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Properties

Label 28.0.17249157379...0000.3
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $101.97$
Ramified primes $2, 5, 29$
Class number $711168$ (GRH)
Class group $[2, 4, 4, 4, 5556]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87247081, 131163708, 12443600, -221014676, -20350779, 62481562, 133893346, -142945196, 92192786, -127633640, 167465338, -171136062, 157184593, -131361794, 96760101, -62540558, 35780211, -18188118, 8217087, -3301012, 1179882, -374022, 104543, -25422, 5336, -936, 135, -14, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 14*x^27 + 135*x^26 - 936*x^25 + 5336*x^24 - 25422*x^23 + 104543*x^22 - 374022*x^21 + 1179882*x^20 - 3301012*x^19 + 8217087*x^18 - 18188118*x^17 + 35780211*x^16 - 62540558*x^15 + 96760101*x^14 - 131361794*x^13 + 157184593*x^12 - 171136062*x^11 + 167465338*x^10 - 127633640*x^9 + 92192786*x^8 - 142945196*x^7 + 133893346*x^6 + 62481562*x^5 - 20350779*x^4 - 221014676*x^3 + 12443600*x^2 + 131163708*x + 87247081)
 
gp: K = bnfinit(x^28 - 14*x^27 + 135*x^26 - 936*x^25 + 5336*x^24 - 25422*x^23 + 104543*x^22 - 374022*x^21 + 1179882*x^20 - 3301012*x^19 + 8217087*x^18 - 18188118*x^17 + 35780211*x^16 - 62540558*x^15 + 96760101*x^14 - 131361794*x^13 + 157184593*x^12 - 171136062*x^11 + 167465338*x^10 - 127633640*x^9 + 92192786*x^8 - 142945196*x^7 + 133893346*x^6 + 62481562*x^5 - 20350779*x^4 - 221014676*x^3 + 12443600*x^2 + 131163708*x + 87247081, 1)
 

Normalized defining polynomial

\( x^{28} - 14 x^{27} + 135 x^{26} - 936 x^{25} + 5336 x^{24} - 25422 x^{23} + 104543 x^{22} - 374022 x^{21} + 1179882 x^{20} - 3301012 x^{19} + 8217087 x^{18} - 18188118 x^{17} + 35780211 x^{16} - 62540558 x^{15} + 96760101 x^{14} - 131361794 x^{13} + 157184593 x^{12} - 171136062 x^{11} + 167465338 x^{10} - 127633640 x^{9} + 92192786 x^{8} - 142945196 x^{7} + 133893346 x^{6} + 62481562 x^{5} - 20350779 x^{4} - 221014676 x^{3} + 12443600 x^{2} + 131163708 x + 87247081 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(172491573797176117219488267129250761475686400000000000000=2^{28}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.97$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(579,·)$, $\chi_{580}(51,·)$, $\chi_{580}(71,·)$, $\chi_{580}(499,·)$, $\chi_{580}(141,·)$, $\chi_{580}(399,·)$, $\chi_{580}(401,·)$, $\chi_{580}(531,·)$, $\chi_{580}(529,·)$, $\chi_{580}(151,·)$, $\chi_{580}(281,·)$, $\chi_{580}(411,·)$, $\chi_{580}(349,·)$, $\chi_{580}(231,·)$, $\chi_{580}(161,·)$, $\chi_{580}(419,·)$, $\chi_{580}(81,·)$, $\chi_{580}(169,·)$, $\chi_{580}(299,·)$, $\chi_{580}(429,·)$, $\chi_{580}(489,·)$, $\chi_{580}(49,·)$, $\chi_{580}(91,·)$, $\chi_{580}(179,·)$, $\chi_{580}(181,·)$, $\chi_{580}(439,·)$, $\chi_{580}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{7}{17} a^{13} + \frac{7}{17} a^{12} - \frac{2}{17} a^{11} + \frac{4}{17} a^{10} - \frac{7}{17} a^{9} + \frac{1}{17} a^{8} - \frac{4}{17} a^{7} - \frac{2}{17} a^{6} + \frac{3}{17} a^{5} - \frac{3}{17} a^{4} - \frac{2}{17} a^{3} - \frac{6}{17} a^{2} - \frac{2}{17}$, $\frac{1}{17} a^{15} - \frac{8}{17} a^{13} - \frac{4}{17} a^{12} + \frac{7}{17} a^{11} + \frac{4}{17} a^{10} + \frac{3}{17} a^{9} + \frac{3}{17} a^{8} + \frac{4}{17} a^{7} + \frac{6}{17} a^{6} + \frac{1}{17} a^{5} - \frac{6}{17} a^{4} - \frac{3}{17} a^{3} - \frac{8}{17} a^{2} - \frac{2}{17} a + \frac{3}{17}$, $\frac{1}{17} a^{16} + \frac{8}{17} a^{13} - \frac{5}{17} a^{12} + \frac{5}{17} a^{11} + \frac{1}{17} a^{10} - \frac{2}{17} a^{9} - \frac{5}{17} a^{8} + \frac{8}{17} a^{7} + \frac{2}{17} a^{6} + \frac{1}{17} a^{5} + \frac{7}{17} a^{4} - \frac{7}{17} a^{3} + \frac{1}{17} a^{2} + \frac{3}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{17} + \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{17} a^{18} + \frac{1}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{19} + \frac{1}{17} a^{3} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{20} + \frac{1}{17} a^{4} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{21} + \frac{1}{17} a^{5} - \frac{1}{17} a^{4}$, $\frac{1}{289} a^{22} + \frac{6}{289} a^{21} + \frac{3}{289} a^{20} - \frac{2}{289} a^{19} - \frac{5}{289} a^{18} + \frac{7}{289} a^{17} - \frac{8}{289} a^{16} - \frac{4}{289} a^{15} + \frac{3}{289} a^{14} - \frac{53}{289} a^{13} + \frac{26}{289} a^{12} - \frac{74}{289} a^{11} - \frac{97}{289} a^{10} - \frac{5}{17} a^{9} + \frac{31}{289} a^{8} - \frac{92}{289} a^{7} - \frac{79}{289} a^{6} + \frac{36}{289} a^{5} + \frac{109}{289} a^{4} - \frac{28}{289} a^{3} - \frac{116}{289} a^{2} - \frac{89}{289} a - \frac{101}{289}$, $\frac{1}{289} a^{23} + \frac{1}{289} a^{21} - \frac{3}{289} a^{20} + \frac{7}{289} a^{19} + \frac{3}{289} a^{18} + \frac{1}{289} a^{17} - \frac{7}{289} a^{16} - \frac{7}{289} a^{15} - \frac{3}{289} a^{14} + \frac{21}{289} a^{13} + \frac{59}{289} a^{12} + \frac{7}{289} a^{11} + \frac{4}{289} a^{10} + \frac{65}{289} a^{9} - \frac{57}{289} a^{8} - \frac{54}{289} a^{7} + \frac{4}{17} a^{6} + \frac{46}{289} a^{5} + \frac{100}{289} a^{4} + \frac{69}{289} a^{3} + \frac{97}{289} a^{2} + \frac{144}{289} a - \frac{23}{289}$, $\frac{1}{699091} a^{24} - \frac{12}{699091} a^{23} - \frac{96}{699091} a^{22} + \frac{1562}{699091} a^{21} - \frac{19135}{699091} a^{20} + \frac{9769}{699091} a^{19} - \frac{11110}{699091} a^{18} + \frac{18104}{699091} a^{17} - \frac{12067}{699091} a^{16} + \frac{337}{11849} a^{15} - \frac{14497}{699091} a^{14} - \frac{99840}{699091} a^{13} - \frac{10346}{699091} a^{12} + \frac{98252}{699091} a^{11} - \frac{154556}{699091} a^{10} - \frac{318465}{699091} a^{9} + \frac{36553}{699091} a^{8} + \frac{238086}{699091} a^{7} - \frac{341828}{699091} a^{6} - \frac{17183}{41123} a^{5} - \frac{53830}{699091} a^{4} + \frac{95876}{699091} a^{3} + \frac{114442}{699091} a^{2} + \frac{37397}{699091} a + \frac{53848}{699091}$, $\frac{1}{699091} a^{25} - \frac{240}{699091} a^{23} + \frac{10}{17051} a^{22} - \frac{23}{41123} a^{21} - \frac{14236}{699091} a^{20} - \frac{17251}{699091} a^{19} + \frac{8153}{699091} a^{18} - \frac{434}{699091} a^{17} - \frac{1552}{699091} a^{16} + \frac{18484}{699091} a^{15} + \frac{14057}{699091} a^{14} + \frac{107510}{699091} a^{13} + \frac{97469}{699091} a^{12} + \frac{325377}{699091} a^{11} - \frac{322602}{699091} a^{10} + \frac{5547}{11849} a^{9} - \frac{269107}{699091} a^{8} + \frac{130070}{699091} a^{7} + \frac{335098}{699091} a^{6} + \frac{18539}{699091} a^{5} + \frac{190130}{699091} a^{4} + \frac{113510}{699091} a^{3} + \frac{300380}{699091} a^{2} + \frac{255874}{699091} a - \frac{217407}{699091}$, $\frac{1}{1062262637920062053570780638299451214565439} a^{26} - \frac{13}{1062262637920062053570780638299451214565439} a^{25} - \frac{2443110477060102718059261694852728}{5561584491728073578904610671724875468929} a^{24} + \frac{5599609213421755429791827804602453226}{1062262637920062053570780638299451214565439} a^{23} - \frac{673403639377124155872152995275439409549}{1062262637920062053570780638299451214565439} a^{22} + \frac{7171323177982415027304127542269096721861}{1062262637920062053570780638299451214565439} a^{21} + \frac{10459974582199964606272627889471564247951}{1062262637920062053570780638299451214565439} a^{20} + \frac{20972986083034678170578723189526852541848}{1062262637920062053570780638299451214565439} a^{19} + \frac{20961832926884221784782111780179155281566}{1062262637920062053570780638299451214565439} a^{18} + \frac{5973338307490959217265205925635566257464}{1062262637920062053570780638299451214565439} a^{17} + \frac{23501830750998438788707016380731955630455}{1062262637920062053570780638299451214565439} a^{16} - \frac{30707640935402440668059517742753380842644}{1062262637920062053570780638299451214565439} a^{15} - \frac{8714875265513482236560489184648106675820}{1062262637920062053570780638299451214565439} a^{14} - \frac{477105721937796538547593874964868749802655}{1062262637920062053570780638299451214565439} a^{13} - \frac{83912561367692695030007099276471326082333}{1062262637920062053570780638299451214565439} a^{12} + \frac{520851636918288349868245332731196169142204}{1062262637920062053570780638299451214565439} a^{11} - \frac{175356005028865672682265845031144023282097}{1062262637920062053570780638299451214565439} a^{10} + \frac{125690744650682074191864776486133132944285}{1062262637920062053570780638299451214565439} a^{9} + \frac{427710433817327287014105476974945177501323}{1062262637920062053570780638299451214565439} a^{8} - \frac{355635685107908953290958969326790470894625}{1062262637920062053570780638299451214565439} a^{7} - \frac{287546035167724077569319792095078975395389}{1062262637920062053570780638299451214565439} a^{6} - \frac{70019255007814551201101540448101193013552}{1062262637920062053570780638299451214565439} a^{5} - \frac{357548435002324002501921881341108983892204}{1062262637920062053570780638299451214565439} a^{4} - \frac{207502148188233810500801178809774877709895}{1062262637920062053570780638299451214565439} a^{3} - \frac{433061119589042155183933712543563635922683}{1062262637920062053570780638299451214565439} a^{2} - \frac{300153323594753035014204493283785191674819}{1062262637920062053570780638299451214565439} a + \frac{1293493476895187939031263814164803360814}{5561584491728073578904610671724875468929}$, $\frac{1}{198579718519083172634440928744284304174703042165919} a^{27} + \frac{93470147}{198579718519083172634440928744284304174703042165919} a^{26} + \frac{2688550696167383675129948486288140170412792}{4843407768758126161815632408397178150602513223559} a^{25} - \frac{1227712070659461415714711505223409561743909}{198579718519083172634440928744284304174703042165919} a^{24} - \frac{850751535235086271647968054945820533625907333}{198579718519083172634440928744284304174