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Properties

Label 27.27.3063651216...3609.6
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![10217831259149790606431, -6676075445617956340881, -7139196829637791316082, 5400420250601345750559, 1286341771105908345240, -1311649284786038411517, -89232717455094542868, 148529522515509508266, 3100750799597879868, -9478115594426938548, -60525165757916574, 377261368227838926, 693982244562516, -9934810805722716, -4637540986830, 179005600103112, 16711859412, -2241127346283, -25092882, 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 - 25092882*x^18 - 2241127346283*x^17 + 16711859412*x^16 + 179005600103112*x^15 - 4637540986830*x^14 - 9934810805722716*x^13 + 693982244562516*x^12 + 377261368227838926*x^11 - 60525165757916574*x^10 - 9478115594426938548*x^9 + 3100750799597879868*x^8 + 148529522515509508266*x^7 - 89232717455094542868*x^6 - 1311649284786038411517*x^5 + 1286341771105908345240*x^4 + 5400420250601345750559*x^3 - 7139196829637791316082*x^2 - 6676075445617956340881*x + 10217831259149790606431)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 25092882*x^18 - 2241127346283*x^17 + 16711859412*x^16 + 179005600103112*x^15 - 4637540986830*x^14 - 9934810805722716*x^13 + 693982244562516*x^12 + 377261368227838926*x^11 - 60525165757916574*x^10 - 9478115594426938548*x^9 + 3100750799597879868*x^8 + 148529522515509508266*x^7 - 89232717455094542868*x^6 - 1311649284786038411517*x^5 + 1286341771105908345240*x^4 + 5400420250601345750559*x^3 - 7139196829637791316082*x^2 - 6676075445617956340881*x + 10217831259149790606431, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 25092882 x^{18} - 2241127346283 x^{17} + 16711859412 x^{16} + 179005600103112 x^{15} - 4637540986830 x^{14} - 9934810805722716 x^{13} + 693982244562516 x^{12} + 377261368227838926 x^{11} - 60525165757916574 x^{10} - 9478115594426938548 x^{9} + 3100750799597879868 x^{8} + 148529522515509508266 x^{7} - 89232717455094542868 x^{6} - 1311649284786038411517 x^{5} + 1286341771105908345240 x^{4} + 5400420250601345750559 x^{3} - 7139196829637791316082 x^{2} - 6676075445617956340881 x + 10217831259149790606431 \)

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$$\chi_{2997}(1,·)$, $\chi_{2997}(1156,·)$, $\chi_{2997}(70,·)$, $\chi_{2997}(904,·)$, $\chi_{2997}(1033,·)$, $\chi_{2997}(2764,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(1765,·)$, $\chi_{2997}(2068,·)$, $\chi_{2997}(2902,·)$, $\chi_{2997}(343,·)$, $\chi_{2997}(1903,·)$, $\chi_{2997}(157,·)$, $\chi_{2997}(1381,·)$, $\chi_{2997}(673,·)$, $\chi_{2997}(34,·)$, $\chi_{2997}(2341,·)$, $\chi_{2997}(766,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(2155,·)$, $\chi_{2997}(1069,·)$, $\chi_{2997}(2671,·)$, $\chi_{2997}(2032,·)$, $\chi_{2997}(1672,·)$, $\chi_{2997}(382,·)$, $\chi_{2997}(2380,·)$, $\chi_{2997}(1342,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11951} a^{9} - \frac{9}{323} a^{7} + \frac{30}{323} a^{5} - \frac{49}{323} a^{3} + \frac{124}{323} a + \frac{2}{17}$, $\frac{1}{11951} a^{10} - \frac{9}{323} a^{8} + \frac{30}{323} a^{6} - \frac{49}{323} a^{4} + \frac{124}{323} a^{2} + \frac{2}{17} a$, $\frac{1}{11951} a^{11} - \frac{60}{323} a^{7} - \frac{72}{323} a^{5} - \frac{43}{323} a^{3} + \frac{2}{17} a^{2} - \frac{52}{323} a + \frac{3}{17}$, $\frac{1}{11951} a^{12} - \frac{60}{323} a^{8} - \frac{72}{323} a^{6} - \frac{43}{323} a^{4} + \frac{2}{17} a^{3} - \frac{52}{323} a^{2} + \frac{3}{17} a$, $\frac{1}{11951} a^{13} - \frac{26}{323} a^{7} + \frac{1}{17} a^{5} + \frac{2}{17} a^{4} + \frac{1}{17} a^{3} + \frac{3}{17} a^{2} + \frac{84}{323} a + \frac{3}{17}$, $\frac{1}{442187} a^{14} - \frac{88}{323} a^{8} + \frac{6}{17} a^{6} - \frac{100}{629} a^{5} + \frac{6}{17} a^{4} + \frac{1}{17} a^{3} + \frac{11}{323} a^{2} + \frac{1}{17} a$, $\frac{1}{442187} a^{15} - \frac{120}{323} a^{7} - \frac{100}{629} a^{6} - \frac{75}{323} a^{5} + \frac{1}{17} a^{4} + \frac{29}{323} a^{3} + \frac{1}{17} a^{2} - \frac{6}{323} a + \frac{1}{17}$, $\frac{1}{442187} a^{16} - \frac{120}{323} a^{8} - \frac{100}{629} a^{7} - \frac{75}{323} a^{6} + \frac{1}{17} a^{5} + \frac{29}{323} a^{4} + \frac{1}{17} a^{3} - \frac{6}{323} a^{2} + \frac{1}{17} a$, $\frac{1}{442187} a^{17} - \frac{100}{629} a^{8} + \frac{1}{19} a^{7} + \frac{1}{17} a^{6} + \frac{9}{19} a^{5} + \frac{1}{17} a^{4} + \frac{8}{19} a^{3} + \frac{1}{17} a^{2} - \frac{155}{323} a + \frac{6}{17}$, $\frac{1}{3518096053119173} a^{18} - \frac{18}{95083677111329} a^{16} + \frac{135}{2569829111117} a^{14} - \frac{546}{69454840841} a^{12} + \frac{47619}{69454840841} a^{10} - \frac{29095938}{5004404058491} a^{9} - \frac{2439558}{69454840841} a^{8} + \frac{261863442}{135254163743} a^{7} + \frac{70205058}{69454840841} a^{6} - \frac{785590326}{3655517939} a^{5} - \frac{1012046940}{69454840841} a^{4} - \frac{603170271}{3655517939} a^{3} + \frac{5616860517}{69454840841} a^{2} - \frac{250294076}{3655517939} a + \frac{95409226}{192395681}$, $\frac{1}{3518096053119173} a^{19} - \frac{18}{95083677111329} a^{17} + \frac{135}{2569829111117} a^{15} - \frac{546}{69454840841} a^{13} + \frac{47619}{69454840841} a^{11} - \frac{29095938}{5004404058491} a^{10} - \frac{2439558}{69454840841} a^{9} + \frac{261863442}{135254163743} a^{8} + \frac{70205058}{69454840841} a^{7} - \frac{785590326}{3655517939} a^{6} - \frac{1012046940}{69454840841} a^{5} - \frac{603170271}{3655517939} a^{4} + \frac{5616860517}{69454840841} a^{3} - \frac{250294076}{3655517939} a^{2} + \frac{95409226}{192395681} a$, $\frac{1}{3518096053119173} a^{20} - \frac{189}{2569829111117} a^{16} + \frac{1884}{69454840841} a^{14} - \frac{316017}{69454840841} a^{12} - \frac{29095938}{5004404058491} a^{11} + \frac{8011417}{2569829111117} a^{10} - \frac{1563403}{135254163743} a^{9} + \frac{8121830445}{69454840841} a^{8} + \frac{1585251279}{3655517939} a^{7} + \frac{13489951638}{69454840841} a^{6} - \frac{569296818}{3655517939} a^{5} + \frac{551154733}{4085578873} a^{4} - \frac{1642869366}{3655517939} a^{3} + \frac{30301221636}{69454840841} a^{2} - \frac{1314790707}{3655517939} a - \frac{4617179}{192395681}$, $\frac{1}{130169553965409401} a^{21} - \frac{189}{95083677111329} a^{17} + \frac{1884}{2569829111117} a^{15} - \frac{8541}{69454840841} a^{13} + \frac{4577083013}{185162950164167} a^{12} + \frac{791208}{69454840841} a^{11} + \frac{156880099}{5004404058491} a^{10} - \frac{49327301}{2569829111117} a^{9} + \frac{1154417047}{7118640197} a^{8} - \frac{12310580997}{69454840841} a^{7} - \frac{524363210}{3655517939} a^{6} + \frac{1593609423}{4085578873} a^{5} - \frac{695304910}{3655517939} a^{4} - \frac{391803585085}{2569829111117} a^{3} + \frac{576216089}{3655517939} a^{2} + \frac{1382633219}{3655517939} a - \frac{3}{17}$, $\frac{1}{130169553965409401} a^{22} - \frac{1518}{2569829111117} a^{16} + \frac{16974}{69454840841} a^{14} + \frac{4577083013}{185162950164167} a^{13} + \frac{103032577}{2569829111117} a^{12} + \frac{156880099}{5004404058491} a^{11} + \frac{785461}{135254163743} a^{10} + \frac{1936975}{135254163743} a^{9} - \frac{1416449381}{69454840841} a^{8} - \frac{534449654}{3655517939} a^{7} - \frac{4058643091}{69454840841} a^{6} + \frac{424290374}{3655517939} a^{5} + \frac{1193445574623}{2569829111117} a^{4} - \frac{3576823}{11317393} a^{3} - \frac{9474716002}{69454840841} a^{2} + \frac{211288807}{3655517939} a + \frac{2447889}{192395681}$, $\frac{1}{2909505169879829655598287199} a^{23} + \frac{3977510}{2125277698962622100510071} a^{22} - \frac{23}{78635274861617017718872627} a^{21} - \frac{217199005}{2125277698962622100510071} a^{20} + \frac{230}{2125277698962622100510071} a^{19} - \frac{73051358}{2125277698962622100510071} a^{18} - \frac{69}{3023154621568452490057} a^{17} - \frac{6745643020011}{57439937809800597311083} a^{16} + \frac{276}{91319455977425432927} a^{15} - \frac{36937608673991546588}{78635274861617017718872627} a^{14} + \frac{123929102331866923}{57439937809800597311083} a^{13} + \frac{19343001495197053258}{2125277698962622100510071} a^{12} + \frac{1202036717651679856}{57439937809800597311083} a^{11} - \frac{431448793142838162}{57439937809800597311083} a^{10} - \frac{1555003278524462014}{57439937809800597311083} a^{9} + \frac{142297081099567922178}{1552430751616232359759} a^{8} - \frac{579211954628902592210}{1552430751616232359759} a^{7} - \frac{14588253711322565014}{41957587881519793507} a^{6} - \frac{21759945630957178092502}{57439937809800597311083} a^{5} - \frac{4640539917597125313}{41957587881519793507} a^{4} + \frac{238347844359855272966}{1552430751616232359759} a^{3} + \frac{8380434722201491776}{41957587881519793507} a^{2} + \frac{416305804693244793}{2208294099027357553} a + \frac{39270287869250679}{116226005211966187}$, $\frac{1}{2909505169879829655598287199} a^{24} - \frac{24}{78635274861617017718872627} a^{22} + \frac{8321747}{78635274861617017718872627} a^{21} + \frac{252}{2125277698962622100510071} a^{20} - \frac{174756687}{2125277698962622100510071} a^{19} - \frac{80}{3023154621568452490057} a^{18} + \frac{1572810183}{57439937809800597311083} a^{17} + \frac{18}{4806287156706601733} a^{16} - \frac{40852303418051817899}{78635274861617017718872627} a^{15} - \frac{864}{2468093404795281971} a^{14} + \frac{79281896823092506182}{2125277698962622100510071} a^{13} + \frac{3088670518020564124}{125016335233095417677063} a^{12} + \frac{7161401648058048}{3023154621568452490057} a^{11} - \frac{465167642416421825}{57439937809800597311083} a^{10} + \frac{34459232650847806}{1552430751616232359759} a^{9} + \frac{273648478549421234360}{1552430751616232359759} a^{8} - \frac{578330929828068966}{2468093404795281971} a^{7} - \frac{4015152712736981107415}{57439937809800597311083} a^{6} + \frac{6830257600221166744}{41957587881519793507} a^{5} + \frac{697793825517881161687}{1552430751616232359759} a^{4} - \frac{199306257435211168834}{1552430751616232359759} a^{3} - \frac{14802123125402384344}{41957587881519793507} a^{2} + \frac{754415221258639731}{2208294099027357553} a + \frac{5895028848809933}{116226005211966187}$, $\frac{1}{107651691285553697257136626363} a^{25} + \frac{54603788}{78635274861617017718872627} a^{22} - \frac{300}{78635274861617017718872627} a^{21} + \frac{154082188}{2125277698962622100510071} a^{20} + \frac{4000}{2125277698962622100510071} a^{19} - \frac{180422409}{2125277698962622100510071} a^{18} - \frac{1350}{3023154621568452490057} a^{17} - \frac{3064216284549258772605}{2909505169879829655598287199} a^{16} + \frac{5760}{91319455977425432927} a^{15} - \frac{9721999160251539731}{78635274861617017718872627} a^{14} - \frac{1197060535571160822}{111856720998032742132109} a^{13} + \frac{80107471065988607066}{2125277698962622100510071} a^{12} + \frac{1601605982727082395}{57439937809800597311083} a^{11} - \frac{905866202279254405}{57439937809800597311083} a^{10} - \frac{1765141920087297741}{57439937809800597311083} a^{9} - \frac{110357489638611051530}{1552430751616232359759} a^{8} - \frac{919974624929515729339646}{2125277698962622100510071} a^{7} + \frac{10264757780011640320}{41957587881519793507} a^{6} - \frac{460624936025303321863}{3378819871164741018299} a^{5} - \frac{532645595327339298881}{1552430751616232359759} a^{4} - \frac{19317471966348663963}{41957587881519793507} a^{3} - \frac{19442894540941984997}{41957587881519793507} a^{2} + \frac{220491376654460938}{2208294099027357553} a - \frac{864231728700387}{116226005211966187}$, $\frac{1}{107651691285553697257136626363} a^{26} - \frac{325}{78635274861617017718872627} a^{22} + \frac{216365422}{78635274861617017718872627} a^{21} + \frac{4550}{2125277698962622100510071} a^{20} + \frac{297438501}{2125277698962622100510071} a^{19} - \frac{1625}{3023154621568452490057} a^{18} - \frac{3064008866666756833443}{2909505169879829655598287199} a^{17} + \frac{390}{4806287156706601733} a^{16} - \frac{11979191626762160681}{78635274861617017718872627} a^{15} - \frac{19500}{2468093404795281971} a^{14} + \frac{45141022175582044072}{2125277698962622100510071} a^{13} - \frac{3380507620546583496}{125016335233095417677063} a^{12} - \frac{1632880775392364873}{57439937809800597311083} a^{11} - \frac{708934024533442464}{57439937809800597311083} a^{10} + \frac{13711228013668215}{1552430751616232359759} a^{9} - \frac{40841486294403128512136}{111856720998032742132109} a^{8} + \frac{6038445562058962962}{41957587881519793507} a^{7} - \frac{4775736252885988666719}{57439937809800597311083} a^{6} + \frac{679836482481377296}{41957587881519793507} a^{5} + \frac{584108099933572717149}{1552430751616232359759} a^{4} + \frac{292992366670442258994}{1552430751616232359759} a^{3} - \frac{18319071073581008060}{41957587881519793507} a^{2} - \frac{27471203523401055}{116226005211966187} a + \frac{24312892172284127}{116226005211966187}$

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

\(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.3

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{27}$ $27$ $27$ $27$ 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$37.9.8.3$x^{9} - 148$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.3$x^{9} - 148$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.3$x^{9} - 148$$9$$1$$8$$C_9$$[\ ]_{9}$