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Properties

Label 28.0.365...625.1
Degree $28$
Signature $[0, 14]$
Discriminant $3.655\times 10^{51}$
Root discriminant $69.43$
Ramified primes $3, 5, 29$
Class number $7232$ (GRH)
Class group $[4, 4, 452]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681)
 
gp: K = bnfinit(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -6888, 60245, 16080, 726277, -287718, 3396106, -1974799, 11808392, -9450995, 17701828, -10099944, 13577493, -6804920, 7071411, -2978786, 2508376, -933216, 649914, -204097, 115818, -30908, 14492, -3001, 1049, -152, 47, -5, 1]);
 

\( x^{28} - 5 x^{27} + 47 x^{26} - 152 x^{25} + 1049 x^{24} - 3001 x^{23} + 14492 x^{22} - 30908 x^{21} + 115818 x^{20} - 204097 x^{19} + 649914 x^{18} - 933216 x^{17} + 2508376 x^{16} - 2978786 x^{15} + 7071411 x^{14} - 6804920 x^{13} + 13577493 x^{12} - 10099944 x^{11} + 17701828 x^{10} - 9450995 x^{9} + 11808392 x^{8} - 1974799 x^{7} + 3396106 x^{6} - 287718 x^{5} + 726277 x^{4} + 16080 x^{3} + 60245 x^{2} - 6888 x + 1681 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3654513548980364725264702136425345220781793212890625\)\(\medspace = 3^{14}\cdot 5^{14}\cdot 29^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.43$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(194,·)$, $\chi_{435}(326,·)$, $\chi_{435}(199,·)$, $\chi_{435}(136,·)$, $\chi_{435}(74,·)$, $\chi_{435}(139,·)$, $\chi_{435}(94,·)$, $\chi_{435}(16,·)$, $\chi_{435}(401,·)$, $\chi_{435}(146,·)$, $\chi_{435}(344,·)$, $\chi_{435}(281,·)$, $\chi_{435}(239,·)$, $\chi_{435}(284,·)$, $\chi_{435}(349,·)$, $\chi_{435}(286,·)$, $\chi_{435}(161,·)$, $\chi_{435}(226,·)$, $\chi_{435}(169,·)$, $\chi_{435}(364,·)$, $\chi_{435}(431,·)$, $\chi_{435}(49,·)$, $\chi_{435}(371,·)$, $\chi_{435}(181,·)$, $\chi_{435}(314,·)$, $\chi_{435}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{41} a^{23} - \frac{10}{41} a^{22} - \frac{13}{41} a^{21} + \frac{3}{41} a^{20} + \frac{9}{41} a^{19} + \frac{4}{41} a^{18} - \frac{9}{41} a^{17} + \frac{6}{41} a^{16} - \frac{11}{41} a^{15} + \frac{4}{41} a^{14} - \frac{10}{41} a^{13} - \frac{9}{41} a^{12} - \frac{12}{41} a^{11} - \frac{10}{41} a^{10} - \frac{10}{41} a^{9} - \frac{1}{41} a^{8} + \frac{12}{41} a^{7} - \frac{14}{41} a^{6} + \frac{10}{41} a^{5} - \frac{5}{41} a^{4} - \frac{17}{41} a^{3} + \frac{16}{41} a^{2} + \frac{20}{41} a$, $\frac{1}{41123} a^{24} + \frac{381}{41123} a^{23} - \frac{18929}{41123} a^{22} + \frac{250}{41123} a^{21} - \frac{5460}{41123} a^{20} - \frac{17838}{41123} a^{19} + \frac{6147}{41123} a^{18} + \frac{10550}{41123} a^{17} - \frac{9063}{41123} a^{16} - \frac{10119}{41123} a^{15} + \frac{2743}{41123} a^{14} + \frac{2518}{41123} a^{13} - \frac{17430}{41123} a^{12} + \frac{9443}{41123} a^{11} + \frac{6904}{41123} a^{10} - \frac{915}{2419} a^{9} + \frac{18399}{41123} a^{8} - \frac{19389}{41123} a^{7} + \frac{15569}{41123} a^{6} + \frac{1568}{41123} a^{5} + \frac{10328}{41123} a^{4} - \frac{2777}{41123} a^{3} + \frac{14886}{41123} a^{2} + \frac{234}{697} a + \frac{122}{1003}$, $\frac{1}{41123} a^{25} + \frac{402}{41123} a^{23} + \frac{922}{2419} a^{22} - \frac{18464}{41123} a^{21} + \frac{6272}{41123} a^{20} + \frac{17130}{41123} a^{19} + \frac{12554}{41123} a^{18} + \frac{1441}{41123} a^{17} - \frac{11448}{41123} a^{16} - \frac{440}{2419} a^{15} - \frac{14490}{41123} a^{14} + \frac{10164}{41123} a^{13} - \frac{11653}{41123} a^{12} - \frac{13178}{41123} a^{11} - \frac{14107}{41123} a^{10} - \frac{17981}{41123} a^{9} + \frac{2625}{41123} a^{8} + \frac{638}{41123} a^{7} - \frac{8509}{41123} a^{6} - \frac{11358}{41123} a^{5} + \frac{10063}{41123} a^{4} + \frac{3725}{41123} a^{3} + \frac{17214}{41123} a^{2} + \frac{8660}{41123} a - \frac{344}{1003}$, $\frac{1}{11639412797} a^{26} + \frac{90482}{11639412797} a^{25} + \frac{113690}{11639412797} a^{24} + \frac{44768126}{11639412797} a^{23} + \frac{1385250353}{11639412797} a^{22} + \frac{31643050}{684671341} a^{21} + \frac{3239853883}{11639412797} a^{20} - \frac{2829787740}{11639412797} a^{19} + \frac{2110761710}{11639412797} a^{18} - \frac{3933425257}{11639412797} a^{17} - \frac{1900308217}{11639412797} a^{16} - \frac{2058964356}{11639412797} a^{15} + \frac{2505722651}{11639412797} a^{14} - \frac{4636511259}{11639412797} a^{13} + \frac{1514246237}{11639412797} a^{12} - \frac{1104984636}{11639412797} a^{11} + \frac{4887975349}{11639412797} a^{10} - \frac{28103078}{11639412797} a^{9} + \frac{3754209872}{11639412797} a^{8} + \frac{4805896816}{11639412797} a^{7} - \frac{3642556465}{11639412797} a^{6} - \frac{295968095}{684671341} a^{5} - \frac{5346240212}{11639412797} a^{4} + \frac{4597992423}{11639412797} a^{3} - \frac{2457595563}{11639412797} a^{2} + \frac{523477101}{11639412797} a + \frac{57322540}{283888117}$, $\frac{1}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{27} - \frac{342250261158561326966813007242918852502962824543627804101038865308472}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{26} - \frac{51611978356730267745561010805640710519484376831726703245464587341715555948}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{25} - \frac{422159524421325155754900646090530696963507047012367653785365312442227863}{203770442879374475262967933757452710585022621456743515487233823699077175923129} a^{24} - \frac{79192806391686920987039639228096982154603862498882838220084133661437809148331}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{23} + \frac{3253111534683133897281985569307945237058687366324068907947090692244064811453344}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{22} + \frac{66780140396725649908529891366863405773910043447646002479296968802327346304291}{141603189119565313318333648882297646338744533554686171779264182570545156149971} a^{21} + \frac{3303641437991203164556943948969684360752180144385150559346513352039021405540163}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{20} - \frac{1412876243689404542402253695591303722372781396333023037537870493942577795081581}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{19} + \frac{3079552791871098088017350102896014033668666368444390967689394267701033320278393}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{18} - \frac{2975896637704706270191642838537306791296655077061780602700127229746142622025083}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{17} - \frac{2708445772932169527095636039603108692476537524300782776772349929656612343911855}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{16} - \frac{375499810076160221247903203293498527966001601551829547784526678800152508181191}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{15} + \frac{3169453757700533398984492922218993832574893039674951046987927270261384740901094}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{14} - \frac{1369228714605292436813305647339092099898716893323546945066258780893980710894219}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{13} - \frac{2592170326126589098760864961355694578800561521208862158442157854507106041690193}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{12} + \frac{3284187747248582992430279026446466770667884654832739138871437914302935646587504}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{11} - \frac{56275062847785445857289879056027938253163009630783787321862680498165351114319}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{10} - \frac{77048391031164459194074521126892176135933841765231618101332053390373599740879}{491446362238491381516569722591503596116819263513322596175093339509539071344017} a^{9} - \frac{3173881067055915229578247966111404400325515495518643577019598637412999832033928}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{8} + \frac{1052145915548080785796535287561521262071767683773184383370192238696110429940725}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{7} - \frac{3931931867906553438761211752212606550737482191153673639810497359138782569963942}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{6} + \frac{584159963091762103022165166515550917291690850419008500033730770349867797225494}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{5} - \frac{1555016703501238562659587934462678924047067992065393275062146483478051273395564}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{4} - \frac{1234229660479116113139485766250802117823461617079654132900600054810712427593690}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{3} - \frac{899906502385140151921739593421652572870194183639384651363654766356551884511023}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a^{2} - \frac{175273819496021017019279642656872296162682354812072313313179997002665653164136}{8354588158054353485781685284055561133985927479726484134976586771662164212848289} a + \frac{24062732505229294957240849415742884460283712038450345168897407749957106755142}{203770442879374475262967933757452710585022621456743515487233823699077175923129}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{452}$, which has order $7232$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{2944727907733851313844166695580586247643057933080334230251436957902}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{27} + \frac{14271014244497695456942016188160138110021276736171669922386030024919}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{26} - \frac{135978308529120404887408203190974516593584323418471359924636877640401}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{25} + \frac{425536117233239771730783713724313289079093355274780523470454240434385}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{24} - \frac{3012765312874440015416700186619905078007069272443493500857940345723134}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{23} + \frac{8338710311987213613917630675882671769621993807103639362273825095004280}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{22} - \frac{41151506115984017970877109756340043753316327168979014317153599803844452}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{21} + \frac{83992996979847503273447678914499035181237397705166314022201055558762189}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{20} - \frac{324804460454186693343653891617518807381958859045684362397963682147893897}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{19} + \frac{543906411037652109715607979798839794708718513271825997000468514034362298}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{18} - \frac{1803644658886089076090483947500570091181814785162840151605644744386981660}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{17} + \frac{2423517944561405299035290118522252965963277892766509164949168857041016231}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{16} - \frac{6865459408811625500018690377645442917201354846624570074057046078496714509}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{15} + \frac{7500461609322319923928028421126120383311543812442659654129090193085504130}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{14} - \frac{19101125488377777174193915609101763151345060775211909510523620519838738384}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{13} + \frac{16415555309730099806856068763615671098455285285566516110596563061975971242}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{12} - \frac{35866484987036329775270117766943400957791820891405149091575556794667568989}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{11} + \frac{22663749541833966543233969669138216258227987804229724710839773761551577003}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{10} - \frac{45628561583377037475183550196108885274738643778225967079513540388962845611}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{9} + \frac{18511702236365329041937840439579219312109272307870612947743476562606785588}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{8} - \frac{28075173717772617821653071636187454557705495418108951279776174741925465456}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{7} - \frac{649273056571248661482193461080844355860706103452234065796108746994178172}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{6} - \frac{7680157428893372518999554886997769133836112457788756459639815166016630161}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{5} - \frac{753277665347945426832902039256201334326984014324432946402408520415558682}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{4} - \frac{1653599691071112203219362968187998293682957001362444431775939969844656890}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{3} - \frac{452085557506791951951668125272014802832983106016857957449153709218711097}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{2} - \frac{112584997216705782453532935320254040117798848782008067201574713607840477}{29429157677826837274001452072245634479959326651655814586196951567585917} a + \frac{260942716696870531257097801519003430910692667546008670278151888027091}{717784333605532616439059806640137426340471381747702794785291501648437} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 34681517373.86067 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 7232}{6\sqrt{3654513548980364725264702136425345220781793212890625}}\approx 0.103349558629174$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, 14.0.773792930870360792667.1, 14.0.60452572724246936927109375.1

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 ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ 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.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$