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Properties

Label 28.0.17501529797...3125.1
Degree $28$
Signature $[0, 14]$
Discriminant $5^{21}\cdot 7^{48}$
Root discriminant $93.97$
Ramified primes $5, 7$
Class number $43793$ (GRH)
Class group $[43793]$ (GRH)
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![88529281, 76664532, 168609280, 151478789, 234771859, 114584232, 208289837, 95224249, 196642418, 109113760, 103699603, 53367559, 40448583, 16164904, 10825963, 4058887, 2521988, 902447, 491337, 149541, 74431, 15191, 5964, 875, 350, -21, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 21*x^26 - 21*x^25 + 350*x^24 + 875*x^23 + 5964*x^22 + 15191*x^21 + 74431*x^20 + 149541*x^19 + 491337*x^18 + 902447*x^17 + 2521988*x^16 + 4058887*x^15 + 10825963*x^14 + 16164904*x^13 + 40448583*x^12 + 53367559*x^11 + 103699603*x^10 + 109113760*x^9 + 196642418*x^8 + 95224249*x^7 + 208289837*x^6 + 114584232*x^5 + 234771859*x^4 + 151478789*x^3 + 168609280*x^2 + 76664532*x + 88529281)
 
gp: K = bnfinit(x^28 + 21*x^26 - 21*x^25 + 350*x^24 + 875*x^23 + 5964*x^22 + 15191*x^21 + 74431*x^20 + 149541*x^19 + 491337*x^18 + 902447*x^17 + 2521988*x^16 + 4058887*x^15 + 10825963*x^14 + 16164904*x^13 + 40448583*x^12 + 53367559*x^11 + 103699603*x^10 + 109113760*x^9 + 196642418*x^8 + 95224249*x^7 + 208289837*x^6 + 114584232*x^5 + 234771859*x^4 + 151478789*x^3 + 168609280*x^2 + 76664532*x + 88529281, 1)
 

Normalized defining polynomial

\( x^{28} + 21 x^{26} - 21 x^{25} + 350 x^{24} + 875 x^{23} + 5964 x^{22} + 15191 x^{21} + 74431 x^{20} + 149541 x^{19} + 491337 x^{18} + 902447 x^{17} + 2521988 x^{16} + 4058887 x^{15} + 10825963 x^{14} + 16164904 x^{13} + 40448583 x^{12} + 53367559 x^{11} + 103699603 x^{10} + 109113760 x^{9} + 196642418 x^{8} + 95224249 x^{7} + 208289837 x^{6} + 114584232 x^{5} + 234771859 x^{4} + 151478789 x^{3} + 168609280 x^{2} + 76664532 x + 88529281 \)

