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Properties

Label 36.0.40060199600...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{48}\cdot 5^{18}\cdot 7^{24}$
Root discriminant $70.81$
Ramified primes $2, 3, 5, 7$
Class number $326592$ (GRH)
Class group $[3, 6, 6, 12, 252]$ (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![5041, 0, 378984, 0, 7101906, 0, 61327725, 0, 295502889, 0, 874409358, 0, 1679154720, 0, 2166050013, 0, 1922710668, 0, 1196774341, 0, 529714350, 0, 168143160, 0, 38356241, 0, 6252714, 0, 716880, 0, 56015, 0, 2814, 0, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041)
 
gp: K = bnfinit(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041, 1)
 

Normalized defining polynomial

\( x^{36} + 81 x^{34} + 2814 x^{32} + 56015 x^{30} + 716880 x^{28} + 6252714 x^{26} + 38356241 x^{24} + 168143160 x^{22} + 529714350 x^{20} + 1196774341 x^{18} + 1922710668 x^{16} + 2166050013 x^{14} + 1679154720 x^{12} + 874409358 x^{10} + 295502889 x^{8} + 61327725 x^{6} + 7101906 x^{4} + 378984 x^{2} + 5041 \)

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:  \(4006019960016756356145265577845273975236004675584000000000000000000=2^{36}\cdot 3^{48}\cdot 5^{18}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.81$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(1159,·)$, $\chi_{1260}(919,·)$, $\chi_{1260}(781,·)$, $\chi_{1260}(529,·)$, $\chi_{1260}(1171,·)$, $\chi_{1260}(151,·)$, $\chi_{1260}(1051,·)$, $\chi_{1260}(541,·)$, $\chi_{1260}(799,·)$, $\chi_{1260}(289,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(169,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(949,·)$, $\chi_{1260}(571,·)$, $\chi_{1260}(319,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(1219,·)$, $\chi_{1260}(709,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(331,·)$, $\chi_{1260}(589,·)$, $\chi_{1260}(79,·)$, $\chi_{1260}(211,·)$, $\chi_{1260}(361,·)$, $\chi_{1260}(991,·)$, $\chi_{1260}(739,·)$, $\chi_{1260}(1129,·)$, $\chi_{1260}(109,·)$, $\chi_{1260}(751,·)$, $\chi_{1260}(1009,·)$, $\chi_{1260}(499,·)$, $\chi_{1260}(631,·)$, $\chi_{1260}(121,·)$, $\chi_{1260}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{71} a^{29} + \frac{17}{71} a^{27} - \frac{8}{71} a^{25} - \frac{10}{71} a^{23} + \frac{27}{71} a^{21} + \frac{13}{71} a^{19} - \frac{18}{71} a^{17} + \frac{15}{71} a^{15} + \frac{8}{71} a^{13} + \frac{8}{71} a^{11} - \frac{24}{71} a^{9} + \frac{1}{71} a^{7} - \frac{13}{71} a^{5} - \frac{17}{71} a^{3} + \frac{16}{71} a$, $\frac{1}{71} a^{30} + \frac{17}{71} a^{28} - \frac{8}{71} a^{26} - \frac{10}{71} a^{24} + \frac{27}{71} a^{22} + \frac{13}{71} a^{20} - \frac{18}{71} a^{18} + \frac{15}{71} a^{16} + \frac{8}{71} a^{14} + \frac{8}{71} a^{12} - \frac{24}{71} a^{10} + \frac{1}{71} a^{8} - \frac{13}{71} a^{6} - \frac{17}{71} a^{4} + \frac{16}{71} a^{2}$, $\frac{1}{71} a^{31} - \frac{13}{71} a^{27} - \frac{16}{71} a^{25} - \frac{16}{71} a^{23} - \frac{20}{71} a^{21} - \frac{26}{71} a^{19} - \frac{34}{71} a^{17} - \frac{34}{71} a^{15} + \frac{14}{71} a^{13} - \frac{18}{71} a^{11} - \frac{17}{71} a^{9} - \frac{30}{71} a^{7} - \frac{9}{71} a^{5} + \frac{21}{71} a^{3} + \frac{12}{71} a$, $\frac{1}{5770099} a^{32} - \frac{28577}{5770099} a^{30} - \frac{2814267}{5770099} a^{28} + \frac{2605893}{5770099} a^{26} + \frac{811509}{5770099} a^{24} + \frac{2246043}{5770099} a^{22} - \frac{2178264}{5770099} a^{20} - \frac{315141}{5770099} a^{18} - \frac{1656208}{5770099} a^{16} + \frac{2706964}{5770099} a^{14} - \frac{1412701}{5770099} a^{12} - \frac{38298}{5770099} a^{10} + \frac{2476131}{5770099} a^{8} - \frac{2710405}{5770099} a^{6} + \frac{765144}{5770099} a^{4} + \frac{456621}{5770099} a^{2} + \frac{17077}{81269}$, $\frac{1}{5770099} a^{33} - \frac{28577}{5770099} a^{31} + \frac{30148}{5770099} a^{29} - \frac{969943}{5770099} a^{27} + \frac{1136585}{5770099} a^{25} + \frac{2652388}{5770099} a^{23} - \frac{390346}{5770099} a^{21} + \frac{2041660}{5770099} a^{19} - \frac{924787}{5770099} a^{17} - \frac{11093}{81269} a^{15} - \frac{1737777}{5770099} a^{13} - \frac{363374}{5770099} a^{11} - \frac{2318740}{5770099} a^{9} + \frac{134010}{5770099} a^{7} - \frac{1591657}{5770099} a^{5} - \frac{1737642}{5770099} a^{3} + \frac{562315}{5770099} a$, $\frac{1}{122332055113742639309075453740943711388113321} a^{34} + \frac{687941942244972524071210255701528578}{122332055113742639309075453740943711388113321} a^{32} - \frac{234326635359397707106808408098137517031143}{122332055113742639309075453740943711388113321} a^{30} + \frac{2645958995797650254667182180523062979880881}{122332055113742639309075453740943711388113321} a^{28} + \frac{36076857477287890339681891940609341065055045}{122332055113742639309075453740943711388113321} a^{26} + \frac{17943522392824282543586087553140804059884331}{122332055113742639309075453740943711388113321} a^{24} + \frac{10827584948734060183996482592883498658648167}{122332055113742639309075453740943711388113321} a^{22} - \frac{22508199786981298008558079584243624569974108}{122332055113742639309075453740943711388113321} a^{20} - \frac{51765412550846364331823900019424093433288186}{122332055113742639309075453740943711388113321} a^{18} + \frac{22219198689238892538663529458926883670246177}{122332055113742639309075453740943711388113321} a^{16} + \frac{1088456821745795688562415696040431736230244}{122332055113742639309075453740943711388113321} a^{14} - \frac{59163941110549802840661235868532924468053159}{122332055113742639309075453740943711388113321} a^{12} - \frac{10208788155460422179291270650757264161675620}{122332055113742639309075453740943711388113321} a^{10} - \frac{49673354201564230821223803082840616270216787}{122332055113742639309075453740943711388113321} a^{8} + \frac{44093287853673940798774994130645844855548277}{122332055113742639309075453740943711388113321} a^{6} - \frac{1634997252689604533257930504615406075887754}{122332055113742639309075453740943711388113321} a^{4} - \frac{12914728577367547310933000680698518237140628}{122332055113742639309075453740943711388113321} a^{2} + \frac{790426084575091801153937862095542921137647}{1722986691742854074775710616069629737860751}$, $\frac{1}{122332055113742639309075453740943711388113321} a^{35} + \frac{687941942244972524071210255701528578}{122332055113742639309075453740943711388113321} a^{33} - \frac{234326635359397707106808408098137517031143}{122332055113742639309075453740943711388113321} a^{31} - \frac{800014387688057894884239051616196495840621}{122332055113742639309075453740943711388113321} a^{29} - \frac{22504690041969148202692269005758070022210489}{122332055113742639309075453740943711388113321} a^{27} + \frac{45511309460709947739997457410254879865656347}{122332055113742639309075453740943711388113321} a^{25} + \frac{45287318783591141679510694914276093415863187}{122332055113742639309075453740943711388113321} a^{23} + \frac{6782573972647221262629000888940080973658659}{122332055113742639309075453740943711388113321} a^{21} + \frac{25768988577582069033083077703709244770445609}{122332055113742639309075453740943711388113321} a^{19} - \frac{38085335521761000078486342103510157154880108}{122332055113742639309075453740943711388113321} a^{17} - \frac{50601143930539826554708902786048460399592286}{122332055113742639309075453740943711388113321} a^{15} + \frac{35600326935307171272002848015296711114288146}{122332055113742639309075453740943711388113321} a^{13} - \frac{37776575223346087375702640507871339967447636}{122332055113742639309075453740943711388113321} a^{11} + \frac{33030007002092764768010306488501611147099261}{122332055113742639309075453740943711388113321} a^{9} + \frac{40647314470188232649223572898506585379826775}{122332055113742639309075453740943711388113321} a^{7} + \frac{43162656732624601410910545513194967108491772}{122332055113742639309075453740943711388113321} a^{5} + \frac{45666818941889491231441160265668892850124906}{122332055113742639309075453740943711388113321} a^{3} + \frac{984677869060187489106848494555395789228905}{122332055113742639309075453740943711388113321} a$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{12}\times C_{252}$, which has order $326592$ (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:  \( \frac{11668155635671881393}{1002736895565255980039} a^{35} + \frac{938276035513231596047}{1002736895565255980039} a^{33} + \frac{32283807410423355517893}{1002736895565255980039} a^{31} + \frac{634654985108940857242377}{1002736895565255980039} a^{29} + \frac{7992411865836211603767424}{1002736895565255980039} a^{27} + \frac{68269950421701000071832360}{1002736895565255980039} a^{25} + \frac{5739546355554253770546201}{14123054867116281409} a^{23} + \frac{1722948118930681427303238362}{1002736895565255980039} a^{21} + \frac{5170537122883937374236160608}{1002736895565255980039} a^{19} + \frac{10932935199281709383236891581}{1002736895565255980039} a^{17} + \frac{16026747093101357680474632097}{1002736895565255980039} a^{15} + \frac{15884373575992061212839665505}{1002736895565255980039} a^{13} + \frac{10292704807702966549762934568}{1002736895565255980039} a^{11} + \frac{4183467902858993711269190544}{1002736895565255980039} a^{9} + \frac{1006607747918671501487159052}{1002736895565255980039} a^{7} + \frac{130616866199337710650813773}{1002736895565255980039} a^{5} + \frac{7645229074990333655777267}{1002736895565255980039} a^{3} + \frac{112176035263440902615685}{1002736895565255980039} a \) (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:  \( 13624539961495.691 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(i, \sqrt{5})\), 6.0.419904.1, 6.0.153664.1, 6.0.1008189504.1, 6.0.1008189504.2, 6.0.52488000.1, 6.6.820125.1, 6.0.19208000.1, 6.6.300125.1, 6.0.126023688000.14, 6.6.1969120125.1, 6.0.126023688000.1, 6.6.1969120125.2, 9.9.62523502209.1, 12.0.2754990144000000.1, 12.0.368947264000000.1, 12.0.15881969937121344000000.1, 12.0.15881969937121344000000.2, 18.0.1024770265180753855691096064.1, 18.0.2001504424181159874396672000000000.16, 18.18.7635133454060210702501953125.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$$[2]^{6}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed