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Properties

Label 36.0.34493191016...9664.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 7^{24}\cdot 13^{30}$
Root discriminant $62.05$
Ramified primes $2, 7, 13$
Class number $139968$ (GRH)
Class group $[3, 6, 6, 36, 36]$ (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, 126, 0, 4025, 0, 59787, 0, 496906, 0, 2527930, 0, 8297859, 0, 18181863, 0, 27188388, 0, 28134360, 0, 20316834, 0, 10287051, 0, 3655689, 0, 907822, 0, 155517, 0, 17888, 0, 1311, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 55*x^34 + 1311*x^32 + 17888*x^30 + 155517*x^28 + 907822*x^26 + 3655689*x^24 + 10287051*x^22 + 20316834*x^20 + 28134360*x^18 + 27188388*x^16 + 18181863*x^14 + 8297859*x^12 + 2527930*x^10 + 496906*x^8 + 59787*x^6 + 4025*x^4 + 126*x^2 + 1)
 
gp: K = bnfinit(x^36 + 55*x^34 + 1311*x^32 + 17888*x^30 + 155517*x^28 + 907822*x^26 + 3655689*x^24 + 10287051*x^22 + 20316834*x^20 + 28134360*x^18 + 27188388*x^16 + 18181863*x^14 + 8297859*x^12 + 2527930*x^10 + 496906*x^8 + 59787*x^6 + 4025*x^4 + 126*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 55 x^{34} + 1311 x^{32} + 17888 x^{30} + 155517 x^{28} + 907822 x^{26} + 3655689 x^{24} + 10287051 x^{22} + 20316834 x^{20} + 28134360 x^{18} + 27188388 x^{16} + 18181863 x^{14} + 8297859 x^{12} + 2527930 x^{10} + 496906 x^{8} + 59787 x^{6} + 4025 x^{4} + 126 x^{2} + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34493191016101639909923950785592661673806077335330090596029169664=2^{36}\cdot 7^{24}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.05$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(107,·)$, $\chi_{364}(261,·)$, $\chi_{364}(51,·)$, $\chi_{364}(263,·)$, $\chi_{364}(9,·)$, $\chi_{364}(277,·)$, $\chi_{364}(23,·)$, $\chi_{364}(25,·)$, $\chi_{364}(155,·)$, $\chi_{364}(29,·)$, $\chi_{364}(289,·)$, $\chi_{364}(347,·)$, $\chi_{364}(165,·)$, $\chi_{364}(295,·)$, $\chi_{364}(43,·)$, $\chi_{364}(303,·)$, $\chi_{364}(179,·)$, $\chi_{364}(309,·)$, $\chi_{364}(183,·)$, $\chi_{364}(53,·)$, $\chi_{364}(205,·)$, $\chi_{364}(207,·)$, $\chi_{364}(337,·)$, $\chi_{364}(211,·)$, $\chi_{364}(79,·)$, $\chi_{364}(95,·)$, $\chi_{364}(225,·)$, $\chi_{364}(81,·)$, $\chi_{364}(361,·)$, $\chi_{364}(235,·)$, $\chi_{364}(113,·)$, $\chi_{364}(233,·)$, $\chi_{364}(121,·)$, $\chi_{364}(191,·)$, $\chi_{364}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{3} a^{26} + \frac{1}{3}$, $\frac{1}{3} a^{27} + \frac{1}{3} a$, $\frac{1}{3} a^{28} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{29} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{30} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{31} + \frac{1}{3} a^{5}$, $\frac{1}{2811} a^{32} - \frac{41}{937} a^{30} + \frac{329}{2811} a^{28} - \frac{446}{2811} a^{26} + \frac{145}{937} a^{24} - \frac{29}{937} a^{22} - \frac{37}{937} a^{20} - \frac{368}{937} a^{18} - \frac{143}{937} a^{16} + \frac{202}{937} a^{14} - \frac{17}{937} a^{12} - \frac{285}{937} a^{10} + \frac{98}{937} a^{8} - \frac{851}{2811} a^{6} + \frac{74}{937} a^{4} - \frac{916}{2811} a^{2} + \frac{355}{2811}$, $\frac{1}{2811} a^{33} - \frac{41}{937} a^{31} + \frac{329}{2811} a^{29} - \frac{446}{2811} a^{27} + \frac{145}{937} a^{25} - \frac{29}{937} a^{23} - \frac{37}{937} a^{21} - \frac{368}{937} a^{19} - \frac{143}{937} a^{17} + \frac{202}{937} a^{15} - \frac{17}{937} a^{13} - \frac{285}{937} a^{11} + \frac{98}{937} a^{9} - \frac{851}{2811} a^{7} + \frac{74}{937} a^{5} - \frac{916}{2811} a^{3} + \frac{355}{2811} a$, $\frac{1}{428429008139745054445897026033} a^{34} - \frac{838555497020151424860146}{47603223126638339382877447337} a^{32} + \frac{4732383756805825547833828387}{47603223126638339382877447337} a^{30} + \frac{67674937319080603903127709449}{428429008139745054445897026033} a^{28} + \frac{48064278564862478666417346394}{428429008139745054445897026033} a^{26} + \frac{36687672702705053337608836234}{142809669379915018148632342011} a^{24} + \frac{53760334201008407196104248012}{142809669379915018148632342011} a^{22} - \frac{13091900624764751983944931325}{142809669379915018148632342011} a^{20} + \frac{41333773785159391026