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Properties

Label 36.0.22879832815...0000.3
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{90}\cdot 5^{18}$
Root discriminant $69.71$
Ramified primes $2, 3, 5$
Class number $1406$ (GRH)
Class group $[1406]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33396841, 0, -468342, 0, 3128841, 0, -8101332, 0, 10819656, 0, -9226998, 0, 7211022, 0, -7198686, 0, 7458714, 0, -6254878, 0, 3996135, 0, -1937520, 0, 712530, 0, -197316, 0, 40455, 0, -5952, 0, 594, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 + 594*x^32 - 5952*x^30 + 40455*x^28 - 197316*x^26 + 712530*x^24 - 1937520*x^22 + 3996135*x^20 - 6254878*x^18 + 7458714*x^16 - 7198686*x^14 + 7211022*x^12 - 9226998*x^10 + 10819656*x^8 - 8101332*x^6 + 3128841*x^4 - 468342*x^2 + 33396841)
 
gp: K = bnfinit(x^36 - 36*x^34 + 594*x^32 - 5952*x^30 + 40455*x^28 - 197316*x^26 + 712530*x^24 - 1937520*x^22 + 3996135*x^20 - 6254878*x^18 + 7458714*x^16 - 7198686*x^14 + 7211022*x^12 - 9226998*x^10 + 10819656*x^8 - 8101332*x^6 + 3128841*x^4 - 468342*x^2 + 33396841, 1)
 

Normalized defining polynomial

\( x^{36} - 36 x^{34} + 594 x^{32} - 5952 x^{30} + 40455 x^{28} - 197316 x^{26} + 712530 x^{24} - 1937520 x^{22} + 3996135 x^{20} - 6254878 x^{18} + 7458714 x^{16} - 7198686 x^{14} + 7211022 x^{12} - 9226998 x^{10} + 10819656 x^{8} - 8101332 x^{6} + 3128841 x^{4} - 468342 x^{2} + 33396841 \)

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:  \(2287983281592785286172874500859947211827825606656000000000000000000=2^{36}\cdot 3^{90}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.71$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(540=2^{2}\cdot 3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{540}(1,·)$, $\chi_{540}(389,·)$, $\chi_{540}(391,·)$, $\chi_{540}(269,·)$, $\chi_{540}(271,·)$, $\chi_{540}(149,·)$, $\chi_{540}(151,·)$, $\chi_{540}(539,·)$, $\chi_{540}(29,·)$, $\chi_{540}(31,·)$, $\chi_{540}(419,·)$, $\chi_{540}(421,·)$, $\chi_{540}(299,·)$, $\chi_{540}(301,·)$, $\chi_{540}(179,·)$, $\chi_{540}(181,·)$, $\chi_{540}(59,·)$, $\chi_{540}(61,·)$, $\chi_{540}(449,·)$, $\chi_{540}(451,·)$, $\chi_{540}(329,·)$, $\chi_{540}(331,·)$, $\chi_{540}(209,·)$, $\chi_{540}(211,·)$, $\chi_{540}(89,·)$, $\chi_{540}(91,·)$, $\chi_{540}(479,·)$, $\chi_{540}(481,·)$, $\chi_{540}(359,·)$, $\chi_{540}(361,·)$, $\chi_{540}(239,·)$, $\chi_{540}(241,·)$, $\chi_{540}(119,·)$, $\chi_{540}(121,·)$, $\chi_{540}(509,·)$, $\chi_{540}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2584} a^{18} - \frac{9}{1292} a^{16} + \frac{135}{2584} a^{14} - \frac{273}{1292} a^{12} + \frac{1287}{2584} a^{10} + \frac{401}{1292} a^{8} - \frac{599}{1292} a^{6} - \frac{135}{646} a^{4} + \frac{81}{2584} a^{2} + \frac{985}{2584}$, $\frac{1}{2584} a^{19} - \frac{9}{1292} a^{17} + \frac{135}{2584} a^{15} - \frac{273}{1292} a^{13} + \frac{1287}{2584} a^{11} + \frac{401}{1292} a^{9} - \frac{599}{1292} a^{7} - \frac{135}{646} a^{5} + \frac{81}{2584} a^{3} + \frac{985}{2584} a$, $\frac{1}{2584} a^{20} - \frac{189}{2584} a^{16} - \frac{175}{646} a^{14} - \frac{789}{2584} a^{12} + \frac{89}{323} a^{10} + \frac{159}{1292} a^{8} + \frac{144}{323} a^{6} + \frac{41}{152} a^{4} - \frac{141}{2584} a^{2} - \frac{179}{1292}$, $\frac{1}{2584} a^{21} - \frac{189}{2584} a^{17} - \frac{175}{646} a^{15} - \frac{789}{2584} a^{13} + \frac{89}{323} a^{11} + \frac{159}{1292} a^{9} + \frac{144}{323} a^{7} + \frac{41}{152} a^{5} - \frac{141}{2584} a^{3} - \frac{179}{1292} a$, $\frac{1}{2584} a^{22} + \frac{533}{1292} a^{16} - \frac{557}{1292} a^{14} + \frac{439}{1292} a^{12} + \frac{35}{136} a^{10} + \frac{137}{1292} a^{8} - \frac{917}{2584} a^{6} + \frac{61}{136} a^{4} - \frac{553}{2584} a^{2} + \frac{117}{2584}$, $\frac{1}{2584} a^{23} + \frac{533}{1292} a^{17} - \frac{557}{1292} a^{15} + \frac{439}{1292} a^{13} + \frac{35}{136} a^{11} + \frac{137}{1292} a^{9} - \frac{917}{2584} a^{7} + \frac{61}{136} a^{5} - \frac{553}{2584} a^{3} + \frac{117}{2584} a$, $\frac{1}{2584} a^{24} - \frac{7}{1292} a^{16} - \frac{6}{17} a^{14} - \frac{1283}{2584} a^{12} + \frac{109}{646} a^{10} - \frac{545}{2584} a^{8} - \frac{853}{2584} a^{6} - \frac{1145}{2584} a^{4} - \frac{957}{2584} a^{2} - \frac{453}{1292}$, $\frac{1}{2584} a^{25} - \frac{7}{1292} a^{17} - \frac{6}{17} a^{15} - \frac{1283}{2584} a^{13} + \frac{109}{646} a^{11} - \frac{545}{2584} a^{9} - \frac{853}{2584} a^{7} - \frac{1145}{2584} a^{5} - \frac{957}{2584} a^{3} - \frac{453}{1292} a$, $\frac{1}{2584} a^{26} - \frac{291}{646} a^{16} + \frac{607}{2584} a^{14} + \frac{4}{19} a^{12} - \frac{615}{2584} a^{10} + \frac{39}{2584} a^{8} + \frac{9}{136} a^{6} - \frac{45}{152} a^{4} + \frac{3}{34} a^{2} + \frac{435}{1292}$, $\frac{1}{14932936} a^{27} - \frac{27}{14932936} a^{25} + \frac{81}{3733234} a^{23} - \frac{2277}{14932936} a^{21} - \frac{1163}{14932936} a^{19} + \frac{175725}{14932936} a^{17} - \frac{745281}{7466468} a^{15} + \frac{775752}{1866617} a^{13} + \frac{41299}{3733234} a^{11} - \frac{59581}{7466468} a^{9} + \frac{36398}{98243} a^{7} + \frac{1402705}{14932936} a^{5} + \frac{6091885}{14932936} a^{3} - \frac{3039781}{14932936} a$, $\frac{1}{86267571272} a^{28} - \frac{2903}{43133785636} a^{26} + \frac{75289}{43133785636} a^{24} - \frac{865099}{43133785636} a^{22} + \frac{11579953}{86267571272} a^{20} - \frac{16606491}{86267571272} a^{18} - \frac{56908650}{10783446409} a^{16} + \frac{79541817}{1627690024} a^{14} - \frac{2228722938}{10783446409} a^{12} - \frac{5357389067}{43133785636} a^{10} + \frac{42580818235}{86267571272} a^{8} - \frac{8521686097}{86267571272} a^{6} - \frac{25826535109}{86267571272} a^{4} - \frac{18362393989}{43133785636} a^{2} - \frac{5701897}{14927768}$, $\frac{1}{86267571272} a^{29} - \frac{29}{86267571272} a^{27} - \frac{5401}{86267571272} a^{25} + \frac{3725}{2270199244} a^{23} - \frac{393569}{21566892818} a^{21} + \frac{591773}{5074563016} a^{19} - \frac{2160051}{4540398488} a^{17} + \frac{816}{629911} a^{15} - \frac{6058635}{2537281508} a^{13} + \frac{33206418379}{86267571272} a^{11} - \frac{17600154945}{86267571272} a^{9} - \frac{973883867}{5074563016} a^{7} - \frac{4468895143}{10783446409} a^{5} + \frac{34477635}{2537281508} a^{3} - \frac{17627573845}{86267571272} a$, $\frac{1}{86267571272} a^{30} - \frac{173775}{86267571272} a^{26} + \frac{563539}{10783446409} a^{24} + \frac{3755137}{21566892818} a^{22} + \frac{3006487}{21566892818} a^{20} + \frac{1441915}{10783446409} a^{18} + \frac{12651010741}{43133785636} a^{16} + \frac{8406275807}{86267571272} a^{14} - \frac{22138839347}{86267571272} a^{12} - \frac{6694904409}{86267571272} a^{10} + \frac{1500326695}{21566892818} a^{8} + \frac{11644908669}{86267571272} a^{6} + \frac{16191537601}{86267571272} a^{4} + \frac{5980882857}{86267571272} a^{2} + \frac{7435057}{14927768}$, $\frac{1}{86267571272} a^{31} - \frac{465}{86267571272} a^{27} - \frac{85529}{43133785636} a^{25} + \frac{129483}{2537281508} a^{23} - \frac{15362809}{86267571272} a^{21} + \frac{1285936}{10783446409} a^{19} - \frac{2166544773}{86267571272} a^{17} + \frac{25772642601}{86267571272} a^{15} - \frac{1024102563}{10783446409} a^{13} - \frac{20797947889}{86267571272} a^{11} - \frac{7158403305}{43133785636} a^{9} + \frac{7049293615}{86267571272} a^{7} - \frac{7683191045}{43133785636} a^{5} + \frac{821276869}{2537281508} a^{3} + \frac{5504227507}{86267571272} a$, $\frac{1}{86267571272} a^{32} - \frac{358856}{10783446409} a^{26} + \frac{3825313}{43133785636} a^{24} + \frac{3681799}{21566892818} a^{22} - \frac{791189}{5074563016} a^{20} - \frac{814915}{10783446409} a^{18} - \frac{3777431351}{86267571272} a^{16} - \frac{8068246601}{86267571272} a^{14} + \frac{98634965}{1627690024} a^{12} - \frac{177539037}{5074563016} a^{10} - \frac{16938727571}{43133785636} a^{8} + \frac{4775647495}{21566892818} a^{6} - \frac{14758741711}{43133785636} a^{4} - \frac{16454497267}{43133785636} a^{2} + \frac{1851761}{3731942}$, $\frac{1}{86267571272} a^{33} + \frac{321}{86267571272} a^{27} - \frac{3100371}{86267571272} a^{25} + \frac{149971}{1268640754} a^{23} - \frac{3793279}{43133785636} a^{21} - \frac{7160567}{86267571272} a^{19} + \frac{9070545063}{43133785636} a^{17} - \frac{18013040143}{86267571272} a^{15} + \frac{205606039}{1627690024} a^{13} + \frac{35142133383}{86267571272} a^{11} + \frac{17614110237}{43133785636} a^{9} - \frac{15113750935}{43133785636} a^{7} - \frac{11353006747}{86267571272} a^{5} + \frac{5231516253}{86267571272} a^{3} + \frac{27704408363}{86267571272} a$, $\frac{1}{86267571272} a^{34} - \frac{1236645}{86267571272} a^{26} - \frac{4752227}{86267571272} a^{24} + \frac{1705309}{10783446409} a^{22} + \frac{1853277}{10783446409} a^{20} + \frac{479947}{21566892818} a^{18} - \frac{35860137039}{86267571272} a^{16} - \frac{136546417}{1627690024} a^{14} + \frac{1774682488}{10783446409} a^{12} + \frac{4305218921}{86267571272} a^{10} - \frac{4667664351}{43133785636} a^{8} - \frac{18659694475}{86267571272} a^{6} - \frac{29037644525}{86267571272} a^{4} - \frac{10441924015}{86267571272} a^{2} + \frac{76465}{1865971}$, $\frac{1}{86267571272} a^{35} - \frac{367}{86267571272} a^{27} - \frac{2373225}{43133785636} a^{25} + \frac{3393287}{21566892818} a^{23} + \frac{2092491}{43133785636} a^{21} - \frac{304357}{86267571272} a^{19} - \frac{25603940089}{86267571272} a^{17} - \frac{108413385}{813845012} a^{15} - \frac{17863405407}{86267571272} a^{13} + \frac{647955082}{10783446409} a^{11} + \frac{33142646117}{86267571272} a^{9} - \frac{10567424611}{21566892818} a^{7} - \frac{16754847663}{43133785636} a^{5} + \frac{37990779}{267082264} a^{3} + \frac{29175033001}{86267571272} a$

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

Class group and class number

$C_{1406}$, which has order $1406$ (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:  \( \frac{1}{439204} a^{27} - \frac{27}{439204} a^{25} + \frac{81}{109801} a^{23} - \frac{2277}{439204} a^{21} + \frac{10395}{439204} a^{19} - \frac{1701}{23116} a^{17} + \frac{918}{5779} a^{15} - \frac{1377}{5779} a^{13} + \frac{53703}{219602} a^{11} - \frac{36465}{219602} a^{9} + \frac{7722}{109801} a^{7} - \frac{7371}{439204} a^{5} + \frac{819}{439204} a^{3} - \frac{27}{439204} a \) (order $4$)
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:  \( 5100074719720140.0 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(i, \sqrt{15})\), 6.0.419904.1, 6.6.157464000.1, 6.0.2460375.1, \(\Q(\zeta_{27})^+\), 12.0.24794911296000000.2, 18.0.258151783382020583032356864.7, 18.18.1512608105754026853705216000000000.1, 18.0.5770142004982097067662109375.1

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 R $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ $18^{2}$

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
5Data not computed