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Properties

Label 36.0.13889291995...1712.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 37^{35}$
Root discriminant $94.66$
Ramified primes $2, 37$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9699328, 0, 276430848, 0, 2349662208, 0, 9398648832, 0, 21538570240, 0, 31524634624, 0, 31524634624, 0, 22517596160, 0, 11838295040, 0, 4673011200, 0, 1401903360, 0, 321384960, 0, 56242368, 0, 7450912, 0, 734080, 0, 52096, 0, 2516, 0, 74, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 74*x^34 + 2516*x^32 + 52096*x^30 + 734080*x^28 + 7450912*x^26 + 56242368*x^24 + 321384960*x^22 + 1401903360*x^20 + 4673011200*x^18 + 11838295040*x^16 + 22517596160*x^14 + 31524634624*x^12 + 31524634624*x^10 + 21538570240*x^8 + 9398648832*x^6 + 2349662208*x^4 + 276430848*x^2 + 9699328)
 
gp: K = bnfinit(x^36 + 74*x^34 + 2516*x^32 + 52096*x^30 + 734080*x^28 + 7450912*x^26 + 56242368*x^24 + 321384960*x^22 + 1401903360*x^20 + 4673011200*x^18 + 11838295040*x^16 + 22517596160*x^14 + 31524634624*x^12 + 31524634624*x^10 + 21538570240*x^8 + 9398648832*x^6 + 2349662208*x^4 + 276430848*x^2 + 9699328, 1)
 

Normalized defining polynomial

\( x^{36} + 74 x^{34} + 2516 x^{32} + 52096 x^{30} + 734080 x^{28} + 7450912 x^{26} + 56242368 x^{24} + 321384960 x^{22} + 1401903360 x^{20} + 4673011200 x^{18} + 11838295040 x^{16} + 22517596160 x^{14} + 31524634624 x^{12} + 31524634624 x^{10} + 21538570240 x^{8} + 9398648832 x^{6} + 2349662208 x^{4} + 276430848 x^{2} + 9699328 \)

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:  \(138892919952333446776057851184385905517238171566853781889085447929331712=2^{54}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.66$
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:  $2, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(296=2^{3}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{296}(1,·)$, $\chi_{296}(5,·)$, $\chi_{296}(9,·)$, $\chi_{296}(13,·)$, $\chi_{296}(109,·)$, $\chi_{296}(145,·)$, $\chi_{296}(277,·)$, $\chi_{296}(73,·)$, $\chi_{296}(25,·)$, $\chi_{296}(29,·)$, $\chi_{296}(261,·)$, $\chi_{296}(133,·)$, $\chi_{296}(33,·)$, $\chi_{296}(165,·)$, $\chi_{296}(41,·)$, $\chi_{296}(45,·)$, $\chi_{296}(49,·)$, $\chi_{296}(137,·)$, $\chi_{296}(61,·)$, $\chi_{296}(117,·)$, $\chi_{296}(65,·)$, $\chi_{296}(69,·)$, $\chi_{296}(289,·)$, $\chi_{296}(201,·)$, $\chi_{296}(205,·)$, $\chi_{296}(81,·)$, $\chi_{296}(121,·)$, $\chi_{296}(93,·)$, $\chi_{296}(225,·)$, $\chi_{296}(233,·)$, $\chi_{296}(237,·)$, $\chi_{296}(125,·)$, $\chi_{296}(245,·)$, $\chi_{296}(169,·)$, $\chi_{296}(249,·)$, $\chi_{296}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.3241792.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.46640037043754870505472.1, \(\Q(\zeta_{37})^+\)

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 ${\href{/LocalNumberField/3.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $36$ $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R $18^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ $18^{2}$ $36$

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
37Data not computed