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Properties

Label 36.36.9006764330...9616.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 3^{54}\cdot 7^{30}$
Root discriminant $52.60$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -144, 0, 4956, 0, -63358, 0, 405648, 0, -1514853, 0, 3636879, 0, -5982741, 0, 7034958, 0, -6080856, 0, 3932379, 0, -1920270, 0, 709280, 0, -196911, 0, 40425, 0, -5951, 0, 594, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 1)
 
gp: K = bnfinit(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 36 x^{34} + 594 x^{32} - 5951 x^{30} + 40425 x^{28} - 196911 x^{26} + 709280 x^{24} - 1920270 x^{22} + 3932379 x^{20} - 6080856 x^{18} + 7034958 x^{16} - 5982741 x^{14} + 3636879 x^{12} - 1514853 x^{10} + 405648 x^{8} - 63358 x^{6} + 4956 x^{4} - 144 x^{2} + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90067643300370785938616861622694756230952958181429238736879616=2^{36}\cdot 3^{54}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.60$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(5,·)$, $\chi_{252}(11,·)$, $\chi_{252}(17,·)$, $\chi_{252}(19,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(31,·)$, $\chi_{252}(37,·)$, $\chi_{252}(41,·)$, $\chi_{252}(95,·)$, $\chi_{252}(71,·)$, $\chi_{252}(173,·)$, $\chi_{252}(179,·)$, $\chi_{252}(55,·)$, $\chi_{252}(185,·)$, $\chi_{252}(187,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(139,·)$, $\chi_{252}(199,·)$, $\chi_{252}(205,·)$, $\chi_{252}(209,·)$, $\chi_{252}(85,·)$, $\chi_{252}(89,·)$, $\chi_{252}(223,·)$, $\chi_{252}(101,·)$, $\chi_{252}(103,·)$, $\chi_{252}(107,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(115,·)$, $\chi_{252}(169,·)$, $\chi_{252}(121,·)$, $\chi_{252}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

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:  \( 33651165194729250000 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\zeta_{36})^+\), 6.6.3024568512.2, 6.6.4148928.1, 6.6.3024568512.1, 6.6.144027072.1, 6.6.6751269.1, 6.6.7057326528.1, 6.6.330812181.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{21})^+\), 6.6.7057326528.2, 6.6.330812181.2, 9.9.62523502209.1, 12.12.186694177220038656.1, 12.12.448252719505312813056.2, \(\Q(\zeta_{84})^+\), 12.12.448252719505312813056.1, 18.18.27668797159880354103659593728.1, 18.18.351496200956998572502045949952.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed
7Data not computed