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Properties

Label 34.0.55821122927...5648.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,2^{34}\cdot 137^{33}$
Root discriminant $237.09$
Ramified primes $2, 137$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![315276593, 0, 5059513709, 0, 33472345530, 0, 119691266971, 0, 255949505268, 0, 344863850882, 0, 303988539239, 0, 181494741221, 0, 75401516247, 0, 22175081533, 0, 4651829657, 0, 695129232, 0, 73241022, 0, 5332588, 0, 259341, 0, 7946, 0, 137, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 + 137*x^32 + 7946*x^30 + 259341*x^28 + 5332588*x^26 + 73241022*x^24 + 695129232*x^22 + 4651829657*x^20 + 22175081533*x^18 + 75401516247*x^16 + 181494741221*x^14 + 303988539239*x^12 + 344863850882*x^10 + 255949505268*x^8 + 119691266971*x^6 + 33472345530*x^4 + 5059513709*x^2 + 315276593)
 
gp: K = bnfinit(x^34 + 137*x^32 + 7946*x^30 + 259341*x^28 + 5332588*x^26 + 73241022*x^24 + 695129232*x^22 + 4651829657*x^20 + 22175081533*x^18 + 75401516247*x^16 + 181494741221*x^14 + 303988539239*x^12 + 344863850882*x^10 + 255949505268*x^8 + 119691266971*x^6 + 33472345530*x^4 + 5059513709*x^2 + 315276593, 1)
 