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:  \(17501529797217428894629579082505064100647449493408203125=5^{21}\cdot 7^{48}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.97$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(64,·)$, $\chi_{245}(1,·)$, $\chi_{245}(197,·)$, $\chi_{245}(134,·)$, $\chi_{245}(71,·)$, $\chi_{245}(8,·)$, $\chi_{245}(204,·)$, $\chi_{245}(141,·)$, $\chi_{245}(78,·)$, $\chi_{245}(211,·)$, $\chi_{245}(148,·)$, $\chi_{245}(22,·)$, $\chi_{245}(218,·)$, $\chi_{245}(92,·)$, $\chi_{245}(29,·)$, $\chi_{245}(162,·)$, $\chi_{245}(99,·)$, $\chi_{245}(36,·)$, $\chi_{245}(232,·)$, $\chi_{245}(169,·)$, $\chi_{245}(106,·)$, $\chi_{245}(43,·)$, $\chi_{245}(239,·)$, $\chi_{245}(176,·)$, $\chi_{245}(113,·)$, $\chi_{245}(183,·)$, $\chi_{245}(57,·)$, $\chi_{245}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{31} a^{21} + \frac{1}{31} a^{20} - \frac{2}{31} a^{19} - \frac{14}{31} a^{18} + \frac{1}{31} a^{17} - \frac{13}{31} a^{16} + \frac{12}{31} a^{15} + \frac{7}{31} a^{14} + \frac{13}{31} a^{13} - \frac{9}{31} a^{12} + \frac{13}{31} a^{11} - \frac{13}{31} a^{10} + \frac{15}{31} a^{9} - \frac{8}{31} a^{8} - \frac{12}{31} a^{7} - \frac{15}{31} a^{6} - \frac{9}{31} a^{5} + \frac{8}{31} a^{3} + \frac{5}{31} a^{2} + \frac{6}{31} a + \frac{12}{31}$, $\frac{1}{31} a^{22} - \frac{3}{31} a^{20} - \frac{12}{31} a^{19} + \frac{15}{31} a^{18} - \frac{14}{31} a^{17} - \frac{6}{31} a^{16} - \frac{5}{31} a^{15} + \frac{6}{31} a^{14} + \frac{9}{31} a^{13} - \frac{9}{31} a^{12} + \frac{5}{31} a^{11} - \frac{3}{31} a^{10} + \frac{8}{31} a^{9} - \frac{4}{31} a^{8} - \frac{3}{31} a^{7} + \frac{6}{31} a^{6} + \frac{9}{31} a^{5} + \frac{8}{31} a^{4} - \frac{3}{31} a^{3} + \frac{1}{31} a^{2} + \frac{6}{31} a - \frac{12}{31}$, $\frac{1}{31} a^{23} - \frac{9}{31} a^{20} + \frac{9}{31} a^{19} + \frac{6}{31} a^{18} - \frac{3}{31} a^{17} - \frac{13}{31} a^{16} + \frac{11}{31} a^{15} - \frac{1}{31} a^{14} - \frac{1}{31} a^{13} + \frac{9}{31} a^{12} + \frac{5}{31} a^{11} + \frac{10}{31} a^{9} + \frac{4}{31} a^{8} + \frac{1}{31} a^{7} - \frac{5}{31} a^{6} + \frac{12}{31} a^{5} - \frac{3}{31} a^{4} - \frac{6}{31} a^{3} - \frac{10}{31} a^{2} + \frac{6}{31} a + \frac{5}{31}$, $\frac{1}{589} a^{24} - \frac{8}{589} a^{23} + \frac{2}{589} a^{22} + \frac{1}{589} a^{21} + \frac{271}{589} a^{20} - \frac{203}{589} a^{19} - \frac{68}{589} a^{18} - \frac{7}{589} a^{17} + \frac{35}{589} a^{16} + \frac{83}{589} a^{15} - \frac{10}{31} a^{14} + \frac{10}{589} a^{13} + \frac{135}{589} a^{12} + \frac{286}{589} a^{11} - \frac{2}{589} a^{10} - \frac{34}{589} a^{9} - \frac{119}{589} a^{8} + \frac{16}{589} a^{7} - \frac{241}{589} a^{6} + \frac{294}{589} a^{5} + \frac{127}{589} a^{4} - \frac{136}{589} a^{3} - \frac{48}{589} a^{2} - \frac{250}{589} a - \frac{192}{589}$, $\frac{1}{23472132450001537584856290962665266050293} a^{25} + \frac{113580954103054582842954307610571894}{241980746907232346235631865594487278869} a^{24} + \frac{180221570835609074273380905855466732366}{23472132450001537584856290962665266050293} a^{23} + \frac{231364403507922185251097924602734948057}{23472132450001537584856290962665266050293} a^{22} - \frac{232539230476508655325629702067815189498}{23472132450001537584856290962665266050293} a^{21} - \frac{6367332912905855649891534095948203101023}{23472132450001537584856290962665266050293} a^{20} - \frac{4068602823604866675093794010094424206505}{23472132450001537584856290962665266050293} a^{19} - \frac{1349409721218772106013184247583674568183}{23472132450001537584856290962665266050293} a^{18} - \frac{7943518455401825401779003555168142471787}{23472132450001537584856290962665266050293} a^{17} - \frac{52328954186228715650094739311626224428}{757165562903275405963106160085976324203} a^{16} - \frac{393367798075998593114179373812215389034}{23472132450001537584856290962665266050293} a^{15} + \frac{1559615342344061825183185580128069076985}{23472132450001537584856290962665266050293} a^{14} - \frac{2788660799411082389694304324183748851404}{23472132450001537584856290962665266050293} a^{13} + \frac{2761000110490917740317304962116869266530}{23472132450001537584856290962665266050293} a^{12} - \frac{2771722546526333471881255514586463785794}{23472132450001537584856290962665266050293} a^{11} - \frac{7327168411096208730014484366869336954668}{23472132450001537584856290962665266050293} a^{10} - \frac{1244852992746365644432223340122818866290}{23472132450001537584856290962665266050293} a^{9} + \frac{5357089175817639192359951240563330051403}{23472132450001537584856290962665266050293} a^{8} - \frac{1480529313699068726262457872371620424738}{23472132450001537584856290962665266050293} a^{7} + \frac{7788901848180873546474510977104447378800}{23472132450001537584856290962665266050293} a^{6} - \frac{4169190127786255041041231261966125544885}{23472132450001537584856290962665266050293} a^{5} + \frac{7277594629148578587075641426319351367582}{23472132450001537584856290962665266050293} a^{4} + \frac{3667424222610538076380293976780214428395}{23472132450001537584856290962665266050293} a^{3} + \frac{6935756991960088542608986919097768619212}{23472132450001537584856290962665266050293} a^{2} - \frac{616171393793490575325104037962972429191}{1235375392105344083413488998035014002647} a - \frac{120614363161395438944886625939326858551}{241980746907232346235631865594487278869}$, $\frac{1}{2276796847650149145731060223378530806878421} a^{26} + \frac{1318110015397596925672937275238850883839}{2276796847650149145731060223378530806878421} a^{24} - \frac{14322036696594052042214180972566058554481}{2276796847650149145731060223378530806878421} a^{23} + \frac{27680310323349535439131682780015868560528}{2276796847650149145731060223378530806878421} a^{22} + \frac{21743145858461319705324684380229432972697}{2276796847650149145731060223378530806878421} a^{21} + \frac{908991395220869445628868441814164445881306}{2276796847650149145731060223378530806878421} a^{20} - \frac{281169857874427620881912682718916527037762}{2276796847650149145731060223378530806878421} a^{19} + \frac{465548836966025344009798841332944221671176}{2276796847650149145731060223378530806878421} a^{18} - \frac{157960261761897191022805082663680131682875}{2276796847650149145731060223378530806878421} a^{17} + \frac{550175277977930492603963027096159058944110}{2276796847650149145731060223378530806878421} a^{16} - \frac{251715854534457765142031108913417338982793}{2276796847650149145731060223378530806878421} a^{15} + \frac{528280963687061700611672929271585052855036}{2276796847650149145731060223378530806878421} a^{14} + \frac{346298567141023600474051074315759219085414}{2276796847650149145731060223378530806878421} a^{13} - \frac{935565845257687338435808590145096788908850}{2276796847650149145731060223378530806878421} a^{12} + \frac{561251499465029338933259972968445483971408}{2276796847650149145731060223378530806878421} a^{11} - \frac{518793945798705208815859805370390288509561}{2276796847650149145731060223378530806878421} a^{10} - \frac{780434946707128105734051836498994259883266}{2276796847650149145731060223378530806878421} a^{9} + \frac{461560339601062158929666681577491666674972}{2276796847650149145731060223378530806878421} a^{8} + \frac{914648682607545701684641906760956835351252}{2276796847650149145731060223378530806878421} a^{7} + \frac{492276609668974481590518488219961107402019}{2276796847650149145731060223378530806878421} a^{6} + \frac{1095637152780760158526796580760865981670888}{2276796847650149145731060223378530806878421} a^{5} - \frac{501469197022247489996176123752320174689174}{2276796847650149145731060223378530806878421} a^{4} - \frac{141182080269301087155997717877993102686255}{2276796847650149145731060223378530806878421} a^{3} - \frac{307101723273170138339359898744840508014038}{2276796847650149145731060223378530806878421} a^{2} - \frac{6701940930908063211264242545010358135703}{23472132450001537584856290962665266050293} a - \frac{80294124981845837949023536772098227879}{241980746907232346235631865594487278869}$, $\frac{1}{220849294222064467135912841667717488267206837} a^{27} + \frac{21}{220849294222064467135912841667717488267206837} a^{25} + \frac{173316448091992663175637388144375554266518}{220849294222064467135912841667717488267206837} a^{24} + \frac{1570569965431035811591666412066044023448233}{220849294222064467135912841667717488267206837} a^{23} + \frac{1272364431482276442717751310067450143209287}{220849294222064467135912841667717488267206837} a^{22} + \frac{845640738692232937909146061359232915110080}{220849294222064467135912841667717488267206837} a^{21} + \frac{42083424042916985557026573279131914186565494}{220849294222064467135912841667717488267206837} a^{20} - \frac{16264693624816315729323661458053793702263533}{220849294222064467135912841667717488267206837} a^{19} + \frac{38543606748657952626205514384684915880032293}{220849294222064467135912841667717488267206837} a^{18} + \frac{1907328686084311144375676768301767948835080}{7124170781356918294706865860248951234426027} a^{17} - \frac{7677344191909281261455012184383607233039205}{220849294222064467135912841667717488267206837} a^{16} + \frac{11939028150701666009402108820086131083850772}{220849294222064467135912841667717488267206837} a^{15} + \frac{5758645277150911746675579270392558644850873}{220849294222064467135912841667717488267206837} a^{14} - \frac{82610414537746230617266146673953572630976597}{220849294222064467135912841667717488267206837} a^{13} - \frac{41763884842868019087452786612542565659838460}{220849294222064467135912841667717488267206837} a^{12} + \frac{47584524162539649883124533804910257149005753}{220849294222064467135912841667717488267206837} a^{11} - \frac{7369999230041096361710562473776423897443062}{220849294222064467135912841667717488267206837} a^{10} + \frac{51508593478533745027606491621004051952011352}{220849294222064467135912841667717488267206837} a^{9} + \frac{83958118167710066174332104667247685575912}{220849294222064467135912841667717488267206837} a^{8} - \frac{97739825903334182302973701529985363579421738}{220849294222064467135912841667717488267206837} a^{7} - \frac{99753157743235171858763881812396284888480092}{220849294222064467135912841667717488267206837} a^{6} + \frac{103594423236898442270034809533863013384627815}{220849294222064467135912841667717488267206837} a^{5} + \frac{6852841666803734857316042148858135826380869}{220849294222064467135912841667717488267206837} a^{4} - \frac{647320186324215746549063963111474388105658}{7124170781356918294706865860248951234426027} a^{3} + \frac{1103321436681035352071421759409832937890592}{2276796847650149145731060223378530806878421} a^{2} - \frac{4101515892574461869797068762628572227991}{23472132450001537584856290962665266050293} a + \frac{18134320852455741760247069128184841445}{241980746907232346235631865594487278869}$

