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Properties

Label 27.27.7186093556...4649.1
Degree $27$
Signature $[27, 0]$
Discriminant $757^{26}$
Root discriminant $592.19$
Ramified prime $757$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-17999855161, -23951734380, 495707735418, 189048734793, -4239629686794, 6524981901, 13574520952295, 395018931867, -17383913593462, -4155692643926, 8291321681691, 2852693916452, -1710585763116, -690978198420, 178480393873, 81671565089, -9934444834, -5296136455, 291920915, 197554784, -4222885, -4319419, 20889, 54397, 103, -364, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 364*x^25 + 103*x^24 + 54397*x^23 + 20889*x^22 - 4319419*x^21 - 4222885*x^20 + 197554784*x^19 + 291920915*x^18 - 5296136455*x^17 - 9934444834*x^16 + 81671565089*x^15 + 178480393873*x^14 - 690978198420*x^13 - 1710585763116*x^12 + 2852693916452*x^11 + 8291321681691*x^10 - 4155692643926*x^9 - 17383913593462*x^8 + 395018931867*x^7 + 13574520952295*x^6 + 6524981901*x^5 - 4239629686794*x^4 + 189048734793*x^3 + 495707735418*x^2 - 23951734380*x - 17999855161)
 
gp: K = bnfinit(x^27 - x^26 - 364*x^25 + 103*x^24 + 54397*x^23 + 20889*x^22 - 4319419*x^21 - 4222885*x^20 + 197554784*x^19 + 291920915*x^18 - 5296136455*x^17 - 9934444834*x^16 + 81671565089*x^15 + 178480393873*x^14 - 690978198420*x^13 - 1710585763116*x^12 + 2852693916452*x^11 + 8291321681691*x^10 - 4155692643926*x^9 - 17383913593462*x^8 + 395018931867*x^7 + 13574520952295*x^6 + 6524981901*x^5 - 4239629686794*x^4 + 189048734793*x^3 + 495707735418*x^2 - 23951734380*x - 17999855161, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 364 x^{25} + 103 x^{24} + 54397 x^{23} + 20889 x^{22} - 4319419 x^{21} - 4222885 x^{20} + 197554784 x^{19} + 291920915 x^{18} - 5296136455 x^{17} - 9934444834 x^{16} + 81671565089 x^{15} + 178480393873 x^{14} - 690978198420 x^{13} - 1710585763116 x^{12} + 2852693916452 x^{11} + 8291321681691 x^{10} - 4155692643926 x^{9} - 17383913593462 x^{8} + 395018931867 x^{7} + 13574520952295 x^{6} + 6524981901 x^{5} - 4239629686794 x^{4} + 189048734793 x^{3} + 495707735418 x^{2} - 23951734380 x - 17999855161 \)