Normalized defining polynomial

\( x^{34} + 137 x^{32} + 7946 x^{30} + 259341 x^{28} + 5332588 x^{26} + 73241022 x^{24} + 695129232 x^{22} + 4651829657 x^{20} + 22175081533 x^{18} + 75401516247 x^{16} + 181494741221 x^{14} + 303988539239 x^{12} + 344863850882 x^{10} + 255949505268 x^{8} + 119691266971 x^{6} + 33472345530 x^{4} + 5059513709 x^{2} + 315276593 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-558211229270783900245717215377860069008727720516819891947659530349296970941595648=-\,2^{34}\cdot 137^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.09$
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:  $2, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(548=2^{2}\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{548}(1,·)$, $\chi_{548}(133,·)$, $\chi_{548}(193,·)$, $\chi_{548}(393,·)$, $\chi_{548}(87,·)$, $\chi_{548}(397,·)$, $\chi_{548}(15,·)$, $\chi_{548}(533,·)$, $\chi_{548}(151,·)$, $\chi_{548}(153,·)$, $\chi_{548}(155,·)$, $\chi_{548}(389,·)$, $\chi_{548}(547,·)$, $\chi_{548}(99,·)$, $\chi_{548}(415,·)$, $\chi_{548}(159,·)$, $\chi_{548}(445,·)$, $\chi_{548}(63,·)$, $\chi_{548}(449,·)$, $\chi_{548}(323,·)$, $\chi_{548}(197,·)$, $\chi_{548}(73,·)$, $\chi_{548}(333,·)$, $\chi_{548}(461,·)$, $\chi_{548}(209,·)$, $\chi_{548}(339,·)$, $\chi_{548}(215,·)$, $\chi_{548}(475,·)$, $\chi_{548}(395,·)$, $\chi_{548}(351,·)$, $\chi_{548}(225,·)$, $\chi_{548}(355,·)$, $\chi_{548}(485,·)$, $\chi_{548}(103,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{37} a^{23} + \frac{6}{37} a^{21} + \frac{11}{37} a^{19} - \frac{15}{37} a^{17} - \frac{15}{37} a^{15} - \frac{4}{37} a^{13} - \frac{1}{37} a^{11} - \frac{6}{37} a^{9} - \frac{11}{37} a^{7} + \frac{15}{37} a^{5} + \frac{15}{37} a^{3} + \frac{4}{37} a$, $\frac{1}{37} a^{24} + \frac{6}{37} a^{22} + \frac{11}{37} a^{20} - \frac{15}{37} a^{18} - \frac{15}{37} a^{16} - \frac{4}{37} a^{14} - \frac{1}{37} a^{12} - \frac{6}{37} a^{10} - \frac{11}{37} a^{8} + \frac{15}{37} a^{6} + \frac{15}{37} a^{4} + \frac{4}{37} a^{2}$, $\frac{1}{1517} a^{25} + \frac{19}{1517} a^{23} - \frac{59}{1517} a^{21} - \frac{501}{1517} a^{19} + \frac{604}{1517} a^{17} - \frac{236}{1517} a^{15} + \frac{539}{1517} a^{13} + \frac{721}{1517} a^{11} - \frac{311}{1517} a^{9} + \frac{316}{1517} a^{7} - \frac{160}{1517} a^{5} + \frac{384}{1517} a^{3} + \frac{385}{1517} a$, $\frac{1}{1517} a^{26} + \frac{19}{1517} a^{24} - \frac{59}{1517} a^{22} - \frac{501}{1517} a^{20} + \frac{604}{1517} a^{18} - \frac{236}{1517} a^{16} + \frac{539}{1517} a^{14} + \frac{721}{1517} a^{12} - \frac{311}{1517} a^{10} + \frac{316}{1517} a^{8} - \frac{160}{1517} a^{6} + \frac{384}{1517} a^{4} + \frac{385}{1517} a^{2}$, $\frac{1}{1517} a^{27} - \frac{10}{1517} a^{23} + \frac{46}{1517} a^{21} - \frac{537}{1517} a^{19} + \frac{342}{1517} a^{17} + \frac{390}{1517} a^{15} - \frac{541}{1517} a^{13} + \frac{750}{1517} a^{11} + \frac{731}{1517} a^{9} - \frac{55}{1517} a^{7} + \frac{472}{1517} a^{5} + \frac{756}{1517} a^{3} + \frac{393}{1517} a$, $\frac{1}{1517} a^{28} - \frac{10}{1517} a^{24} + \frac{46}{1517} a^{22} - \frac{537}{1517} a^{20} + \frac{342}{1517} a^{18} + \frac{390}{1517} a^{16} - \frac{541}{1517} a^{14} + \frac{750}{1517} a^{12} + \frac{731}{1517} a^{10} - \frac{55}{1517} a^{8} + \frac{472}{1517} a^{6} + \frac{756}{1517} a^{4} + \frac{393}{1517} a^{2}$, $\frac{1}{1517} a^{29} - \frac{10}{1517} a^{23} + \frac{431}{1517} a^{21} + \frac{211}{1517} a^{19} - \frac{499}{1517} a^{17} - \frac{728}{1517} a^{15} - \frac{461}{1517} a^{13} + \frac{602}{1517} a^{11} - \frac{172}{1517} a^{9} + \frac{270}{1517} a^{7} + \frac{17}{1517} a^{5} + \frac{543}{1517} a^{3} - \frac{168}{1517} a$, $\frac{1}{1517} a^{30} - \frac{10}{1517} a^{24} + \frac{431}{1517} a^{22} + \frac{211}{1517} a^{20} - \frac{499}{1517} a^{18} - \frac{728}{1517} a^{16} - \frac{461}{1517} a^{14} + \frac{602}{1517} a^{12} - \frac{172}{1517} a^{10} + \frac{270}{1517} a^{8} + \frac{17}{1517} a^{6} + \frac{543}{1517} a^{4} - \frac{168}{1517} a^{2}$, $\frac{1}{1517} a^{31} + \frac{6}{1517} a^{23} + \frac{482}{1517} a^{21} - \frac{138}{1517} a^{19} - \frac{633}{1517} a^{17} + \frac{336}{1517} a^{15} - \frac{650}{1517} a^{13} + \frac{68}{1517} a^{11} - \frac{667}{1517} a^{9} - \frac{677}{1517} a^{7} + \frac{337}{1517} a^{5} + \frac{515}{1517} a^{3} - \frac{127}{1517} a$, $\frac{1}{358832496707247857227819717711818099906948536854841271018278929} a^{32} + \frac{24599509294534413679981680788309330137924245616821358180919}{358832496707247857227819717711818099906948536854841271018278929} a^{30} - \frac{98581672951981707915112868655891787677137762689871937393983}{358832496707247857227819717711818099906948536854841271018278929} a^{28} + \frac{18451255949447852051934258922779824613603454791617668775343}{358832496707247857227819717711818099906948536854841271018278929} a^{26} - \frac{2525328279622608000878719202123196683820304131690839938209287}{358832496707247857227819717711818099906948536854841271018278929} a^{24} + \frac{156716210669026734421697001126462838057764912377588790911633695}{358832496707247857227819717711818099906948536854841271018278929} a^{22} - \frac{61143480591796699264248335433863417077655226033141661533484605}{358832496707247857227819717711818099906948536854841271018278929} a^{20} + \frac{168541940958425869608936128935763525327683741660531157013978473}{358832496707247857227819717711818099906948536854841271018278929} a^{18} - \frac{153798102541599949764184018692597345191475793385404430842508824}{358832496707247857227819717711818099906948536854841271018278929} a^{16} + \frac{59476550623678819681410464557756782841477875582282076230665787}{358832496707247857227819717711818099906948536854841271018278929} a^{14} + \frac{17606475493266717890944598443521915087137313062387495902539168}{358832496707247857227819717711818099906948536854841271018278929} a^{12} + \frac{100198070896237120761281196691652968828464719180593987930148103}{358832496707247857227819717711818099906948536854841271018278929} a^{10} - \frac{134840608639801829309515913638900486459303089862818413042435314}{358832496707247857227819717711818099906948536854841271018278929} a^{8} - \frac{61809129244084198967281051086765557507438702200937028131217433}{358832496707247857227819717711818099906948536854841271018278929} a^{6} - \frac{75285118304683276455623013297189387195094345560489601463693451}{358832496707247857227819717711818099906948536854841271018278929} a^{4} - \frac{106807460762912016414215703061116160484493196059164737630996501}{358832496707247857227819717711818099906948536854841271018278929} a^{2} + \frac{192319835318727345565186952261586144603308732020523973973}{6392996431563859274667635584311462878493266169980603093201}$, $\frac{1}{358832496707247857227819717711818099906948536854841271018278929} a^{33} + \frac{24599509294534413679981680788309330137924245616821358180919}{358832496707247857227819717711818099906948536854841271018278929} a^{31} - \frac{98581672951981707915112868655891787677137762689871937393983}{358832496707247857227819717711818099906948536854841271018278929} a^{29} + \frac{18451255949447852051934258922779824613603454791617668775343}{358832496707247857227819717711818099906948536854841271018278929} a^{27} + \frac{76621268023882723911008480691568707726455199491265520723520}{358832496707247857227819717711818099906948536854841271018278929} a^{25} + \frac{2491564753980193279614960290533107576993359292976721891252771}{358832496707247857227819717711818099906948536854841271018278929} a^{23} - \frac{1298640995927412594084598729123813072079761415953235978030044}{358832496707247857227819717711818099906948536854841271018278929} a^{21} - \frac{145820872570863782504295515651584038796465635715925038887994300}{358832496707247857227819717711818099906948536854841271018278929} a^{19} + \frac{166714773554854134971277891326857846221784106046573105235123311}{358832496707247857227819717711818099906948536854841271018278929} a^{17} - \frac{11485709766680018267400290428100455109797378904502618103865313}{358832496707247857227819717711818099906948536854841271018278929} a^{15} + \frac{81709050712557534838036980447412953369789292948783003118065595}{358832496707247857227819717711818099906948536854841271018278929} a^{13} + \frac{26870401826199654880843416539600489612146956210916470451132633}{358832496707247857227819717711818099906948536854841271018278929} a^{11} - \frac{80909290743129112468419739849648985616333896452862045348191678}{358832496707247857227819717711818099906948536854841271018278929} a^{9} + \frac{130025514677852526287670689891668509087508735761670928886464974}{358832496707247857227819717711818099906948536854841271018278929} a^{7} + \frac{41802611339408806159914732429475055424509824342705144188282864}{358832496707247857227819717711818099906948536854841271018278929} a^{5} - \frac{9589164028120408424344968730491744491246097412269706392688894}{358832496707247857227819717711818099906948536854841271018278929} a^{3} - \frac{4349566219136852450186314827629295868422891892930782977146885}{9698175586682374519670803181400489186674284779860574892385917} a$

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:  $16$
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

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{-137}) \), 17.17.15400296222263289476715621650663041.1

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 $17^{2}$ $34$ $34$ $34$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ $17^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{34}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{17}$ $17^{2}$ $17^{2}$ $34$ $34$

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
2Data not computed
137Data not computed