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Properties

Label 36.36.2287983281...0000.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 3^{90}\cdot 5^{18}$
Root discriminant $69.71$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (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![9, 0, -729, 0, 17901, 0, -206064, 0, 1328724, 0, -5318379, 0, 14084001, 0, -25700409, 0, 33159726, 0, -30715038, 0, 20569950, 0, -9962190, 0, 3468195, 0, -857304, 0, 147510, 0, -17118, 0, 1269, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 54*x^34 + 1269*x^32 - 17118*x^30 + 147510*x^28 - 857304*x^26 + 3468195*x^24 - 9962190*x^22 + 20569950*x^20 - 30715038*x^18 + 33159726*x^16 - 25700409*x^14 + 14084001*x^12 - 5318379*x^10 + 1328724*x^8 - 206064*x^6 + 17901*x^4 - 729*x^2 + 9)
 
gp: K = bnfinit(x^36 - 54*x^34 + 1269*x^32 - 17118*x^30 + 147510*x^28 - 857304*x^26 + 3468195*x^24 - 9962190*x^22 + 20569950*x^20 - 30715038*x^18 + 33159726*x^16 - 25700409*x^14 + 14084001*x^12 - 5318379*x^10 + 1328724*x^8 - 206064*x^6 + 17901*x^4 - 729*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{36} - 54 x^{34} + 1269 x^{32} - 17118 x^{30} + 147510 x^{28} - 857304 x^{26} + 3468195 x^{24} - 9962190 x^{22} + 20569950 x^{20} - 30715038 x^{18} + 33159726 x^{16} - 25700409 x^{14} + 14084001 x^{12} - 5318379 x^{10} + 1328724 x^{8} - 206064 x^{6} + 17901 x^{4} - 729 x^{2} + 9 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
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$$\chi_{540}(1,·)$, $\chi_{540}(131,·)$, $\chi_{540}(11,·)$, $\chi_{540}(529,·)$, $\chi_{540}(409,·)$, $\chi_{540}(539,·)$, $\chi_{540}(289,·)$, $\chi_{540}(419,·)$, $\chi_{540}(421,·)$, $\chi_{540}(169,·)$, $\chi_{540}(299,·)$, $\chi_{540}(301,·)$, $\chi_{540}(431,·)$, $\chi_{540}(49,·)$, $\chi_{540}(179,·)$, $\chi_{540}(181,·)$, $\chi_{540}(311,·)$, $\chi_{540}(59,·)$, $\chi_{540}(61,·)$, $\chi_{540}(191,·)$, $\chi_{540}(71,·)$, $\chi_{540}(469,·)$, $\chi_{540}(349,·)$, $\chi_{540}(479,·)$, $\chi_{540}(481,·)$, $\chi_{540}(229,·)$, $\chi_{540}(359,·)$, $\chi_{540}(361,·)$, $\chi_{540}(491,·)$, $\chi_{540}(109,·)$, $\chi_{540}(239,·)$, $\chi_{540}(241,·)$, $\chi_{540}(371,·)$, $\chi_{540}(119,·)$, $\chi_{540}(121,·)$, $\chi_{540}(251,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18}$, $\frac{1}{3} a^{19}$, $\frac{1}{3} a^{20}$, $\frac{1}{3} a^{21}$, $\frac{1}{3} a^{22}$, $\frac{1}{3} a^{23}$, $\frac{1}{3} a^{24}$, $\frac{1}{3} a^{25}$, $\frac{1}{3} a^{26}$, $\frac{1}{3} a^{27}$, $\frac{1}{3} a^{28}$, $\frac{1}{3} a^{29}$, $\frac{1}{3} a^{30}$, $\frac{1}{3} a^{31}$, $\frac{1}{140937} a^{32} + \frac{7202}{46979} a^{30} - \frac{559}{46979} a^{28} + \frac{13849}{140937} a^{26} + \frac{1207}{140937} a^{24} + \frac{3307}{140937} a^{22} - \frac{7394}{140937} a^{20} - \frac{19928}{140937} a^{18} - \frac{5083}{46979} a^{16} - \frac{10693}{46979} a^{14} - \frac{12868}{46979} a^{12} - \frac{21211}{46979} a^{10} + \frac{22458}{46979} a^{8} + \frac{21236}{46979} a^{6} - \frac{23438}{46979} a^{4} - \frac{1360}{46979} a^{2} - \frac{22984}{46979}$, $\frac{1}{140937} a^{33} + \frac{7202}{46979} a^{31} - \frac{559}{46979} a^{29} + \frac{13849}{140937} a^{27} + \frac{1207}{140