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

Class group and class number

$C_{43793}$, which has order $43793$ (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{290106923447510490333593123640811}{119831413034218376091108432809396358256759} a^{27} - \frac{222843898368442712553926096345220}{119831413034218376091108432809396358256759} a^{26} - \frac{5982297509771884723917896991172275}{119831413034218376091108432809396358256759} a^{25} + \frac{1559907288579098987877482674416540}{119831413034218376091108432809396358256759} a^{24} - \frac{93910132837434566448767022434814795}{119831413034218376091108432809396358256759} a^{23} - \frac{331865355846985373095135521073090029}{119831413034218376091108432809396358256759} a^{22} - \frac{19350278508326442206765916032643270}{1235375392105344083413488998035014002647} a^{21} - \frac{5617504701046296008737507518194392065}{119831413034218376091108432809396358256759} a^{20} - \frac{23965821331333627669182833340145466640}{119831413034218376091108432809396358256759} a^{19} - \frac{57109768395072945050452905739803188145}{119831413034218376091108432809396358256759} a^{18} - \frac{162585524651539498303083618489133783348}{119831413034218376091108432809396358256759} a^{17} - \frac{339149564437032811060971999055108973475}{119831413034218376091108432809396358256759} a^{16} - \frac{824014432726582307249367169113520375575}{119831413034218376091108432809396358256759} a^{15} - \frac{1535779069756829357581526167433682954285}{119831413034218376091108432809396358256759} a^{14} - \frac{3472177280572165452682075426400310189240}{119831413034218376091108432809396358256759} a^{13} - \frac{6122875332542490493865343529440531814861}{119831413034218376091108432809396358256759} a^{12} - \frac{12834680587142962194248036414440410520305}{119831413034218376091108432809396358256759} a^{11} - \frac{668091132718253726419059177143185500060}{3865529452716721809390594606754721234089} a^{10} - \frac{32295699566789186643433762488879752899905}{119831413034218376091108432809396358256759} a^{9} - \frac{41796109594368492237215737700170257040545}{119831413034218376091108432809396358256759} a^{8} - \frac{52576199986282719317765056168875721954752}{119831413034218376091108432809396358256759} a^{7} - \frac{44035663530304765956261085250974439837400}{119831413034218376091108432809396358256759} a^{6} - \frac{32505985078694750902884819367238263331370}{119831413034218376091108432809396358256759} a^{5} - \frac{55957769272500218160826806013525601406495}{119831413034218376091108432809396358256759} a^{4} - \frac{429808924766367991366573198905138480855}{1235375392105344083413488998035014002647} a^{3} - \frac{180884074966070294093427641726119682189642}{119831413034218376091108432809396358256759} a^{2} - \frac{2491227650835413194318478312567300685}{12735828784591176117664835031288804151} a - \frac{2271466426394400605954881527221522895}{12735828784591176117664835031288804151} \) (order $10$)
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:  \( 1494941023922.2002 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 7.7.13841287201.1, 14.14.14967283701606751125078125.1

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 $28$ $28$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
5Data not computed
7Data not computed