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Properties

Label 36.0.22879832815...0000.2
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 $923742$ (GRH)
Class group $[923742]$ (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![33373729, 0, -467694, 0, -3111399, 0, -7915284, 0, -9773136, 0, -5645574, 0, 901446, 0, 5638626, 0, 7250706, 0, 6243322, 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 + 6243322*x^18 + 7250706*x^16 + 5638626*x^14 + 901446*x^12 - 5645574*x^10 - 9773136*x^8 - 7915284*x^6 - 3111399*x^4 - 467694*x^2 + 33373729)
 
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 + 6243322*x^18 + 7250706*x^16 + 5638626*x^14 + 901446*x^12 - 5645574*x^10 - 9773136*x^8 - 7915284*x^6 - 3111399*x^4 - 467694*x^2 + 33373729, 1)
 

Normalized defining polynomial

\( x^{36} + 36 x^{34} + 594 x^{32} + 5952 x^{30} + 40455 x^{28} + 197316 x^{26} + 712530 x^{24} + 1937520 x^{22} + 3996135 x^{20} + 6243322 x^{18} + 7250706 x^{16} + 5638626 x^{14} + 901446 x^{12} - 5645574 x^{10} - 9773136 x^{8} - 7915284 x^{6} - 3111399 x^{4} - 467694 x^{2} + 33373729 \)