481865087}{142809669379915018148632342011} a^{18} - \frac{60823616248012574978045089784}{142809669379915018148632342011} a^{16} + \frac{37311437039394162442070974439}{142809669379915018148632342011} a^{14} - \frac{31664776609242030356003401622}{142809669379915018148632342011} a^{12} + \frac{22213611649593706627078975663}{47603223126638339382877447337} a^{10} - \frac{12343760860124962320423453431}{428429008139745054445897026033} a^{8} + \frac{45595351614248090227431312314}{142809669379915018148632342011} a^{6} + \frac{69098952691207623312439841777}{142809669379915018148632342011} a^{4} + \frac{81860817773901832270765295102}{428429008139745054445897026033} a^{2} - \frac{172010188419672783368880316217}{428429008139745054445897026033}$, $\frac{1}{428429008139745054445897026033} a^{35} - \frac{838555497020151424860146}{47603223126638339382877447337} a^{33} + \frac{4732383756805825547833828387}{47603223126638339382877447337} a^{31} + \frac{67674937319080603903127709449}{428429008139745054445897026033} a^{29} + \frac{48064278564862478666417346394}{428429008139745054445897026033} a^{27} + \frac{36687672702705053337608836234}{142809669379915018148632342011} a^{25} + \frac{53760334201008407196104248012}{142809669379915018148632342011} a^{23} - \frac{13091900624764751983944931325}{142809669379915018148632342011} a^{21} + \frac{41333773785159391026481865087}{142809669379915018148632342011} a^{19} - \frac{60823616248012574978045089784}{142809669379915018148632342011} a^{17} + \frac{37311437039394162442070974439}{142809669379915018148632342011} a^{15} - \frac{31664776609242030356003401622}{142809669379915018148632342011} a^{13} + \frac{22213611649593706627078975663}{47603223126638339382877447337} a^{11} - \frac{12343760860124962320423453431}{428429008139745054445897026033} a^{9} + \frac{45595351614248090227431312314}{142809669379915018148632342011} a^{7} + \frac{69098952691207623312439841777}{142809669379915018148632342011} a^{5} + \frac{81860817773901832270765295102}{428429008139745054445897026033} a^{3} - \frac{172010188419672783368880316217}{428429008139745054445897026033} a$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{36}\times C_{36}$, which has order $139968$ (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:  \( -\frac{1589620738669101764}{5791131897569746179} a^{35} - \frac{87687161295932572615}{5791131897569746179} a^{33} - \frac{2097687417390278278025}{5791131897569746179} a^{31} - \frac{9582156977711942649889}{1930377299189915393} a^{29} - \frac{83732708101292303405887}{1930377299189915393} a^{27} - \frac{491606290485781757727428}{1930377299189915393} a^{25} - \frac{1991534650963886384946798}{1930377299189915393} a^{23} - \frac{5633094669063603638015342}{1930377299189915393} a^{21} - \frac{11146762736309379646807859}{1930377299189915393} a^{19} - \frac{15341727593986249765465804}{1930377299189915393} a^{17} - \frac{14486638799037617510862556}{1930377299189915393} a^{15} - \frac{9158350776663471144796391}{1930377299189915393} a^{13} - \frac{3713707091093352310483578}{1930377299189915393} a^{11} - \frac{2669349204981514843105781}{5791131897569746179} a^{9} - \frac{308997332142749953826581}{5791131897569746179} a^{7} - \frac{3727940945647470557936}{5791131897569746179} a^{5} + \frac{621409252376036385603}{1930377299189915393} a^{3} + \frac{25167805450447728655}{1930377299189915393} a \) (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:  \( 4866030378143.887 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, \(\Q(i, \sqrt{13})\), 6.0.1827904.1, 6.0.153664.1, 6.0.4388797504.2, 6.0.4388797504.1, \(\Q(\zeta_{13})^+\), 6.0.23762752.1, 6.6.5274997.1, 6.0.337599808.1, 6.6.891474493.1, 6.0.57054367552.1, 6.6.891474493.2, 6.0.57054367552.2, 9.9.567869252041.1, 12.0.564668382613504.1, 12.0.113973630361636864.1, 12.0.3255200856758710472704.1, 12.0.3255200856758710472704.2, 18.0.84535014172552012147112280064.1, 18.18.708478645847689707516501157.1, 18.0.185723426137096770687205679300608.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$$[2]^{6}$
7Data not computed
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$