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Properties

Label 28.0.69632758551...0625.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{21}\cdot 29^{27}$
Root discriminant $148.92$
Ramified primes $3, 5, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9200187361, -67387744281, 188267474927, -260927965161, 212356160468, -140023586864, 104743069711, -56641660997, 46647651823, -20170505866, 12213543141, -6840651052, 2539665866, -1476922586, 497984202, -210052337, 85342680, -20452367, 11305361, -1451393, 1100638, -80998, 75865, -3394, 3394, -88, 88, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 88*x^26 - 88*x^25 + 3394*x^24 - 3394*x^23 + 75865*x^22 - 80998*x^21 + 1100638*x^20 - 1451393*x^19 + 11305361*x^18 - 20452367*x^17 + 85342680*x^16 - 210052337*x^15 + 497984202*x^14 - 1476922586*x^13 + 2539665866*x^12 - 6840651052*x^11 + 12213543141*x^10 - 20170505866*x^9 + 46647651823*x^8 - 56641660997*x^7 + 104743069711*x^6 - 140023586864*x^5 + 212356160468*x^4 - 260927965161*x^3 + 188267474927*x^2 - 67387744281*x + 9200187361)
 
gp: K = bnfinit(x^28 - x^27 + 88*x^26 - 88*x^25 + 3394*x^24 - 3394*x^23 + 75865*x^22 - 80998*x^21 + 1100638*x^20 - 1451393*x^19 + 11305361*x^18 - 20452367*x^17 + 85342680*x^16 - 210052337*x^15 + 497984202*x^14 - 1476922586*x^13 + 2539665866*x^12 - 6840651052*x^11 + 12213543141*x^10 - 20170505866*x^9 + 46647651823*x^8 - 56641660997*x^7 + 104743069711*x^6 - 140023586864*x^5 + 212356160468*x^4 - 260927965161*x^3 + 188267474927*x^2 - 67387744281*x + 9200187361, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 88 x^{26} - 88 x^{25} + 3394 x^{24} - 3394 x^{23} + 75865 x^{22} - 80998 x^{21} + 1100638 x^{20} - 1451393 x^{19} + 11305361 x^{18} - 20452367 x^{17} + 85342680 x^{16} - 210052337 x^{15} + 497984202 x^{14} - 1476922586 x^{13} + 2539665866 x^{12} - 6840651052 x^{11} + 12213543141 x^{10} - 20170505866 x^{9} + 46647651823 x^{8} - 56641660997 x^{7} + 104743069711 x^{6} - 140023586864 x^{5} + 212356160468 x^{4} - 260927965161 x^{3} + 188267474927 x^{2} - 67387744281 x + 9200187361 \)