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$$\chi_{540}(1,·)$, $\chi_{540}(131,·)$, $\chi_{540}(389,·)$, $\chi_{540}(11,·)$, $\chi_{540}(269,·)$, $\chi_{540}(19,·)$, $\chi_{540}(149,·)$, $\chi_{540}(29,·)$, $\chi_{540}(421,·)$, $\chi_{540}(71,·)$, $\chi_{540}(301,·)$, $\chi_{540}(431,·)$, $\chi_{540}(371,·)$, $\chi_{540}(181,·)$, $\chi_{540}(311,·)$, $\chi_{540}(61,·)$, $\chi_{540}(319,·)$, $\chi_{540}(449,·)$, $\chi_{540}(139,·)$, $\chi_{540}(199,·)$, $\chi_{540}(329,·)$, $\chi_{540}(439,·)$, $\chi_{540}(79,·)$, $\chi_{540}(209,·)$, $\chi_{540}(89,·)$, $\chi_{540}(481,·)$, $\chi_{540}(379,·)$, $\chi_{540}(259,·)$, $\chi_{540}(361,·)$, $\chi_{540}(491,·)$, $\chi_{540}(191,·)$, $\chi_{540}(241,·)$, $\chi_{540}(499,·)$, $\chi_{540}(121,·)$, $\chi_{540}(251,·)$, $\chi_{540}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2584} a^{18} + \frac{9}{1292} a^{16} + \frac{135}{2584} a^{14} + \frac{273}{1292} a^{12} + \frac{1287}{2584} a^{10} - \frac{401}{1292} a^{8} - \frac{599}{1292} a^{6} + \frac{135}{646} a^{4} + \frac{81}{2584} a^{2} + \frac{989}{2584}$, $\frac{1}{2584} a^{19} + \frac{9}{1292} a^{17} + \frac{135}{2584} a^{15} + \frac{273}{1292} a^{13} + \frac{1287}{2584} a^{11} - \frac{401}{1292} a^{9} - \frac{599}{1292} a^{7} + \frac{135}{646} a^{5} + \frac{81}{2584} a^{3} + \frac{989}{2584} a$, $\frac{1}{2584} a^{20} - \frac{189}{2584} a^{16} + \frac{175}{646} a^{14} - \frac{789}{2584} a^{12} - \frac{89}{323} a^{10} + \frac{159}{1292} a^{8} - \frac{144}{323} a^{6} + \frac{41}{152} a^{4} - \frac{469}{2584} a^{2} + \frac{143}{1292}$, $\frac{1}{2584} a^{21} - \frac{189}{2584} a^{17} + \frac{175}{646} a^{15} - \frac{789}{2584} a^{13} - \frac{89}{323} a^{11} + \frac{159}{1292} a^{9} - \frac{144}{323} a^{7} + \frac{41}{152} a^{5} - \frac{469}{2584} a^{3} + \frac{143}{1292} a$, $\frac{1}{2584} a^{22} - \frac{533}{1292} a^{16} - \frac{557}{1292} a^{14} - \frac{439}{1292} a^{12} + \frac{35}{136} a^{10} - \frac{137}{1292} a^{8} - \frac{917}{2584} a^{6} + \frac{815}{2584} a^{4} + \frac{91}{2584} a^{2} + \frac{873}{2584}$, $\frac{1}{2584} a^{23} - \frac{533}{1292} a^{17} - \frac{557}{1292} a^{15} - \frac{439}{1292} a^{13} + \frac{35}{136} a^{11} - \frac{137}{1292} a^{9} - \frac{917}{2584} a^{7} + \frac{815}{2584} a^{5} + \frac{91}{2584} a^{3} + \frac{873}{2584} a$, $\frac{1}{2584} a^{24} - \frac{7}{1292} a^{16} + \frac{6}{17} a^{14} - \frac{1283}{2584} a^{12} - \frac{109}{646} a^{10} - \frac{545}{2584} a^{8} + \frac{243}{2584} a^{6} - \frac{501}{2584} a^{4} - \frac{637}{2584} a^{2} + \frac{1}{1292}$, $\frac{1}{2584} a^{25} - \frac{7}{1292} a^{17} + \frac{6}{17} a^{15} - \frac{1283}{2584} a^{13} - \frac{109}{646} a^{11} - \frac{545}{2584} a^{9} + \frac{243}{2584} a^{7} - \frac{501}{2584} a^{5} - \frac{637}{2584} a^{3} + \frac{1}{1292} a$, $\frac{1}{2584} a^{26} + \frac{291}{646} a^{16} + \frac{607}{2584} a^{14} - \frac{4}{19} a^{12} - \frac{615}{2584} a^{10} - \frac{649}{2584} a^{8} + \frac{815}{2584} a^{6} - \frac{829}{2584} a^{4} + \frac{142}{323} a^{2} + \frac{463}{1292}$, $\frac{1}{14927768} a^{27} + \frac{27}{14927768} a^{25} + \frac{81}{3731942} a^{23} + \frac{2277}{14927768} a^{21} - \frac{61}{785672} a^{19} - \frac{175653}{14927768} a^{17} - \frac{745011}{7463884} a^{15} - \frac{775479}{1865971} a^{13} + \frac{20647}{1865971} a^{11} + \frac{13449}{1865971} a^{9} + \frac{2713303}{7463884} a^{7} - \frac{1714175}{14927768} a^{5} + \frac{5743157}{14927768} a^{3} - \frac{4945085}{14927768} a$, $\frac{1}{86267571272} a^{28} - \frac{2875}{43133785636} a^{26} - \frac{74939}{43133785636} a^{24} - \frac{862523}{43133785636} a^{22} - \frac{11555159}{86267571272} a^{20} - \frac{16524563}{86267571272} a^{18} + \frac{113864781}{21566892818} a^{16} + \frac{4216026381}{86267571272} a^{14} + \frac{4457534055}{21566892818} a^{12} - \frac{5357252931}{43133785636} a^{10} - \frac{42513911533}{86267571272} a^{8} - \frac{7987480801}{86267571272} a^{6} + \frac{27161952799}{86267571272} a^{4} - \frac{17828229265}{43133785636} a^{2} + \frac{2201797}{14932936}$, $\frac{1}{86267571272} a^{29} + \frac{29}{86267571272} a^{27} + \frac{6155}{86267571272} a^{25} + \frac{73675}{43133785636} a^{23} + \frac{200453}{10783446409} a^{21} + \frac{10162859}{86267571272} a^{19} + \frac{2173449}{4540398488} a^{17} + \frac{43422}{33385283