937} a^{25} + \frac{3307}{140937} a^{23} - \frac{7394}{140937} a^{21} - \frac{19928}{140937} a^{19} - \frac{5083}{46979} a^{17} - \frac{10693}{46979} a^{15} - \frac{12868}{46979} a^{13} - \frac{21211}{46979} a^{11} + \frac{22458}{46979} a^{9} + \frac{21236}{46979} a^{7} - \frac{23438}{46979} a^{5} - \frac{1360}{46979} a^{3} - \frac{22984}{46979} a$, $\frac{1}{1950373313128716520625853} a^{34} + \frac{3916204508399265346}{1950373313128716520625853} a^{32} - \frac{84693961811878515263289}{650124437709572173541951} a^{30} + \frac{2433754794560254880590}{650124437709572173541951} a^{28} + \frac{79094222736092254160482}{650124437709572173541951} a^{26} - \frac{89344677118432713887382}{650124437709572173541951} a^{24} - \frac{54614399591277962382928}{1950373313128716520625853} a^{22} - \frac{193192510789772788122760}{1950373313128716520625853} a^{20} - \frac{71228309448387769920533}{650124437709572173541951} a^{18} - \frac{44127168558631452398728}{650124437709572173541951} a^{16} - \frac{279944454330365720119186}{650124437709572173541951} a^{14} + \frac{43144318813691081409297}{650124437709572173541951} a^{12} + \frac{6954469836539135511981}{650124437709572173541951} a^{10} - \frac{177650558965933283931843}{650124437709572173541951} a^{8} - \frac{231936919520910243747759}{650124437709572173541951} a^{6} + \frac{62534302822989068806072}{650124437709572173541951} a^{4} + \frac{125671291535299703970597}{650124437709572173541951} a^{2} - \frac{176282377497728609484305}{650124437709572173541951}$, $\frac{1}{1950373313128716520625853} a^{35} + \frac{3916204508399265346}{1950373313128716520625853} a^{33} - \frac{84693961811878515263289}{650124437709572173541951} a^{31} + \frac{2433754794560254880590}{650124437709572173541951} a^{29} + \frac{79094222736092254160482}{650124437709572173541951} a^{27} - \frac{89344677118432713887382}{650124437709572173541951} a^{25} - \frac{54614399591277962382928}{1950373313128716520625853} a^{23} - \frac{193192510789772788122760}{1950373313128716520625853} a^{21} - \frac{71228309448387769920533}{650124437709572173541951} a^{19} - \frac{44127168558631452398728}{650124437709572173541951} a^{17} - \frac{279944454330365720119186}{650124437709572173541951} a^{15} + \frac{43144318813691081409297}{650124437709572173541951} a^{13} + \frac{6954469836539135511981}{650124437709572173541951} a^{11} - \frac{177650558965933283931843}{650124437709572173541951} a^{9} - \frac{231936919520910243747759}{650124437709572173541951} a^{7} + \frac{62534302822989068806072}{650124437709572173541951} a^{5} + \frac{125671291535299703970597}{650124437709572173541951} a^{3} - \frac{176282377497728609484305}{650124437709572173541951} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:  \( 6642656411972985000000 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{36})^+\), 6.6.157464000.1, 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.12.24794911296000000.1, \(\Q(\zeta_{108})^+\), 18.18.1512608105754026853705216000000000.1, 18.18.1923380668327365689220703125.1

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 $18^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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