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Properties

Label 28.0.26333545371...9456.2
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 3^{14}\cdot 29^{24}$
Root discriminant $87.82$
Ramified primes $2, 3, 29$
Class number $2752$ (GRH)
Class group $[4, 4, 172]$ (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![6379363, 848352, 12345988, -2328948, 7481301, -4114816, -3574880, -1040484, 8299336, 16836040, -8566886, -22214196, 8847842, 16118448, -5496216, -7607912, 2269775, 2365228, -639052, -489536, 127316, 64264, -16614, -5092, 1315, 220, -56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 4*x^27 - 56*x^26 + 220*x^25 + 1315*x^24 - 5092*x^23 - 16614*x^22 + 64264*x^21 + 127316*x^20 - 489536*x^19 - 639052*x^18 + 2365228*x^17 + 2269775*x^16 - 7607912*x^15 - 5496216*x^14 + 16118448*x^13 + 8847842*x^12 - 22214196*x^11 - 8566886*x^10 + 16836040*x^9 + 8299336*x^8 - 1040484*x^7 - 3574880*x^6 - 4114816*x^5 + 7481301*x^4 - 2328948*x^3 + 12345988*x^2 + 848352*x + 6379363)
 
gp: K = bnfinit(x^28 - 4*x^27 - 56*x^26 + 220*x^25 + 1315*x^24 - 5092*x^23 - 16614*x^22 + 64264*x^21 + 127316*x^20 - 489536*x^19 - 639052*x^18 + 2365228*x^17 + 2269775*x^16 - 7607912*x^15 - 5496216*x^14 + 16118448*x^13 + 8847842*x^12 - 22214196*x^11 - 8566886*x^10 + 16836040*x^9 + 8299336*x^8 - 1040484*x^7 - 3574880*x^6 - 4114816*x^5 + 7481301*x^4 - 2328948*x^3 + 12345988*x^2 + 848352*x + 6379363, 1)
 

Normalized defining polynomial

\( x^{28} - 4 x^{27} - 56 x^{26} + 220 x^{25} + 1315 x^{24} - 5092 x^{23} - 16614 x^{22} + 64264 x^{21} + 127316 x^{20} - 489536 x^{19} - 639052 x^{18} + 2365228 x^{17} + 2269775 x^{16} - 7607912 x^{15} - 5496216 x^{14} + 16118448 x^{13} + 8847842 x^{12} - 22214196 x^{11} - 8566886 x^{10} + 16836040 x^{9} + 8299336 x^{8} - 1040484 x^{7} - 3574880 x^{6} - 4114816 x^{5} + 7481301 x^{4} - 2328948 x^{3} + 12345988 x^{2} + 848352 x + 6379363 \)

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:  \(2633354537185343913419055876339673429183385802381459456=2^{42}\cdot 3^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.82$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(696=2^{3}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(451,·)$, $\chi_{696}(571,·)$, $\chi_{696}(257,·)$, $\chi_{696}(523,·)$, $\chi_{696}(401,·)$, $\chi_{696}(83,·)$, $\chi_{696}(139,·)$, $\chi_{696}(25,·)$, $\chi_{696}(355,·)$, $\chi_{696}(281,·)$, $\chi_{696}(107,·)$, $\chi_{696}(161,·)$, $\chi_{696}(587,·)$, $\chi_{696}(547,·)$, $\chi_{696}(529,·)$, $\chi_{696}(65,·)$, $\chi_{696}(227,·)$, $\chi_{696}(169,·)$, $\chi_{696}(683,·)$, $\chi_{696}(49,·)$, $\chi_{696}(691,·)$, $\chi_{696}(625,·)$, $\chi_{696}(545,·)$, $\chi_{696}(233,·)$, $\chi_{696}(371,·)$, $\chi_{696}(313,·)$, $\chi_{696}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{34} a^{20} - \frac{1}{17} a^{19} + \frac{7}{34} a^{18} - \frac{5}{34} a^{17} + \frac{7}{34} a^{16} - \frac{1}{34} a^{15} + \frac{3}{17} a^{14} + \frac{1}{17} a^{13} - \frac{4}{17} a^{12} + \frac{3}{34} a^{11} - \frac{11}{34} a^{10} - \frac{1}{17} a^{9} - \frac{11}{34} a^{8} + \frac{7}{17} a^{7} + \frac{13}{34} a^{6} - \frac{5}{34} a^{5} - \frac{15}{34} a^{4} - \frac{5}{34} a^{3} + \frac{3}{34} a^{2} - \frac{5}{17} a + \frac{3}{17}$, $\frac{1}{34} a^{21} + \frac{3}{34} a^{19} - \frac{4}{17} a^{18} - \frac{3}{34} a^{17} - \frac{2}{17} a^{16} + \frac{2}{17} a^{15} - \frac{3}{34} a^{14} - \frac{2}{17} a^{13} - \frac{13}{34} a^{12} - \frac{5}{34} a^{11} + \frac{5}{17} a^{10} - \frac{15}{34} a^{9} + \frac{9}{34} a^{8} + \frac{7}{34} a^{7} + \frac{2}{17} a^{6} + \frac{9}{34} a^{5} + \frac{8}{17} a^{4} - \frac{7}{34} a^{3} + \frac{13}{34} a^{2} - \frac{7}{17} a - \frac{5}{34}$, $\frac{1}{34} a^{22} - \frac{1}{17} a^{19} - \frac{7}{34} a^{18} - \frac{3}{17} a^{17} - \frac{5}{34} a^{14} - \frac{1}{17} a^{13} - \frac{15}{34} a^{12} - \frac{8}{17} a^{11} - \frac{8}{17} a^{10} - \frac{1}{17} a^{9} - \frac{11}{34} a^{8} - \frac{2}{17} a^{7} - \frac{13}{34} a^{6} + \frac{7}{17} a^{5} - \frac{13}{34} a^{4} - \frac{3}{17} a^{3} - \frac{3}{17} a^{2} + \frac{4}{17} a - \frac{1}{34}$, $\frac{1}{34} a^{23} + \frac{3}{17} a^{19} + \frac{4}{17} a^{18} + \frac{7}{34} a^{17} - \frac{3}{34} a^{16} - \frac{7}{34} a^{15} - \frac{7}{34} a^{14} + \frac{3}{17} a^{13} + \frac{1}{17} a^{12} - \frac{5}{17} a^{11} - \frac{7}{34} a^{10} - \frac{15}{34} a^{9} + \frac{4}{17} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} - \frac{3}{17} a^{5} + \frac{15}{34} a^{4} - \frac{8}{17} a^{3} - \frac{3}{34} a^{2} + \frac{13}{34} a + \frac{6}{17}$, $\frac{1}{6494} a^{24} - \frac{65}{6494} a^{23} - \frac{39}{6494} a^{22} - \frac{19}{6494} a^{21} - \frac{24}{3247} a^{20} + \frac{103}{3247} a^{19} - \frac{1027}{6494} a^{18} + \frac{43}{3247} a^{17} - \frac{233}{6494} a^{16} + \frac{434}{3247} a^{15} + \frac{627}{6494} a^{14} - \frac{1379}{6494} a^{13} + \frac{2739}{6494} a^{12} - \frac{998}{3247} a^{11} + \frac{1482}{3247} a^{10} - \frac{412}{3247} a^{9} + \frac{1270}{3247} a^{8} + \frac{1410}{3247} a^{7} - \frac{1294}{3247} a^{6} + \frac{1539}{6494} a^{5} + \frac{2521}{6494} a^{4} - \frac{353}{3247} a^{3} + \frac{1631}{6494} a^{2} + \frac{2653}{6494} a + \frac{654}{3247}$, $\frac{1}{6494} a^{25} - \frac{31}{3247} a^{23} - \frac{71}{6494} a^{22} + \frac{27}{3247} a^{21} - \frac{49}{6494} a^{20} - \frac{790}{3247} a^{19} + \frac{568}{3247} a^{18} + \frac{1537}{6494} a^{17} + \frac{406}{3247} a^{16} + \frac{893}{6494} a^{15} + \frac{985}{6494} a^{14} + \frac{1919}{6494} a^{13} - \frac{127}{3247} a^{12} + \frac{597}{3247} a^{11} - \frac{1301}{3247} a^{10} + \frac{657}{3247} a^{9} - \frac{1115}{6494} a^{8} - \frac{2457}{6494} a^{7} + \frac{58}{191} a^{6} - \frac{292}{3247} a^{5} - \frac{3011}{6494} a^{4} + \frac{1268}{3247} a^{3} - \frac{11}{6494} a^{2} + \frac{1404}{3247} a + \frac{299}{3247}$, $\frac{1}{3599086970614119876903635618} a^{26} + \frac{49581960114192473106949}{1799543485307059938451817809} a^{25} + \frac{75533022716916651098263}{3599086970614119876903635618} a^{24} - \frac{1320282094485905242352175}{105855499135709408144224577} a^{23} + \frac{21508574738027011155029584}{1799543485307059938451817809} a^{22} + \frac{24166124631309320004733858}{1799543485307059938451817809} a^{21} + \frac{14861944285994952208529211}{3599086970614119876903635618} a^{20} - \frac{2008870859501296526572723}{12453588133612871546379362} a^{19} + \frac{17170750435770118997529836}{1799543485307059938451817809} a^{18} + \frac{28465771909041746395191841}{211710998271418816288449154} a^{17} + \frac{282765258748701294800091013}{3599086970614119876903635618} a^{16} - \frac{333534632125932266662504963}{1799543485307059938451817809} a^{15} + \frac{230963462195114217784979768}{1799543485307059938451817809} a^{14} + \frac{130742101774943947291195087}{3599086970614119876903635618} a^{13} + \frac{1736324690613273443050072647}{3599086970614119876903635618} a^{12} + \frac{844214453714663253928417873}{1799543485307059938451817809} a^{11} - \frac{1747384297267603424862352959}{3599086970614119876903635618} a^{10} + \frac{466476894689384404776423573}{1799543485307059938451817809} a^{9} + \frac{518686014603585381701109439}{1799543485307059938451817809} a^{8} - \frac{322840617114094445091293489}{3599086970614119876903635618} a^{7} + \frac{460341955353067806462615662}{1799543485307059938451817809} a^{6} + \frac{1372384549140185904893260939}{3599086970614119876903635618} a^{5} + \frac{353002518390177857312155141}{3599086970614119876903635618} a^{4} + \frac{848193369321851136898436216}{1799543485307059938451817809} a^{3} + \frac{928690741409357563334889283}{3599086970614119876903635618} a^{2} + \frac{734489333604922186260298718}{1799543485307059938451817809} a + \frac{626147531538887010862326482}{1799543485307059938451817809}$, $\frac{1}{177714390890257752095272485429726298150471836709200688676211578771666} a^{27} + \frac{15116600093895118531479459166839491484}{88857195445128876047636242714863149075235918354600344338105789385833} a^{26} - \frac{1198613693720589168976400660157116210817277417216149785882982485}{177714390890257752095272485429726298150471836709200688676211578771666} a^{25} - \frac{3618995256486662159936770498612659709154371462493812119857486648}{88857195445128876047636242714863149075235918354600344338105789385833} a^{24} + \frac{11118678985233663452977091419221053650786361232917540440992230818}{88857195445128876047636242714863149075235918354600344338105789385833} a^{23} - \frac{1313410285559893812524034545100007640700769896719569802939665110339}{177714390890257752095272485429726298150471836709200688676211578771666} a^{22} - \frac{63330544860570302396863089769337323647749762209461127890059800787}{10453787699426926593839557966454488126498343335835334628012445810098} a^{21} + \frac{2560216821502129300757211370249217929515008392874562533108863089937}{177714390890257752095272485429726298150471836709200688676211578771666} a^{20} - \frac{8972375286768181828377881327030326903647925274256067675227180708100}{88857195445128876047636242714863149075235918354600344338105789385833} a^{19} + \frac{4916920715299796565105946037977974845643616205356033606810326511315}{177714390890257752095272485429726298150471836709200688676211578771666} a^{18} - \frac{5749021095520527844007938042398153555092433770488784018706725228024}{88857195445128876047636242714863149075235918354600344338105789385833} a^{17} + \frac{15453030078773741672004782177626171215181416654546284872470414375823}{88857195445128876047636242714863149075235918354600344338105789385833} a^{16} - \frac{7985116331172405746301555410641400740871863353588955632179466323906}{88857195445128876047636242714863149075235918354600344338105789385833} a^{15} - \frac{5399967527900717774997871964699790531202542282457548370905335042549}{88857195445128876047636242714863149075235918354600344338105789385833} a^{14} - \frac{27587053478328099272471539639353267348991001740547305109303112388169}{88857195445128876047636242714863149075235918354600344338105789385833} a^{13} - \frac{4079467035677105963546864543578142776193470169010290533454712663085}{177714390890257752095272485429726298150471836709200688676211578771666} a^{12} + \frac{33764800248208383480648424142491207834223676871379594432058571010508}{88857195445128876047636242714863149075235918354600344338105789385833} a^{11} - \frac{56143365570036973601184737374401109599588199818082655653573948630733}{177714390890257752095272485429726298150471836709200688676211578771666} a^{10} - \frac{251852163235554718413395070884366157556644248978532989928681883133}{1027250814394553480319494135431943919944923911613876813157292362842} a^{9} + \frac{67197606698220544590628196905395565342025326163444983489153342883913}{177714390890257752095272485429726298150471836709200688676211578771666} a^{8} - \frac{23807831527780878378357409401830232423769831782736244607697931182121}{88857195445128876047636242714863149075235918354600344338105789385833} a^{7} + \frac{7651968271081383289818513892253545418540506060394418071511939157451}{