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:  \(6963275855162665256600064094162323796066183958530426025390625=3^{14}\cdot 5^{21}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.92$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(128,·)$, $\chi_{435}(1,·)$, $\chi_{435}(2,·)$, $\chi_{435}(4,·)$, $\chi_{435}(113,·)$, $\chi_{435}(8,·)$, $\chi_{435}(137,·)$, $\chi_{435}(77,·)$, $\chi_{435}(256,·)$, $\chi_{435}(143,·)$, $\chi_{435}(16,·)$, $\chi_{435}(17,·)$, $\chi_{435}(274,·)$, $\chi_{435}(68,·)$, $\chi_{435}(154,·)$, $\chi_{435}(218,·)$, $\chi_{435}(286,·)$, $\chi_{435}(32,·)$, $\chi_{435}(289,·)$, $\chi_{435}(34,·)$, $\chi_{435}(226,·)$, $\chi_{435}(64,·)$, $\chi_{435}(362,·)$, $\chi_{435}(109,·)$, $\chi_{435}(272,·)$, $\chi_{435}(136,·)$, $\chi_{435}(308,·)$, $\chi_{435}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{59} a^{18} - \frac{1}{59} a^{17} + \frac{5}{59} a^{16} - \frac{5}{59} a^{15} - \frac{22}{59} a^{14} + \frac{22}{59} a^{13} + \frac{22}{59} a^{12} - \frac{22}{59} a^{11} + \frac{1}{59} a^{10} - \frac{1}{59} a^{9} + \frac{25}{59} a^{8} - \frac{25}{59} a^{7} + \frac{21}{59} a^{6} - \frac{21}{59} a^{5} - \frac{4}{59} a^{4} + \frac{4}{59} a^{3} + \frac{22}{59} a^{2} - \frac{22}{59} a$, $\frac{1}{59} a^{19} + \frac{4}{59} a^{17} - \frac{27}{59} a^{15} - \frac{15}{59} a^{13} - \frac{21}{59} a^{11} + \frac{24}{59} a^{9} - \frac{4}{59} a^{7} - \frac{25}{59} a^{5} + \frac{26}{59} a^{3} - \frac{22}{59} a$, $\frac{1}{59} a^{20} + \frac{4}{59} a^{17} + \frac{12}{59} a^{16} + \frac{20}{59} a^{15} + \frac{14}{59} a^{14} - \frac{29}{59} a^{13} + \frac{9}{59} a^{12} + \frac{29}{59} a^{11} + \frac{20}{59} a^{10} + \frac{4}{59} a^{9} + \frac{14}{59} a^{8} - \frac{18}{59} a^{7} + \frac{9}{59} a^{6} + \frac{25}{59} a^{5} - \frac{17}{59} a^{4} - \frac{16}{59} a^{3} + \frac{8}{59} a^{2} + \frac{29}{59} a$, $\frac{1}{59} a^{21} + \frac{16}{59} a^{17} - \frac{25}{59} a^{15} - \frac{20}{59} a^{13} - \frac{10}{59} a^{11} + \frac{18}{59} a^{9} - \frac{9}{59} a^{7} + \frac{8}{59} a^{5} - \frac{8}{59} a^{3} + \frac{29}{59} a$, $\frac{1}{59} a^{22} + \frac{16}{59} a^{17} + \frac{13}{59} a^{16} + \frac{21}{59} a^{15} - \frac{22}{59} a^{14} + \frac{2}{59} a^{13} - \frac{8}{59} a^{12} - \frac{2}{59} a^{11} + \frac{2}{59} a^{10} + \frac{16}{59} a^{9} + \frac{4}{59} a^{8} - \frac{13}{59} a^{7} + \frac{26}{59} a^{6} - \frac{18}{59} a^{5} - \frac{3}{59} a^{4} - \frac{5}{59} a^{3} - \frac{28}{59} a^{2} - \frac{2}{59} a$, $\frac{1}{59} a^{23} + \frac{29}{59} a^{17} - \frac{1}{59} a^{15} - \frac{6}{59} a^{13} + \frac{20}{59} a^{9} + \frac{13}{59} a^{7} - \frac{21}{59} a^{5} + \frac{26}{59} a^{3} - \frac{2}{59} a$, $\frac{1}{3481} a^{24} - \frac{1}{3481} a^{23} + \frac{8}{3481} a^{22} - \frac{8}{3481} a^{21} - \frac{26}{3481} a^{20} + \frac{26}{3481} a^{19} + \frac{9}{3481} a^{18} - \frac{1661}{3481} a^{17} + \frac{222}{3481} a^{16} + \frac{486}{3481} a^{15} - \frac{224}{3481} a^{14} + \frac{1463}{3481} a^{13} - \frac{384}{3481} a^{12} + \frac{1387}{3481} a^{11} + \frac{735}{3481} a^{10} - \frac{263}{3481} a^{9} - \frac{406}{3481} a^{8} + \frac{52}{3481} a^{7} - \frac{408}{3481} a^{6} - \frac{1362}{3481} a^{5} + \frac{1055}{3481} a^{4} + \frac{1305}{3481} a^{3} - \frac{1641}{3481} a^{2} - \frac{1545}{3481} a - \frac{4}{59}$, $\frac{1}{3481} a^{25} + \frac{7}{3481} a^{23} + \frac{25}{3481} a^{21} - \frac{24}{3481} a^{19} + \frac{1098}{3481} a^{17} - \frac{25}{59} a^{16} - \frac{918}{3481} a^{15} - \frac{5}{59} a^{14} - \frac{1163}{3481} a^{13} - \frac{16}{59} a^{12} + \frac{1237}{3481} a^{11} - \frac{23}{59} a^{10} + \frac{806}{3481} a^{9} - \frac{14}{59} a^{8} - \frac{179}{3481} a^{7} + \frac{27}{59} a^{6} - \frac{1723}{3481} a^{5} - \frac{13}{59} a^{4} + \frac{785}{3481} a^{3} - \frac{28}{59} a^{2} - \frac{306}{3481} a - \frac{4}{59}$, $\frac{1}{205379} a^{26} - \frac{1}{205379} a^{25} - \frac{11}{205379} a^{24} + \frac{11}{205379} a^{23} + \frac{1002}{205379} a^{22} - \frac{1002}{205379} a^{21} + \frac{1034}{205379} a^{20} + \frac{795}{205379} a^{19} - \frac{775}{205379} a^{18} - \frac{19344}{205379} a^{17} - \frac{7746}{205379} a^{16} - \frac{60163}{205379} a^{15} - \frac{76545}{205379} a^{14} - \frac{46942}{205379} a^{13} + \frac{51337}{205379} a^{12} - \frac{55998}{205379} a^{11} + \frac{4745}{205379} a^{10} - \frac{27991}{205379} a^{9} - \frac{67329}{205379} a^{8} + \frac{23551}{205379} a^{7} + \frac{12996}{205379} a^{6} + \frac{96095}{205379} a^{5} + \frac{7342}{205379} a^{4} - \frac{27520}{205379} a^{3} + \frac{71004}{205379} a^{2} + \frac{100686}{205379} a + \frac{1728}{3481}$, $\frac{1}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{27} - \frac{10585839516229780357052498580984552685083619873958356938828013354030640642815287623334888520691559751232379229670532255600101339914146298}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{26} - \frac{295688033796854035434073908165578476420539331070139176346137922623326225646142599057276408756405282503750288197307356734759550932016696249}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{25} - \frac{640619636903278164893215356165066241188578073205477130357198029694542820942508445592241215588531654073487005475987215151908573194184976162}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{24} - \frac{10668141647630550585720495560820600513476471301260497409172814969183953085602250709407879375714516704516318101133241268560620231135390617895}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{23} - \frac{15899741602683435959594597234378110166