $\chi_{504}(499,·)$, $\chi_{504}(373,·)$, $\chi_{504}(247,·)$, $\chi_{504}(121,·)$, $\chi_{504}(379,·)$, $\chi_{504}(253,·)$, $\chi_{504}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1537} a^{24} + \frac{139}{1537} a^{20} - \frac{33}{1537} a^{16} + \frac{340}{1537} a^{12} + \frac{723}{1537} a^{8} + \frac{150}{1537} a^{4} + \frac{49}{1537}$, $\frac{1}{1537} a^{25} + \frac{139}{1537} a^{21} - \frac{33}{1537} a^{17} + \frac{340}{1537} a^{13} + \frac{723}{1537} a^{9} + \frac{150}{1537} a^{5} + \frac{49}{1537} a$, $\frac{1}{1537} a^{26} + \frac{139}{1537} a^{22} - \frac{33}{1537} a^{18} + \frac{340}{1537} a^{14} + \frac{723}{1537} a^{10} + \frac{150}{1537} a^{6} + \frac{49}{1537} a^{2}$, $\frac{1}{1537} a^{27} + \frac{139}{1537} a^{23} - \frac{33}{1537} a^{19} + \frac{340}{1537} a^{15} + \frac{723}{1537} a^{11} + \frac{150}{1537} a^{7} + \frac{49}{1537} a^{3}$, $\frac{1}{12296} a^{28} - \frac{498}{1537} a^{20} - \frac{729}{1537} a^{16} + \frac{523}{1537} a^{12} + \frac{329}{1537} a^{8} + \frac{666}{1537} a^{4} - \frac{663}{12296}$, $\frac{1}{12296} a^{29} - \frac{498}{1537} a^{21} - \frac{729}{1537} a^{17} + \frac{523}{1537} a^{13} + \frac{329}{1537} a^{9} + \frac{666}{1537} a^{5} - \frac{663}{12296} a$, $\frac{1}{12296} a^{30} - \frac{498}{1537} a^{22} - \frac{729}{1537} a^{18} + \frac{523}{1537} a^{14} + \frac{329}{1537} a^{10} + \frac{666}{1537} a^{6} - \frac{663}{12296} a^{2}$, $\frac{1}{12296} a^{31} - \frac{498}{1537} a^{23} - \frac{729}{1537} a^{19} + \frac{523}{1537} a^{15} + \frac{329}{1537} a^{11} + \frac{666}{1537} a^{7} - \frac{663}{12296} a^{3}$, $\frac{1}{29602381262764908776384} a^{32} + \frac{704480871763353273}{29602381262764908776384} a^{28} + \frac{805965838864293305}{3700297657845613597048} a^{24} - \frac{222697941470102849347}{3700297657845613597048} a^{20} + \frac{830660127185587026241}{1850148828922806798524} a^{16} + \frac{53476474018532892665}{462537207230701699631} a^{12} + \frac{1707071167133203870047}{3700297657845613597048} a^{8} + \frac{5370287505039882213569}{29602381262764908776384} a^{4} + \frac{7335764073165972550353}{29602381262764908776384}$, $\frac{1}{29602381262764908776384} a^{33} + \frac{704480871763353273}{29602381262764908776384} a^{29} + \frac{805965838864293305}{3700297657845613597048} a^{25} - \frac{222697941470102849347}{3700297657845613597048} a^{21} + \frac{830660127185587026241}{1850148828922806798524} a^{17} + \frac{53476474018532892665}{462537207230701699631} a^{13} + \frac{1707071167133203870047}{3700297657845613597048} a^{9} + \frac{5370287505039882213569}{29602381262764908776384} a^{5} + \frac{7335764073165972550353}{29602381262764908776384} a$, $\frac{1}{29602381262764908776384} a^{34} + \frac{704480871763353273}{29602381262764908776384} a^{30} + \frac{805965838864293305}{3700297657845613597048} a^{26} - \frac{222697941470102849347}{3700297657845613597048} a^{22} + \frac{830660127185587026241}{1850148828922806798524} a^{18} + \frac{53476474018532892665}{462537207230701699631} a^{14} + \frac{1707071167133203870047}{3700297657845613597048} a^{10} + \frac{5370287505039882213569}{29602381262764908776384} a^{6} + \frac{7335764073165972550353}{29602381262764908776384} a^{2}$, $\frac{1}{29602381262764908776384} a^{35} + \frac{704480871763353273}{29602381262764908776384} a^{31} + \frac{805965838864293305}{3700297657845613597048} a^{27} - \frac{222697941470102849347}{3700297657845613597048} a^{23} + \frac{830660127185587026241}{1850148828922806798524} a^{19} + \frac{53476474018532892665}{462537207230701699631} a^{15} + \frac{1707071167133203870047}{3700297657845613597048} a^{11} + \frac{5370287505039882213569}{29602381262764908776384} a^{7} + \frac{7335764073165972550353}{29602381262764908776384} a^{3}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{18}\times C_{126}$, which has order $9072$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{777169578454749605}{116544808121121688096} a^{33} - \frac{181842965177485526397}{116544808121121688096} a^{29} - \frac{1602190315831148653893}{14568101015140211012} a^{25} - \frac{41071031422755870665389}{14568101015140211012} a^{21} - \frac{166692176757014898869253}{7284050507570105506} a^{17} - \frac{67564173576078929817462}{3642025253785052753} a^{13} - \frac{68551006951128593547867}{14568101015140211012} a^{9} - \frac{46568446236894731217381}{116544808121121688096} a^{5} - \frac{780433761781366821813}{116544808121121688096} a \) (order $8$)
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:  \( 160258501280890.78 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{8})\), 6.0.419904.1, 6.0.153664.1, 6.0.1008189504.1, 6.0.1008189504.2, 6.6.3359232.1, 6.0.3359232.1, 6.6.1229312.1, 6.0.1229312.1, 6.6.8065516032.1, 6.0.8065516032.1, 6.6.8065516032.2, 6.0.8065516032.2, 9.9.62523502209.1, 12.0.722204136308736.1, 12.0.96717311574016.1, 12.0.4163363127196737601536.1, 12.0.4163363127196737601536.2, 18.0.1024770265180753855691096064.1, 18.18.524682375772545974113841184768.1, 18.0.524682375772545974113841184768.4

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
3Data not computed
7Data not computed