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:  \(718609355602660210319584878812136494786740645292210033727402940660065354649=757^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $592.19$
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:  $757$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(757\)
Dirichlet character group:    $\lbrace$$\chi_{757}(1,·)$, $\chi_{757}(3,·)$, $\chi_{757}(100,·)$, $\chi_{757}(9,·)$, $\chi_{757}(10,·)$, $\chi_{757}(76,·)$, $\chi_{757}(270,·)$, $\chi_{757}(143,·)$, $\chi_{757}(81,·)$, $\chi_{757}(530,·)$, $\chi_{757}(729,·)$, $\chi_{757}(538,·)$, $\chi_{757}(27,·)$, $\chi_{757}(90,·)$, $\chi_{757}(30,·)$, $\chi_{757}(159,·)$, $\chi_{757}(673,·)$, $\chi_{757}(674,·)$, $\chi_{757}(228,·)$, $\chi_{757}(684,·)$, $\chi_{757}(429,·)$, $\chi_{757}(477,·)$, $\chi_{757}(243,·)$, $\chi_{757}(53,·)$, $\chi_{757}(300,·)$, $\chi_{757}(505,·)$, $\chi_{757}(508,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{47} a^{21} - \frac{10}{47} a^{19} + \frac{5}{47} a^{18} - \frac{13}{47} a^{17} + \frac{17}{47} a^{16} - \frac{21}{47} a^{15} - \frac{9}{47} a^{13} - \frac{1}{47} a^{12} + \frac{2}{47} a^{11} - \frac{12}{47} a^{10} - \frac{11}{47} a^{9} - \frac{19}{47} a^{8} - \frac{21}{47} a^{7} + \frac{3}{47} a^{6} + \frac{19}{47} a^{5} - \frac{1}{47} a^{4} - \frac{15}{47} a^{3} - \frac{19}{47} a^{2} + \frac{4}{47} a + \frac{7}{47}$, $\frac{1}{2773} a^{22} + \frac{648}{2773} a^{20} + \frac{522}{2773} a^{19} - \frac{1329}{2773} a^{18} - \frac{547}{2773} a^{17} - \frac{21}{2773} a^{16} + \frac{18}{59} a^{15} - \frac{150}{2773} a^{14} - \frac{894}{2773} a^{13} + \frac{49}{2773} a^{12} - \frac{1234}{2773} a^{11} + \frac{647}{2773} a^{10} - \frac{489}{2773} a^{9} - \frac{867}{2773} a^{8} + \frac{97}{2773} a^{7} + \frac{348}{2773} a^{6} + \frac{281}{2773} a^{5} - \frac{250}{2773} a^{4} - \frac{19}{2773} a^{3} + \frac{286}{2773} a^{2} + \frac{947}{2773} a + \frac{24}{59}$, $\frac{1}{2773} a^{23} - \frac{1}{2773} a^{21} + \frac{522}{2773} a^{20} - \frac{385}{2773} a^{19} - \frac{1019}{2773} a^{18} + \frac{97}{2773} a^{17} + \frac{905}{2773} a^{16} - \frac{386}{2773} a^{15} - \frac{894}{2773} a^{14} + \frac{344}{2773} a^{13} - \frac{585}{2773} a^{12} - \frac{651}{2773} a^{11} - \frac{1020}{2773} a^{10} + \frac{726}{2773} a^{9} + \frac{1336}{2773} a^{8} + \frac{112}{2773} a^{7} + \frac{1107}{2773} a^{6} + \frac{1284}{2773} a^{5} + \frac{630}{2773} a^{4} - \frac{1071}{2773} a^{3} - \frac{587}{2773} a^{2} + \frac{1305}{2773} a + \frac{17}{47}$, $\frac{1}{2773} a^{24} - \frac{9}{2773} a^{21} + \frac{263}{2773} a^{20} - \frac{733}{2773} a^{19} - \frac{1114}{2773} a^{18} - \frac{1058}{2773} a^{17} - \frac{1115}{2773} a^{16} + \frac{11}{2773} a^{15} + \frac{194}{2773} a^{14} + \frac{527}{2773} a^{13} - \frac{71}{2773} a^{12} - \frac{543}{2773} a^{11} - \frac{574}{2773} a^{10} + \frac{1142}{2773} a^{9} + \frac{1015}{2773} a^{8} + \frac{1263}{2773} a^{7} + \frac{39}{2773} a^{6} - \frac{859}{2773} a^{5} - \frac{790}{2773} a^{4} - \frac{960}{2773} a^{3} + \frac{588}{2773} a^{2} - \frac{174}{2773} a + \frac{184}{2773}$, $\frac{1}{63675686339} a^{25} + \frac{1511490}{63675686339} a^{24} + \frac{297803}{63675686339} a^{23} - \frac{1021810}{63675686339} a^{22} + \frac{195018935}{63675686339} a^{21} - \frac{5181995755}{63675686339} a^{20} + \frac{140784292}{63675686339} a^{19} + \frac{6330682698}{63675686339} a^{18} + \frac{22728943405}{63675686339} a^{17} + \frac{7644259072}{63675686339} a^{16} + \frac{14514279484}{63675686339} a^{15} + \frac{19339539234}{63675686339} a^{14} + \frac{14256144793}{63675686339} a^{13} - \frac{28598606717}{63675686339} a^{12} - \frac{28888856804}{63675686339} a^{11} + \frac{17392082459}{63675686339} a^{10} + \frac{20945631628}{63675686339} a^{9} - \frac{3662230446}{63675686339} a^{8} - \frac{2246745377}{63675686339} a^{7} + \frac{15807894001}{63675686339} a^{6} - \frac{21955434054}{63675686339} a^{5} + \frac{4870633455}{63675686339} a^{4} + \frac{23556298013}{63675686339} a^{3} - \frac{1365882180}{63675686339} a^{2} + \frac{861749936}{63675686339} a + \frac{17564770}{501383357}$, $\frac{1}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{26} + \frac{566707414140845746918900089583261455649654472279038408205613863954629337995649158464575870063110065566580881498687522503034297720710944300717640}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{25} - \frac{92842697940396484773776115148132212164586747168470369766634939881700838535576957110237373787550952467007666891460183900191108452710272781065099133876019}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{24} - \frac{102933749809311762512645514270726194446863498668571065506833049266386165818459238026999315744224154716619316744301352109745262516906725611641273295561366}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{23} + \frac{43513339653999694110953854881883098630800273834850728325164634098292204719645559514157065910777207464863236434215264009276250943513197183710514569641618}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{22} + \frac{4670054618288583955608038920489320333727671164427885616784156996056464973808319280449341539450703818762873331999788685051287238768882193512311669270925929}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{21} - \frac{279992771060860922403444429786488884714929440611628719692247758216561811427055278942135071026152937712467112072053433015534002032716233567817477567006490953}