836413609211485427320690336344078786793219153051618342590673770516374994964246704913948239727344304515}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{22} - \frac{10271158074763811690040871473181847868466665001312394570927291726781367376356986771740732289717192228729876722576970523787432245799561323357}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{21} + \frac{38625686170850664762105189671430210603168858620523147276517364560463534012442505523343388883663978653246953372100978070755494946051422967383}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{20} - \frac{23411951529817468402328883062471091568860495757120383560863383507263122537114739435229431296603784613710780156758082272022486819804171985345}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{19} + \frac{769039675312402607546173891033978799969593343375742045044957759536698579162101187904013989486962079339545669412096263924301508581213697587}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{18} + \frac{1123596811197784140750560521645962333142599984234894881332440337370707484914118174222072771858814118055698324258171173708032779811466077390387}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{17} + \frac{2357941684717753683692368292754356845813141077572250887396252022802498589458436549934478212819033071042583435977953761010378864129332844371537}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{16} + \frac{317916764260199617406974553435825146193729798559436254681590910523764334729750680164125177503737339885550635571354647936286584425685801629669}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{15} - \frac{1857752526468022553923600332100373986362687400895209904768786650378731647230801587969666368802209732707429949421539664567809420548011970217422}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{14} + \frac{1512302792448173077544180800394137256220551164215917712117055344573861314862945697016848320916574930906061390187736071137642723528618660660562}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{13} - \frac{1031833662117481434587457108491424537666166629902008788309558436752975191644794446189627873514243150724211334848994361221684540231835543241056}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{12} - \frac{1988881692310163212456894687110987454462017450342928329520716115100238088073007558391942983800098204899482233645612617141790659403316743580063}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{11} + \frac{2017773372277068064231970090267690953732372976883823906756716750847098884785014641200732563316755180074435949326375672734235602127235836113850}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{10} - \frac{1158805935748870579917193751465994970172786502340897192338060451312279145957426377459202554770691683611951662111117527990953365288815506073577}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{9} - \frac{1110021007459305080324658160750133321932926600057800228507328949928206125329307023688087154994037138807376113739047002585181705738306846162096}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{8} + \frac{2551452608083882117651015797225036740398854233399583428683864120831622379916889964031432014720920067014169554218553288662014553667523639232499}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{7} - \frac{2300356727750362224327157857424755188985508379123141916189928378990311373601178925775144862693236615925287360184360342792604006112909557925808}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{6} - \frac{801109076050249562180003143636184398203147832387754195447784177686457882084841292690727315077379105388172573872222442041378910185809109960117}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{5} + \frac{1423853388214203658441313561204487688884123153005598267563953503307687188546681957581626886932243219207327814054995466442674119332627108957656}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{4} - \frac{1275889967342673416355441884768922898164485203545752501014891996381527663318395200277580060621905714991051706539536138374634634988444567788571}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{3} - \frac{908193708599301223383269806202091490033642398385172557497200104526243911253735250783549384446564700993417386918164019200546485451903170017429}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a^{2} - \frac{1155243339204753542878575209864956067150707235519473416420604251537862051541052298142538878804360059167384899078419015151352150846080479737947}{5126096447027915198597727773235896557317507976947619778501576106968441962053166381980813084042534072139301970714310761852677783966560518913931} a - \frac{15417927009866157098897939032557320268468241962220034578452059228359552600960218543557136109685840674152887900278441799976651497789119401135}{86882990627591783027080131749760958598601830117756267432230103507939694272087565796284967526144645290496643571428995963604708202823059642609}$

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:  $13$
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:  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

$C_{28}$ (as 28T1):

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

Intermediate fields

\(\Q(\sqrt{145}) \), 4.0.27437625.1, 7.7.594823321.1, 14.14.801611618199890796015625.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R R $28$ $28$ $28$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ $28$ $28$ R $28$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5Data not computed
29Data not computed