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Properties

Label 28.0.76140026734...3125.1
Degree $28$
Signature $[0, 14]$
Discriminant $5^{21}\cdot 43^{24}$
Root discriminant $84.01$
Ramified primes $5, 43$
Class number $22709$ (GRH)
Class group $[22709]$ (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![5764801, 823543, 12353145, 7983325, 23880346, -25868717, 44171883, -40181120, 48229826, -34016343, 29065829, -18209154, 13393480, -7532046, 5128339, -2675897, 1682463, -780480, 389755, -148986, 63548, -15390, 5175, -992, 271, -2, 19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 19*x^26 - 2*x^25 + 271*x^24 - 992*x^23 + 5175*x^22 - 15390*x^21 + 63548*x^20 - 148986*x^19 + 389755*x^18 - 780480*x^17 + 1682463*x^16 - 2675897*x^15 + 5128339*x^14 - 7532046*x^13 + 13393480*x^12 - 18209154*x^11 + 29065829*x^10 - 34016343*x^9 + 48229826*x^8 - 40181120*x^7 + 44171883*x^6 - 25868717*x^5 + 23880346*x^4 + 7983325*x^3 + 12353145*x^2 + 823543*x + 5764801)
 
gp: K = bnfinit(x^28 - x^27 + 19*x^26 - 2*x^25 + 271*x^24 - 992*x^23 + 5175*x^22 - 15390*x^21 + 63548*x^20 - 148986*x^19 + 389755*x^18 - 780480*x^17 + 1682463*x^16 - 2675897*x^15 + 5128339*x^14 - 7532046*x^13 + 13393480*x^12 - 18209154*x^11 + 29065829*x^10 - 34016343*x^9 + 48229826*x^8 - 40181120*x^7 + 44171883*x^6 - 25868717*x^5 + 23880346*x^4 + 7983325*x^3 + 12353145*x^2 + 823543*x + 5764801, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 19 x^{26} - 2 x^{25} + 271 x^{24} - 992 x^{23} + 5175 x^{22} - 15390 x^{21} + 63548 x^{20} - 148986 x^{19} + 389755 x^{18} - 780480 x^{17} + 1682463 x^{16} - 2675897 x^{15} + 5128339 x^{14} - 7532046 x^{13} + 13393480 x^{12} - 18209154 x^{11} + 29065829 x^{10} - 34016343 x^{9} + 48229826 x^{8} - 40181120 x^{7} + 44171883 x^{6} - 25868717 x^{5} + 23880346 x^{4} + 7983325 x^{3} + 12353145 x^{2} + 823543 x + 5764801 \)

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:  \(761400267340438731808185095577412659073352813720703125=5^{21}\cdot 43^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.01$
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:  $5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(215=5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{215}(64,·)$, $\chi_{215}(1,·)$, $\chi_{215}(4,·)$, $\chi_{215}(133,·)$, $\chi_{215}(193,·)$, $\chi_{215}(11,·)$, $\chi_{215}(78,·)$, $\chi_{215}(207,·)$, $\chi_{215}(16,·)$, $\chi_{215}(84,·)$, $\chi_{215}(21,·)$, $\chi_{215}(87,·)$, $\chi_{215}(127,·)$, $\chi_{215}(97,·)$, $\chi_{215}(164,·)$, $\chi_{215}(102,·)$, $\chi_{215}(41,·)$, $\chi_{215}(107,·)$, $\chi_{215}(44,·)$, $\chi_{215}(173,·)$, $\chi_{215}(47,·)$, $\chi_{215}(176,·)$, $\chi_{215}(54,·)$, $\chi_{215}(183,·)$, $\chi_{215}(121,·)$, $\chi_{215}(59,·)$, $\chi_{215}(188,·)$, $\chi_{215}(213,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{9} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{18} + \frac{2}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{19} + \frac{2}{7} a^{16} + \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{2}{7} a^{11} + \frac{1}{7} a^{8} + \frac{2}{7} a^{6} + \frac{1}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{21} - \frac{2}{7} a^{16} - \frac{2}{7} a^{11} + \frac{2}{7} a^{6} + \frac{1}{7} a$, $\frac{1}{49} a^{22} - \frac{1}{49} a^{21} - \frac{2}{49} a^{20} - \frac{2}{49} a^{19} - \frac{2}{49} a^{18} + \frac{2}{49} a^{17} + \frac{16}{49} a^{16} - \frac{11}{49} a^{15} + \frac{23}{49} a^{14} - \frac{19}{49} a^{13} + \frac{2}{49} a^{12} - \frac{15}{49} a^{11} - \frac{22}{49} a^{10} - \frac{2}{7} a^{9} - \frac{15}{49} a^{8} + \frac{17}{49} a^{7} - \frac{12}{49} a^{6} - \frac{19}{49} a^{5} - \frac{12}{49} a^{4} - \frac{18}{49} a^{3} + \frac{22}