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Properties

Label 27.27.3063651216...3609.3
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 37^{24}$
Root discriminant $1135.13$
Ramified primes $3, 37$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1431296727003283447513, -840471524754754757895, -6484339376994847164786, 4874690167640696959299, 1168349437296368858520, -1298861255741049657135, -81047663668306668564, 148414315046635735704, 2816328081138069564, -9477769626051942234, -54973370953145502, 377261368227838926, 630325301674068, -9934810805722716, -4212153040590, 179005600103112, 15178929876, -2241127346283, -22791186, 19481903595, 0, -115336881, 0, 443556, 0, -999, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 22791186*x^18 - 2241127346283*x^17 + 15178929876*x^16 + 179005600103112*x^15 - 4212153040590*x^14 - 9934810805722716*x^13 + 630325301674068*x^12 + 377261368227838926*x^11 - 54973370953145502*x^10 - 9477769626051942234*x^9 + 2816328081138069564*x^8 + 148414315046635735704*x^7 - 81047663668306668564*x^6 - 1298861255741049657135*x^5 + 1168349437296368858520*x^4 + 4874690167640696959299*x^3 - 6484339376994847164786*x^2 - 840471524754754757895*x + 1431296727003283447513)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 22791186*x^18 - 2241127346283*x^17 + 15178929876*x^16 + 179005600103112*x^15 - 4212153040590*x^14 - 9934810805722716*x^13 + 630325301674068*x^12 + 377261368227838926*x^11 - 54973370953145502*x^10 - 9477769626051942234*x^9 + 2816328081138069564*x^8 + 148414315046635735704*x^7 - 81047663668306668564*x^6 - 1298861255741049657135*x^5 + 1168349437296368858520*x^4 + 4874690167640696959299*x^3 - 6484339376994847164786*x^2 - 840471524754754757895*x + 1431296727003283447513, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 22791186 x^{18} - 2241127346283 x^{17} + 15178929876 x^{16} + 179005600103112 x^{15} - 4212153040590 x^{14} - 9934810805722716 x^{13} + 630325301674068 x^{12} + 377261368227838926 x^{11} - 54973370953145502 x^{10} - 9477769626051942234 x^{9} + 2816328081138069564 x^{8} + 148414315046635735704 x^{7} - 81047663668306668564 x^{6} - 1298861255741049657135 x^{5} + 1168349437296368858520 x^{4} + 4874690167640696959299 x^{3} - 6484339376994847164786 x^{2} - 840471524754754757895 x + 1431296727003283447513 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30636512167696967158640831949436414926745966008852301931537630975865004722034443609=3^{94}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1135.13$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2997=3^{4}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2997}(2560,·)$, $\chi_{2997}(1,·)$, $\chi_{2997}(1159,·)$, $\chi_{2997}(10,·)$, $\chi_{2997}(1099,·)$, $\chi_{2997}(256,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(16,·)$, $\chi_{2997}(2008,·)$, $\chi_{2997}(2254,·)$, $\chi_{2997}(601,·)$, $\chi_{2997}(1624,·)$, $\chi_{2997}(1561,·)$, $\chi_{2997}(2014,·)$, $\chi_{2997}(160,·)$, $\chi_{2997}(100,·)$, $\chi_{2997}(2599,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(1255,·)$, $\chi_{2997}(2098,·)$, $\chi_{2997}(2158,·)$, $\chi_{2997}(1009,·)$, $\chi_{2997}(562,·)$, $\chi_{2997}(625,·)$, $\chi_{2997}(1015,·)$, $\chi_{2997}(1600,·)$, $\chi_{2997}(2623,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{37} a^{9}$, $\frac{1}{37} a^{10}$, $\frac{1}{37} a^{11}$, $\frac{1}{37} a^{12}$, $\frac{1}{37} a^{13}$, $\frac{1}{1369} a^{14} - \frac{2}{37} a^{5}$, $\frac{1}{1369} a^{15