↑

Properties

Label 36.0.15235205762...5281.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 13^{30}$
Root discriminant $44.05$
Ramified primes $3, 13$
Class number $182$ (GRH)
Class group $[182]$ (GRH)
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![1, 0, 0, 69, 0, 0, 4932, 0, 0, -11599, 0, 0, 36100, 0, 0, 11446, 0, 0, 16733, 0, 0, -2801, 0, 0, 1910, 0, 0, -36, 0, 0, 57, 0, 0, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 4*x^33 + 57*x^30 - 36*x^27 + 1910*x^24 - 2801*x^21 + 16733*x^18 + 11446*x^15 + 36100*x^12 - 11599*x^9 + 4932*x^6 + 69*x^3 + 1)
 
gp: K = bnfinit(x^36 - 4*x^33 + 57*x^30 - 36*x^27 + 1910*x^24 - 2801*x^21 + 16733*x^18 + 11446*x^15 + 36100*x^12 - 11599*x^9 + 4932*x^6 + 69*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 4 x^{33} + 57 x^{30} - 36 x^{27} + 1910 x^{24} - 2801 x^{21} + 16733 x^{18} + 11446 x^{15} + 36100 x^{12} - 11599 x^{9} + 4932 x^{6} + 69 x^{3} + 1 \)

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:  \(152352057627354962655862528959273781104287214136691847895281=3^{54}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.05$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(4,·)$, $\chi_{117}(10,·)$, $\chi_{117}(14,·)$, $\chi_{117}(16,·)$, $\chi_{117}(17,·)$, $\chi_{117}(22,·)$, $\chi_{117}(23,·)$, $\chi_{117}(25,·)$, $\chi_{117}(29,·)$, $\chi_{117}(35,·)$, $\chi_{117}(38,·)$, $\chi_{117}(40,·)$, $\chi_{117}(43,·)$, $\chi_{117}(49,·)$, $\chi_{117}(53,·)$, $\chi_{117}(55,·)$, $\chi_{117}(56,·)$, $\chi_{117}(61,·)$, $\chi_{117}(62,·)$, $\chi_{117}(64,·)$, $\chi_{117}(68,·)$, $\chi_{117}(74,·)$, $\chi_{117}(77,·)$, $\chi_{117}(79,·)$, $\chi_{117}(82,·)$, $\chi_{117}(88,·)$, $\chi_{117}(92,·)$, $\chi_{117}(94,·)$, $\chi_{117}(95,·)$, $\chi_{117}(100,·)$, $\chi_{117}(101,·)$, $\chi_{117}(103,·)$, $\chi_{117}(107,·)$, $\chi_{117}(113,·)$, $\chi_{117}(116,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{18} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{19} - \frac{1}{2} a$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{20} - \frac{1}{2} a^{2}$, $\frac{1}{4293950} a^{30} + \frac{347274}{2146975} a^{27} - \frac{315588}{2146975} a^{24} - \frac{91341}{858790} a^{21} + \frac{771502}{2146975} a^{18} - \frac{336349}{2146975} a^{15} + \frac{219623}{2146975} a^{12} - \frac{55447}{429395} a^{9} + \frac{619338}{2146975} a^{6} - \frac{1028417}{4293950} a^{3} - \frac{862378}{2146975}$, $\frac{1}{4293950} a^{31} + \frac{347274}{2146975} a^{28} - \frac{315588}{2146975} a^{25} - \frac{91341}{858790} a^{22} + \frac{771502}{2146975} a^{19} - \frac{336349}{2146975} a^{16} + \frac{219623}{2146975} a^{13} - \frac{55447}{429395} a^{10} + \frac{619338}{2146975} a^{7} - \frac{1028417}{4293950} a^{4} - \frac{862378}{2146975} a$, $\frac{1}{4293950} a^{32} + \frac{347274}{2146975} a^{29} - \frac{315588}{2146975} a^{26} - \frac{91341}{858790} a^{23} + \frac{771502}{2146975} a^{20} - \frac{336349}{2146975} a^{17} + \frac{219623}{2146975} a^{14} - \frac{55447}{429395} a^{11} + \frac{619338}{2146975} a^{8} - \frac{1028417}{4293950} a^{5} - \frac{862378}{2146975} a^{2}$, $\frac{1}{138007715940066481523050} a^{33} + \frac{11669894013884477}{138007715940066481523050} a^{30} + \frac{30337311672974609030091}{138007715940066481523050} a^{27} - \frac{36648369119587946883909}{138007715940066481523050} a^{24} + \frac{26087333904991841546709}{138007715940066481523050} a^{21} - \frac{48739560173857978814857}{138007715940066481523050} a^{18} - \frac{6239686167378370064973}{69003857970033240761525} a^{15} - \frac{19670328349779731363943}{69003857970033240761525} a^{12} + \frac{29463569820235797457373}{69003857970033240761525} a^{9} + \frac{27595635152852089265087}{138007715940066481523050} a^{6} + \frac{45049047849282799940651}{138007715940066481523050} a^{3} - \frac{34973552607362147832499}{138007715940066481523050}$, $\frac{1}{138007715940066481523050} a^{34} + \frac{11669894013884477}{138007715940066481523050} a^{31} + \frac{30337311672974609030091}{138007715940066481523050} a^{28} - \frac{36648369119587946883909}{138007715940066481523050} a^{25} + \frac{26087333904991841546709}{138007715940066481523050} a^{22} - \frac{48739560173857978814857}{138007715940066481523050} a^{19} - \frac{6239686167378370064973}{69003857970033240761525} a^{16} - \frac{19670328349779731363943}{69003857970033240761525} a^{13} + \frac{29463569820235797457373}{69003857970033240761525} a^{10} + \frac{27595635152852089265087}{138007715940066481523050} a^{7} + \frac{45049047849282799940651}{138007715940066481523050} a^{4} - \frac{34973552607362147832499}{138007715940066481523050} a$, $\frac{1}{138007715940066481523050} a^{35} + \frac{11669894013884477}{138007715940066481523050} a^{32} + \frac{30337311672974609030091}{138007715940066481523050} a^{29} - \frac{36648369119587946883909}{138007715940066481523050} a^{26} + \frac{26087333904991841546709}{138007715940066481523050} a^{23} - \frac{48739560173857978814857}{138007715940066481523050} a^{20} - \frac{6239686167378370064973}{69003857970033240761525} a^{17} - \frac{19670328349779731363943}{69003857970033240761525} a^{14} + \frac{29463569820235797457373}{69003857970033240761525} a^{11} + \frac{27595635152852089265087}{138007715940066481523050} a^{8} + \frac{45049047849282799940651}{138007715940066481523050} a^{5} - \frac{34973552607362147832499}{138007715940066481523050} a^{2}$