703042165919} a^{23} - \frac{316101249525180007755053928490590587968299453153}{198579718519083172634440928744284304174703042165919} a^{22} + \frac{4037697054050971217618913159486374264754964762254}{198579718519083172634440928744284304174703042165919} a^{21} - \frac{5050824828576704296597864902741988094598355316129}{198579718519083172634440928744284304174703042165919} a^{20} + \frac{3230555228392085672287651683195596312965042200625}{198579718519083172634440928744284304174703042165919} a^{19} - \frac{1838256407576669804818217580184335525183650048025}{198579718519083172634440928744284304174703042165919} a^{18} - \frac{4304599046041821625923243419809488978833564721827}{198579718519083172634440928744284304174703042165919} a^{17} + \frac{3825730795610832826730348087100598350223718312823}{198579718519083172634440928744284304174703042165919} a^{16} + \frac{835735294105891453159873562960608249412440044986}{198579718519083172634440928744284304174703042165919} a^{15} - \frac{3668184353226236164945215345692010340332322147203}{198579718519083172634440928744284304174703042165919} a^{14} + \frac{72532414708622525484063273846508836167341758239566}{198579718519083172634440928744284304174703042165919} a^{13} + \frac{63625426498102429800522826533602678681392128776359}{198579718519083172634440928744284304174703042165919} a^{12} + \frac{78462966786746429906621602196724595332525906499511}{198579718519083172634440928744284304174703042165919} a^{11} + \frac{79625853663895909138188677756075105691933910094080}{198579718519083172634440928744284304174703042165919} a^{10} - \frac{78638574227717774604083801504715532586460316894186}{198579718519083172634440928744284304174703042165919} a^{9} - \frac{54618551678796976314370898527680544033809457275901}{198579718519083172634440928744284304174703042165919} a^{8} - \frac{2714685594035600197868787879063543898098858686181}{11681159912887245449084760514369664951453120127407} a^{7} - \frac{90681865336137240000178693364857284050834273183458}{198579718519083172634440928744284304174703042165919} a^{6} + \frac{20392993170938463020208839547822305739195713451747}{198579718519083172634440928744284304174703042165919} a^{5} - \frac{4934165645243455175356975243415270159164468234634}{11681159912887245449084760514369664951453120127407} a^{4} + \frac{1579306399538272785114882985796557670357204026338}{11681159912887245449084760514369664951453120127407} a^{3} + \frac{16422677386144072523364564951517235192628996271744}{198579718519083172634440928744284304174703042165919} a^{2} - \frac{1185031666496251521208797218870690551969819712292}{3365757941001409705668490317699733969062763426541} a + \frac{519058311317623965095018258375075476111354155579}{1039684390152267919552046747352273843846612786209}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{5556}$, which has order $711168$ (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:  $13$
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:  \( 34681517373.86067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-145}) \), \(\Q(\sqrt{5}, \sqrt{-29})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, 14.0.168110140833113738264576.1, 14.0.13133604752587010801920000000.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$