177714390890257752095272485429726298150471836709200688676211578771666} a^{6} - \frac{36880578454536623903163350530212730678596197922727152166874041516928}{88857195445128876047636242714863149075235918354600344338105789385833} a^{5} - \frac{21038411791850267544171934892601576742962334890006690034215312118585}{177714390890257752095272485429726298150471836709200688676211578771666} a^{4} + \frac{67019131183025100622967310517457815311491765532458479305814240560443}{177714390890257752095272485429726298150471836709200688676211578771666} a^{3} - \frac{44035632885985901314177162271904749729514074481113688685403624935738}{88857195445128876047636242714863149075235918354600344338105789385833} a^{2} + \frac{86847305764974031381060892218833170908548200512168455297166084102627}{177714390890257752095272485429726298150471836709200688676211578771666} a - \frac{31644751745387668823490023619964602863840302041780570773403296947729}{88857195445128876047636242714863149075235918354600344338105789385833}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{172}$, which has order $2752$ (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:  \( -\frac{54424713262623576692}{1799543485307059938451817809} a^{27} + \frac{61674929512899441223}{1799543485307059938451817809} a^{26} + \frac{3931314657269809523556}{1799543485307059938451817809} a^{25} - \frac{4397141440787472781208}{1799543485307059938451817809} a^{24} - \frac{119999230028648340256038}{1799543485307059938451817809} a^{23} + \frac{136758569650261394365760}{1799543485307059938451817809} a^{22} + \frac{118402245801675574266285}{105855499135709408144224577} a^{21} - \frac{4849409229364357079724153}{3599086970614119876903635618} a^{20} - \frac{20535891274890667526372492}{1799543485307059938451817809} a^{19} + \frac{26072630682495031067331919}{1799543485307059938451817809} a^{18} + \frac{133399060713770853618996896}{1799543485307059938451817809} a^{17} - \frac{349274786792918829718137131}{3599086970614119876903635618} a^{16} - \frac{564440029566411701320111048}{1799543485307059938451817809} a^{15} + \frac{737657326077574034577857265}{1799543485307059938451817809} a^{14} + \frac{1582564811669656537219319267}{1799543485307059938451817809} a^{13} - \frac{2038054431253614349236851199}{1799543485307059938451817809} a^{12} - \frac{2851404290029709322373609778}{1799543485307059938451817809} a^{11} + \frac{3670905200169903751750344969}{1799543485307059938451817809} a^{10} + \frac{2982948662239823456299290964}{1799543485307059938451817809} a^{9} - \frac{3875760454181689397325989683}{1799543485307059938451817809} a^{8} - \frac{1503209683788338117357428676}{1799543485307059938451817809} a^{7} + \frac{1238354685065364123464850147}{1799543485307059938451817809} a^{6} - \frac{534695249079615565204257206}{1799543485307059938451817809} a^{5} + \frac{3169814821531193177302232827}{3599086970614119876903635618} a^{4} + \frac{46456685648277327450147532}{1799543485307059938451817809} a^{3} + \frac{163098297261283881180002685}{3599086970614119876903635618} a^{2} - \frac{1496767698438569143182098747}{1799543485307059938451817809} a + \frac{1035343214580204313896565113}{3599086970614119876903635618} \) (order $6$)
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:  \( 2588946506907.025 \) (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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 7.7.594823321.1, 14.14.1622761392560638877063184384.1, 14.0.773792930870360792667.1, 14.0.742003380228915810271232.1

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 R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$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}$