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

Class group and class number

$C_{182}$, which has order $182$ (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:  \( \frac{8006916871000372755411}{138007715940066481523050} a^{34} - \frac{15999036483199363731579}{69003857970033240761525} a^{31} + \frac{456287209308044626763661}{138007715940066481523050} a^{28} - \frac{286609093224317281407519}{138007715940066481523050} a^{25} + \frac{7646400165727391002555862}{69003857970033240761525} a^{22} - \frac{22371357471923176533373497}{138007715940066481523050} a^{19} + \frac{66959328678314239772095592}{69003857970033240761525} a^{16} + \frac{46053764808988543654807637}{69003857970033240761525} a^{13} + \frac{144793583721413369005408378}{69003857970033240761525} a^{10} - \frac{91703548894246182669905423}{138007715940066481523050} a^{7} + \frac{19787860378510520360433823}{69003857970033240761525} a^{4} + \frac{553675180639213151010041}{138007715940066481523050} a \) (order $18$)
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:  \( 20980577392492.816 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.169.1, 3.3.13689.1, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\zeta_{9})\), 6.0.562166163.1, 6.0.771147.1, 6.0.562166163.2, 6.6.14414517.1, 6.0.43243551.1, 6.6.2436053373.2, 6.0.7308160119.2, \(\Q(\zeta_{13})^+\), 6.0.10024911.1, 6.6.2436053373.1, 6.0.7308160119.1, 9.9.2565164201769.1, 12.0.1870004703089601.1, 12.0.53409204324942094161.1, 12.0.100498840557921.1, 12.0.53409204324942094161.2, 18.0.177661819315004155453692747.1, 18.18.14456408038335708501176406117.1, 18.0.390323017035064129531762965159.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
13Data not computed