} - \frac{2}{37} a^{6}$, $\frac{1}{1369} a^{16} - \frac{2}{37} a^{7}$, $\frac{1}{1369} a^{17} - \frac{2}{37} a^{8}$, $\frac{1}{19808029901622710611} a^{18} - \frac{18}{535352159503316503} a^{16} + \frac{135}{14468977283873419} a^{14} - \frac{546}{391053440104687} a^{12} + \frac{47619}{391053440104687} a^{10} - \frac{4954744511799653}{535352159503316503} a^{9} - \frac{2439558}{391053440104687} a^{8} + \frac{1185768754576620}{14468977283873419} a^{7} + \frac{70205058}{391053440104687} a^{6} - \frac{37825302787677}{391053440104687} a^{5} - \frac{1012046940}{391053440104687} a^{4} - \frac{9173534703138}{391053440104687} a^{3} + \frac{5616860517}{391053440104687} a^{2} - \frac{54595140837043}{391053440104687} a - \frac{131332558743384}{391053440104687}$, $\frac{1}{19808029901622710611} a^{19} - \frac{18}{535352159503316503} a^{17} + \frac{135}{14468977283873419} a^{15} - \frac{546}{391053440104687} a^{13} + \frac{47619}{391053440104687} a^{11} - \frac{4954744511799653}{535352159503316503} a^{10} - \frac{2439558}{391053440104687} a^{9} + \frac{1185768754576620}{14468977283873419} a^{8} + \frac{70205058}{391053440104687} a^{7} - \frac{37825302787677}{391053440104687} a^{6} - \frac{1012046940}{391053440104687} a^{5} - \frac{9173534703138}{391053440104687} a^{4} + \frac{5616860517}{391053440104687} a^{3} - \frac{54595140837043}{391053440104687} a^{2} - \frac{131332558743384}{391053440104687} a$, $\frac{1}{19808029901622710611} a^{20} - \frac{189}{14468977283873419} a^{16} + \frac{1884}{391053440104687} a^{14} - \frac{316017}{391053440104687} a^{12} - \frac{4954744511799653}{535352159503316503} a^{11} + \frac{29274696}{391053440104687} a^{10} - \frac{12608434262559}{14468977283873419} a^{9} - \frac{1554540570}{391053440104687} a^{8} + \frac{189126513938385}{391053440104687} a^{7} + \frac{45744521688}{391053440104687} a^{6} - \frac{173405024596052}{391053440104687} a^{5} - \frac{668406401523}{391053440104687} a^{4} + \frac{92685788548041}{391053440104687} a^{3} - \frac{127591729639062}{391053440104687} a^{2} + \frac{7606132265253}{391053440104687} a + \frac{128486460356144}{391053440104687}$, $\frac{1}{732897106360040292607} a^{21} - \frac{189}{535352159503316503} a^{17} + \frac{1884}{14468977283873419} a^{15} - \frac{8541}{391053440104687} a^{13} - \frac{135175540066660424}{19808029901622710611} a^{12} + \frac{791208}{391053440104687} a^{11} - \frac{1185768754576620}{535352159503316503} a^{10} - \frac{42014610}{391053440104687} a^{9} + \frac{3317554034775881}{14468977283873419} a^{8} + \frac{1236338424}{391053440104687} a^{7} + \frac{143279544239718}{391053440104687} a^{6} - \frac{18065037879}{391053440104687} a^{5} + \frac{150471187838207}{391053440104687} a^{4} + \frac{263461710465625}{14468977283873419} a^{3} + \frac{190447785247287}{391053440104687} a^{2} - \frac{17665416752790}{391053440104687} a$, $\frac{1}{732897106360040292607} a^{22} - \frac{1518}{14468977283873419} a^{16} + \frac{16974}{391053440104687} a^{14} - \frac{135175540066660424}{19808029901622710611} a^{13} - \frac{3026970}{391053440104687} a^{12} - \frac{1185768754576620}{535352159503316503} a^{11} + \frac{290985057}{391053440104687} a^{10} - \frac{75650605575354}{14468977283873419} a^{9} - \frac{15823490670}{391053440104687} a^{8} + \frac{179952979235247}{391053440104687} a^{7} + \frac{472878932715}{391053440104687} a^{6} - \frac{9745695618642}{391053440104687} a^{5} + \frac{1603673163085}{14468977283873419} a^{4} + \frac{172683783371921}{391053440104687} a^{3} + \frac{21613288842591}{391053440104687} a^{2} - \frac{115662331267187}{391053440104687} a + \frac{175947513425451}{391053440104687}$, $\frac{1}{415165909959711549908065388394161} a^{23} + \frac{3519914088}{11220700269181393240758524010653} a^{22} - \frac{23}{11220700269181393240758524010653} a^{21} - \frac{887799041}{303262169437334952452933081369} a^{20} + \frac{230}{303262169437334952452933081369} a^{19} - \frac{1794911758}{303262169437334952452933081369} a^{18} - \frac{1311}{8196274849657701417646840037} a^{17} - \frac{191458807561251}{8196274849657701417646840037} a^{16} + \frac{4692}{221520941882640578855320001} a^{15} - \frac{559949596484582862479959}{56957869386707579902327533049} a^{14} + \frac{3654007094744844309812856917}{303262169437334952452933081369} a^{13} + \frac{1544341985273328217276704799}{303262169437334952452933081369} a^{12} - \frac{32856017816169434282142880}{8196274849657701417646840037} a^{11} - \frac{76086650839666617948047517}{8196274849657701417646840037} a^{10} + \frac{83873104484241225019338082}{8196274849657701417646840037} a^{9} - \frac{94563246696115637479933257}{221520941882640578855320001} a^{8} + \frac{14939326987618617302209883}{221520941882640578855320001} a^{7} + \frac{672201092344913220668875}{5987052483314610239332973} a^{6} + \frac{928388484167989561125736477}{8196274849657701417646840037} a^{5} - \frac{99756099442545423114834332}{221520941882640578855320001} a^{4} + \frac{84538929370947382189042401}{221520941882640578855320001} a^{3} - \frac{2615052156703781309299381}{5987052483314610239332973} a^{2} + \frac{1272494277997937912765898}{5987052483314610239332973} a + \frac{1510095700513492144215768}{5987052483314610239332973}$, $\frac{1}{415165909959711549908065388394161} a^{24} - \frac{24}{11220700269181393240758524010653} a^{22} - \frac{7553738355}{11220700269181393240758524010653} a^{21} + \frac{252}{303262169437334952452933081369} a^{20} + \frac{5527883665}{303262169437334952452933081369} a^{19} - \frac{1520}{8196274849657701417646840037} a^{18} + \frac{1328154643125}{8196274849657701417646840037} a^{17} + \frac{5814}{221520941882640578855320001} a^{16} - \frac{110310180019034770815706166}{11220700269181393240758524010653} a^{15} - \frac{14688}{5987052483314610239332973} a^{14} + \frac{1654652693154773221526935860}{303262169437334952452933081369} a^{13} - \frac{1905046326432795338440172010}{303262169437334952452933081369} a^{12} + \frac{40526236659042754212238110}{8196274849657701417646840037} a^{11} - \frac{83610398681880504817104669}{8196274849657701417646840037} a^{10} - \frac{1095604316965409666439656}{221520941882640578855320001} a^{9} - \frac{46805585567916007079945658}{221520941882640578855320001} a^{8} - \frac{928570980521706153747257}{5987052483314610239332973} a^{7} + \frac{2946383266391197912174530647}{8196274849657701417646840037} a^{6} + \frac{600951023751129710165614}{5987052483314610239332973} a^{5} - \frac{84552386599298365523974559}{221520941882640578855320001} a^{4} + \frac{29912524563055652056379996}{221520941882640578855320001} a^{3} - \frac{1743737794627376903739387}{5987052483314610239332973} a^{2} + \frac{1382761870632338310334651}{5987052483314610239332973} a + \frac{2852663029323796663835065}{5987052483314610239332973}$, $\frac{1}{15361138668509327346598419370583957} a^{25} - \frac{3862520874}{11220700269181393240758524010653} a^{22} - \frac{300}{11220700269181393240758524010653} a^{21} - \frac{3364999758}{303262169437334952452933081369} a^{20} + \frac{4000}{303262169437334952452933081369} a^{19} - \frac{968462103}{303262169437334952452933081369} a^{18} - \frac{25650}{8196274849657701417646840037} a^{17} + \frac{131030087427368844461343426507}{415165909959711549908065388394161} a^{16} + \frac{97920}{221520941882640578855320001} a^{15} - \frac{992791748786534525109506022}{11220700269181393240758524010653} a^{14} - \frac{2784120135926149304368162080}{303262169437334952452933081369} a^{13} + \frac{2547480213790492915923909327}{303262169437334952452933081369} a^{12} - \frac{14097881515530102546882659}{8196274849657701417646840037} a^{11} - \frac{67016274979804633645341497}{8196274849657701417646840037} a^{10} - \frac{32807686202333262099005021}{8196274849657701417646840037} a^{9} - \frac{73147536545206204165987278}{221520941882640578855320001} a^{8} - \frac{12689577282340056673821721446}{303262169437334952452933081369} a^{7} + \frac{1545013334530492691869987}{5987052483314610239332973} a^{6} + \frac{2673540253780075336337410512}{8196274849657701417646840037} a^{5} - \frac{52933637723037516496696403}{221520941882640578855320001} a^{4} + \frac{2854150622077862276970109}{5987052483314610239332973} a^{3} + \frac{731282157006255896900706}{5987052483314610239332973} a^{2} + \frac{2096723790861165041269569}{5987052483314610239332973} a + \frac{2114928831592914987747682}{5987052483314610239332973}$, $\frac{1}{15361138668509327346598419370583957} a^{26} - \frac{325}{11220700269181393240758524010653} a^{22} + \frac{2633611097}{11220700269181393240758524010653} a^{21} + \frac{4550}{303262169437334952452933081369} a^{20} - \frac{1619322676}{303262169437334952452933081369} a^{19} - \frac{30875}{8196274849657701417646840037} a^{18} + \frac{131030087415163506307649937066}{415165909959711549908065388394161} a^{17} + \frac{125970}{221520941882640578855320001} a^{16} - \frac{992791613783568281209855365}{11220700269181393240758524010653} a^{15} - \frac{331500}{5987052483314610239332973} a^{14} - \frac{2891040799674774484730953290}{303262169437334952452933081369} a^{13} - \frac{2668827845839565821990611847}{303262169437334952452933081369} a^{12} - \frac{31430297847893553153529285}{8196274849657701417646840037} a^{11} - \frac{87869116741318709600686051}{8196274849657701417646840037} a^{10} + \frac{384704812670838785667091}{221520941882640578855320001} a^{9} + \frac{60955431144515786501345195997}{303262169437334952452933081369} a^{8} - \frac{2509172287993743499010717}{5987052483314610239332973} a^{7} + \frac{3291664558559516460870867425}{8196274849657701417646840037} a^{6} - \frac{2809533706353372657447948}{5987052483314610239332973} a^{5} - \frac{82656911967988393532067266}{221520941882640578855320001} a^{4} + \frac{32813107079536243854056624}{221520941882640578855320001} a^{3} - \frac{2486458703768800841528095}{5987052483314610239332973} a^{2} + \frac{251417882416674178317709}{5987052483314610239332973} a - \frac{528054066680406276119051}{5987052483314610239332973}$

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:  $26$
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

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.2

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ R $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
3Data not computed
$37$37.9.8.2$x^{9} + 74$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.2$x^{9} + 74$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.2$x^{9} + 74$$9$$1$$8$$C_9$$[\ ]_{9}$