} a^{15} + \frac{6075165}{2537281508} a^{13} + \frac{33206905463}{86267571272} a^{11} + \frac{17600436941}{86267571272} a^{9} - \frac{874878619}{4540398488} a^{7} + \frac{17641894147}{43133785636} a^{5} + \frac{29681712}{10783446409} a^{3} + \frac{17160179941}{86267571272} a$, $\frac{1}{86267571272} a^{30} + \frac{172905}{86267571272} a^{26} + \frac{1123453}{21566892818} a^{24} - \frac{111328}{634320377} a^{22} + \frac{1426205}{10783446409} a^{20} - \frac{6828335}{43133785636} a^{18} + \frac{12648481651}{43133785636} a^{16} - \frac{8414706107}{86267571272} a^{14} - \frac{22148581027}{86267571272} a^{12} + \frac{6687290517}{86267571272} a^{10} + \frac{499293050}{10783446409} a^{8} - \frac{27604235103}{86267571272} a^{6} + \frac{17259683957}{86267571272} a^{4} + \frac{2173681163}{5074563016} a^{2} - \frac{2721851}{14932936}$, $\frac{1}{86267571272} a^{31} - \frac{465}{86267571272} a^{27} - \frac{93589}{43133785636} a^{25} - \frac{2270961}{43133785636} a^{23} - \frac{16115737}{86267571272} a^{21} - \frac{6516269}{43133785636} a^{19} - \frac{2173542093}{86267571272} a^{17} - \frac{236562009}{791445608} a^{15} - \frac{1026105163}{10783446409} a^{13} + \frac{20783889637}{86267571272} a^{11} - \frac{7162513905}{43133785636} a^{9} - \frac{4982420509}{86267571272} a^{7} - \frac{472282397}{43133785636} a^{5} - \frac{4910980953}{21566892818} a^{3} + \frac{1600507731}{4540398488} a$, $\frac{1}{86267571272} a^{32} - \frac{357616}{10783446409} a^{26} - \frac{70421}{813845012} a^{24} + \frac{4092487}{21566892818} a^{22} - \frac{293445}{2270199244} a^{20} - \frac{8805403}{86267571272} a^{18} + \frac{1195251628}{10783446409} a^{16} - \frac{17908395753}{43133785636} a^{14} + \frac{379365664}{10783446409} a^{12} - \frac{11048158037}{43133785636} a^{10} - \frac{1526574811}{43133785636} a^{8} - \frac{12707522691}{43133785636} a^{6} - \frac{1115361045}{4540398488} a^{4} - \frac{4639419956}{10783446409} a^{2} - \frac{3913313}{14932936}$, $\frac{1}{86267571272} a^{33} - \frac{1}{267082264} a^{27} + \frac{3001143}{86267571272} a^{25} + \frac{2104511}{21566892818} a^{23} - \frac{3841755}{43133785636} a^{21} + \frac{7140851}{43133785636} a^{19} - \frac{2075500560}{10783446409} a^{17} - \frac{637607910}{10783446409} a^{15} + \frac{42832384447}{86267571272} a^{13} - \frac{279667153}{2537281508} a^{11} - \frac{4815380595}{43133785636} a^{9} - \frac{1245916411}{43133785636} a^{7} - \frac{183216365}{791445608} a^{5} + \frac{980254711}{21566892818} a^{3} - \frac{207384729}{21566892818} a$, $\frac{1}{86267571272} a^{34} + \frac{1143893}{86267571272} a^{26} - \frac{6607267}{86267571272} a^{24} + \frac{2676443}{86267571272} a^{22} - \frac{6134121}{43133785636} a^{20} - \frac{3653699}{43133785636} a^{18} + \frac{7369587433}{86267571272} a^{16} - \frac{25850315191}{86267571272} a^{14} + \frac{6629948075}{21566892818} a^{12} - \frac{8166698385}{21566892818} a^{10} - \frac{1079227797}{2270199244} a^{8} - \frac{3509100615}{21566892818} a^{6} + \frac{1636775772}{10783446409} a^{4} + \frac{12381050157}{43133785636} a^{2} + \frac{1715717}{14932936}$, $\frac{1}{86267571272} a^{35} - \frac{349}{86267571272} a^{27} - \frac{2058259}{43133785636} a^{25} - \frac{204963}{21566892818} a^{23} - \frac{6827601}{43133785636} a^{21} - \frac{2067780}{10783446409} a^{19} - \frac{9646770531}{86267571272} a^{17} + \frac{582238747}{5074563016} a^{15} + \frac{7896600723}{86267571272} a^{13} - \frac{27799771751}{86267571272} a^{11} - \frac{22935579743}{86267571272} a^{9} - \frac{3865075457}{21566892818} a^{7} - \frac{31685908}{203461253} a^{5} - \frac{6883431271}{43133785636} a^{3} - \frac{10934300779}{43133785636} a$

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

Class group and class number

$C_{923742}$, which has order $923742$ (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:  \( -1 \) (order $2$)
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:  \( 5723956792899.807 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\zeta_{36})^+\), 6.0.52488000.1, 6.0.2460375.1, \(\Q(\zeta_{27})^+\), 12.0.24794911296000000.1, \(\Q(\zeta_{108})^+\), 18.0.504202701918008951235072000000000.2, 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 $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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