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{343} a^{23} - \frac{1}{343} a^{22} + \frac{19}{343} a^{21} - \frac{2}{343} a^{20} - \frac{23}{343} a^{19} - \frac{12}{343} a^{18} - \frac{19}{343} a^{17} + \frac{143}{343} a^{16} - \frac{5}{343} a^{15} + \frac{121}{343} a^{14} - \frac{89}{343} a^{13} - \frac{106}{343} a^{12} + \frac{146}{343} a^{11} + \frac{20}{49} a^{10} - \frac{50}{343} a^{9} - \frac{109}{343} a^{8} - \frac{82}{343} a^{7} - \frac{117}{343} a^{6} - \frac{40}{343} a^{5} + \frac{45}{343} a^{4} - \frac{41}{343} a^{3} - \frac{20}{49} a^{2} + \frac{2}{7} a$, $\frac{1}{2401} a^{24} - \frac{1}{2401} a^{23} + \frac{19}{2401} a^{22} - \frac{2}{2401} a^{21} - \frac{72}{2401} a^{20} + \frac{37}{2401} a^{19} + \frac{30}{2401} a^{18} + \frac{45}{2401} a^{17} + \frac{779}{2401} a^{16} - \frac{810}{2401} a^{15} + \frac{793}{2401} a^{14} - \frac{841}{2401} a^{13} - \frac{638}{2401} a^{12} + \frac{76}{343} a^{11} + \frac{489}{2401} a^{10} - \frac{452}{2401} a^{9} - \frac{1013}{2401} a^{8} - \frac{313}{2401} a^{7} + \frac{9}{2401} a^{6} - \frac{4}{2401} a^{5} - \frac{1119}{2401} a^{4} + \frac{141}{343} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{320887207760476860325632884160500827} a^{25} + \frac{5792472772853199415830543488176}{320887207760476860325632884160500827} a^{24} + \frac{302593369127198068353498616130661}{320887207760476860325632884160500827} a^{23} - \frac{198328859215840478868548833343458}{320887207760476860325632884160500827} a^{22} + \frac{5847746050555267688785592945781478}{320887207760476860325632884160500827} a^{21} - \frac{19239839362248304295279408242056129}{320887207760476860325632884160500827} a^{20} - \frac{19189030073164701350914177986794636}{320887207760476860325632884160500827} a^{19} + \frac{19134321094169650972481484212700474}{320887207760476860325632884160500827} a^{18} - \frac{3645909708658949303468527353292925}{320887207760476860325632884160500827} a^{17} - \frac{92706620187484322378220184306530496}{320887207760476860325632884160500827} a^{16} + \frac{33358705879291104754654714438069821}{320887207760476860325632884160500827} a^{15} - \frac{22377808846492225035973194846091504}{320887207760476860325632884160500827} a^{14} - \frac{132480288712255777754064914454258878}{320887207760476860325632884160500827} a^{13} - \frac{4843791847760661378554254090208185}{45841029680068122903661840594357261} a^{12} - \frac{77222252022606994550130225457457819}{320887207760476860325632884160500827} a^{11} - \frac{42443669938177505114506298625812940}{320887207760476860325632884160500827} a^{10} - \frac{110054870839308916496447221045528986}{320887207760476860325632884160500827} a^{9} + \frac{100232819614447999531257741892227300}{320887207760476860325632884160500827} a^{8} - \frac{71476323505877978590187050094710617}{320887207760476860325632884160500827} a^{7} - \frac{53381030930438114975244793940123906}{320887207760476860325632884160500827} a^{6} - \frac{151304491963610484894280843860627829}{320887207760476860325632884160500827} a^{5} + \frac{10933210417319435646798821290093487}{45841029680068122903661840594357261} a^{4} - \frac{507057930334685146751878066203259}{6548718525724017557665977227765323} a^{3} + \frac{71614869223860654885257769399864}{935531217960573936809425318252189} a^{2} - \frac{62507477022987791731505622913906}{133647316851510562401346474036027} a - \frac{3217005266627796929861361788769}{19092473835930080343049496290861}$, $\frac{1}{2246210454323338022279430189123505789} a^{26} - \frac{1}{2246210454323338022279430189123505789} a^{25} + \frac{411396109309202883291800103302718}{2246210454323338022279430189123505789} a^{24} + \frac{1068534141715078403877644532052973}{2246210454323338022279430189123505789} a^{23} + \frac{6336595825850573495374757327395878}{2246210454323338022279430189123505789} a^{22} + \frac{576199776060382026291595786613983}{8949045634754334750117251749496039} a^{21} - \frac{8801776119567531111815755850633061}{2246210454323338022279430189123505789} a^{20} + \frac{83655305945620845548367634474347451}{2246210454323338022279430189123505789} a^{19} + \frac{24642186392744728875453858553283197}{2246210454323338022279430189123505789} a^{18} - \frac{155721486737518210867955311815552604}{2246210454323338022279430189123505789} a^{17} - \frac{28564408615270797324025912646264383}{2246210454323338022279430189123505789} a^{16} - \frac{384931640837297457229526462019493919}{2246210454323338022279430189123505789} a^{15} - \frac{217610839449868190195769481585678872}{2246210454323338022279430189123505789} a^{14} - \frac{118432833688714449077352930045939953}{320887207760476860325632884160500827} a^{13} - \frac{487671193081274489903758286725380332}{2246210454323338022279430189123505789} a^{12} + \frac{32132726760370752157516607991984764}{2246210454323338022279430189123505789} a^{11} + \frac{799231646060740411392299640417312783}{2246210454323338022279430189123505789} a^{10} - \frac{31697579365072967114524735823103629}{2246210454323338022279430189123505789} a^{9} + \frac{842730640324059011359223039060600409}{2246210454323338022279430189123505789} a^{8} + \frac{922560224948034621199037189876727026}{2246210454323338022279430189123505789} a^{7} - \frac{987117841574702339028902325908135324}{2246210454323338022279430189123505789} a^{6} - \frac{78022086908727972483490274226962075}{320887207760476860325632884160500827} a^{5} + \frac{2247360063989227480840099503595053}{6548718525724017557665977227765323} a^{4} + \frac{2824009315033637988051424949685313}{6548718525724017557665977227765323} a^{3} + \frac{176419081769458401589003548810714}{935531217960573936809425318252189} a^{2} + \frac{22083814453573178810017743454354}{133647316851510562401346474036027} a - \frac{4390614379171897578832673807223}{19092473835930080343049496290861}$, $\frac{1}{15723473180263366155956011323864540523} a^{27} - \frac{1}{15723473180263366155956011323864540523} a^{26} + \frac{19}{15723473180263366155956011323864540523} a^{25} + \frac{1525769005818137881696804550777595}{15723473180263366155956011323864540523} a^{24} - \frac{3554394112741401955237286720160846}{15723473180263366155956011323864540523} a^{23} + \frac{148348497457780833735334463209115188}{15723473180263366155956011323864540523} a^{22} - \frac{158925676283491243070217464894795216}{15723473180263366155956011323864540523} a^{21} + \frac{400605160033169920089024005400182671}{15723473180263366155956011323864540523} a^{20} - \frac{865798968024975584216924973125197043}{15723473180263366155956011323864540523} a^{19} - \frac{1036087313199773056468488686127753487}{15723473180263366155956011323864540523} a^{18} + \frac{42781563161629083061850270167519688}{15723473180263366155956011323864540523} a^{17} + \frac{513982371975584727026179332968221595}{15723473180263366155956011323864540523} a^{16} - \frac{448622676969809786331532089799446058}{15723473180263366155956011323864540523} a^{15} + \frac{199318458705994137315493304410785718}{2246210454323338022279430189123505789} a^{14} + \frac{279477669460051737580301691967046241}{15723473180263366155956011323864540523} a^{13} - \frac{4832651100261352155115574993354606450}{15723473180263366155956011323864540523} a^{12} - \frac{1823457264137419737303301428862645189}{15723473180263366155956011323864540523} a^{11} - \frac{136288919345092181722045961799635692}{15723473180263366155956011323864540523} a^{10} - \frac{4526616763410337280250659992611102290}{15723473180263366155956011323864540523} a^{9} + \frac{3640237787401223115733695738030315509}{15723473180263366155956011323864540523} a^{8} - \frac{3197543633026794224012440981146754252}{15723473180263366155956011323864540523} a^{7} + \frac{946865066663884266072543605892106270}{2246210454323338022279430189123505789} a^{6} + \frac{15038977162219729051051111665043643}{45841029680068122903661840594357261} a^{5} - \frac{16195721652047510684048585302886187}{45841029680068122903661840594357261} a^{4} - \frac{539388579687442073708115676398050}{6548718525724017557665977227765323} a^{3} + \frac{300729858570584326414143542501835}{935531217960573936809425318252189} a^{2} - \frac{56492179864951704278705055241792}{133647316851510562401346474036027} a - \frac{8130805896423644528069900659429}{19092473835930080343049496290861}$

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

Class group and class number

$C_{22709}$, which has order $22709$ (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{70220814453860710580553707086}{2246210454323338022279430189123505789} a^{27} - \frac{120659019081760610783289161490}{2246210454323338022279430189123505789} a^{26} - \frac{1154947606149024845074896498125}{2246210454323338022279430189123505789} a^{25} - \frac{3407557417182082902908974628068}{2246210454323338022279430189123505789} a^{24} - \frac{18981794896580453133774938925984}{2246210454323338022279430189123505789} a^{23} + \frac{19323659388000564487917108289429}{2246210454323338022279430189123505789} a^{22} - \frac{178096248356299103221707191089670}{2246210454323338022279430189123505789} a^{21} + \frac{17645751505787261192992298656953}{320887207760476860325632884160500827} a^{20} - \frac{1661455884553231665811686746839509}{2246210454323338022279430189123505789} a^{19} - \frac{1083283569166520313047937200374943}{2246210454323338022279430189123505789} a^{18} - \frac{951846835754421608798909739219524}{2246210454323338022279430189123505789} a^{17} - \frac{12781578253908014304081934558858070}{2246210454323338022279430189123505789} a^{16} + \frac{13319275271081107839227437618785619}{2246210454323338022279430189123505789} a^{15} - \frac{90050268824623924251463287368066267}{2246210454323338022279430189123505789} a^{14} + \frac{61712352397684241315681986981001065}{2246210454323338022279430189123505789} a^{13} - \frac{269717301971673203334500318033172174}{2246210454323338022279430189123505789} a^{12} + \frac{187014368528249038375671844588167646}{2246210454323338022279430189123505789} a^{11} - \frac{733550900163919810630608118245160595}{2246210454323338022279430189123505789} a^{10} + \frac{603207698864065550681493850891682300}{2246210454323338022279430189123505789} a^{9} - \frac{249859539373104234605028014007978264}{320887207760476860325632884160500827} a^{8} + \frac{22976647791279739766503878722934555}{45841029680068122903661840594357261} a^{7} - \frac{3441128838597687257967575498670369091}{2246210454323338022279430189123505789} a^{6} + \frac{381663061969921831083574236853189}{935531217960573936809425318252189} a^{5} - \frac{106247788101030931987881998489912}{133647316851510562401346474036027} a^{4} - \frac{7777879158849992916672382976973}{19092473835930080343049496290861} a^{3} - \frac{6933381469233825949953618657544}{19092473835930080343049496290861} a^{2} - \frac{30684604813553860367287072480428}{19092473835930080343049496290861} a - \frac{3169176231272924174885515990855}{19092473835930080343049496290861} \) (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:  \( 263819853122.8475 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 7.7.6321363049.1, 14.14.3121846156036138781328125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
43Data not computed