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{20} - \frac{101846074861704068455625823542972584555552567541772447729991676088179369464847288192027029532885607986648455796606804522421624031364669782143496239047479653}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{19} - \frac{90663975612353823284481431027955828151547106324237373423509168572823071111967129313238497729811952589106412517899724371114936097959384614265865730590281354}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{18} + \frac{67980232925865647625824501270548545770373549905563738620773856009216714662686169224658261075941982162280240395082878050605114860957734891235198120381731840}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{17} + \frac{195212141638358972643418086587766745122451567960850457794911537982576168698584795859626411690819046753531540390690451611259814290536205007995222214056648862}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{16} + \frac{708220260202450679327520236315765127753847314639253970808226907509428456917539812060818544080617464211068270975657994669160190788164999108115970507633802}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{15} + \frac{360142295386689819901751887144346431979194909208887329072042453308645809556588529629310903528592502667417972562285405549630246315029798754337349269366797314}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{14} - \frac{42665910187053784738435946512096784215219478876953342635745923534505727479468877866245549659232777219376648852078271872341521515611150715115654950192905330}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{13} - \frac{117240364404830999271903339681521453843689089985539672705805462786899820902106476506291265810898576229462760961679930093609950303091942945362134042082623025}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{12} - \frac{188573026451661305988889822252949622684697583735632296975436814636739630234812332556829417031153582323952602563572024424607610589660676512551059443506394877}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{11} + \frac{385604379940520151160927593253987333120924708622797916169289895608585804567531091960967541743904377624768657737271873938310464520546333939831522775756248858}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{10} + \frac{247571712094231322751075764187638561109046930806220427602879761334805749163795004530938027672659129002874107540423358458646809833507615793834500734451812395}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{9} - \frac{173777090546330349251945884882984452558693451201374089269365365082589575345847568672332161841966744003817154730309115415344147774043185523490460488403514836}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{8} + \frac{82357729376380006595829495674707108191774206313424308271441529247769663328614534196361060133155547817144162721300397927507903475523921066023926632561654378}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{7} - \frac{58066736703337449059769235959308814962297794041995400368023184548929097301656060890817075555872343956724087404014085414209786909987884384536487761542324187}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{6} - \frac{5708553587913938081355496071570181425216598863884503413531856801758632277594386673584520958469309985224385154520823033636718180907160760105801577247726175}{13464886283718386727537826352780590873785902476273740838422418726956225772371081249940642080611211265104842872675471128570729824730075691809416652226899701} a^{5} - \frac{361740746457591854078520690973218950829776040297665933290715779262165620519415536402511966217073502556594164950984334160254179888359575015437740677533740764}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{4} - \frac{247501328252570450785301843968073209013271564964838475740055350503297621723577630245545578195580243612702815546062121770692712108741074795396943610409527373}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{3} - \frac{41407386772515192093536246917606295584360990430901259314037721039209074466141041461375344086153534046509150626586285190218324802152681398517600024408046590}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a^{2} - \frac{53683807379950326632234582436261447126857650456485794196175963621365782052875916585805622389688723980426489322766390054243477013293775462452461823381291131}{794428290739384816924731754814054861553368246100150709466922704890417320569893793746497882756061464641185729487852796585673059659074465816755582481387082359} a + \frac{1941810431676899017133359674971938335799925570915456958883358954078153340459614928558010063542062777217224375935839779108709859748302858561645217535012741}{6255340871963659975785289407984683949239120048032682751708052794412734807636959005877936084693397359379415192817738555792701257158066659974453405365252617}$

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:  $26$
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:  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:  \( 119360706557237020000000000000 \) (assuming GRH)
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

3.3.573049.1, 9.9.107836810944509231272801.1

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 $27$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ $27$ $27$ $27$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{27}$ $27$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{27}$

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
757Data not computed