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Properties

Label 28.0.18020212407...0625.1
Degree $28$
Signature $[0, 14]$
Discriminant $5^{14}\cdot 43^{26}$
Root discriminant $73.50$
Ramified primes $5, 43$
Class number $1421$ (GRH)
Class group $[1421]$ (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![6241, -13272, 27471, 67288, 189716, -51405, -20587, 128151, 994183, 680690, 173519, -846630, -369218, -199373, 217434, -52161, 7013, -13267, 38446, 2322, 2790, -152, 2085, 277, -43, 98, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 5*x^26 + 98*x^25 - 43*x^24 + 277*x^23 + 2085*x^22 - 152*x^21 + 2790*x^20 + 2322*x^19 + 38446*x^18 - 13267*x^17 + 7013*x^16 - 52161*x^15 + 217434*x^14 - 199373*x^13 - 369218*x^12 - 846630*x^11 + 173519*x^10 + 680690*x^9 + 994183*x^8 + 128151*x^7 - 20587*x^6 - 51405*x^5 + 189716*x^4 + 67288*x^3 + 27471*x^2 - 13272*x + 6241)
 
gp: K = bnfinit(x^28 - x^27 + 5*x^26 + 98*x^25 - 43*x^24 + 277*x^23 + 2085*x^22 - 152*x^21 + 2790*x^20 + 2322*x^19 + 38446*x^18 - 13267*x^17 + 7013*x^16 - 52161*x^15 + 217434*x^14 - 199373*x^13 - 369218*x^12 - 846630*x^11 + 173519*x^10 + 680690*x^9 + 994183*x^8 + 128151*x^7 - 20587*x^6 - 51405*x^5 + 189716*x^4 + 67288*x^3 + 27471*x^2 - 13272*x + 6241, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 5 x^{26} + 98 x^{25} - 43 x^{24} + 277 x^{23} + 2085 x^{22} - 152 x^{21} + 2790 x^{20} + 2322 x^{19} + 38446 x^{18} - 13267 x^{17} + 7013 x^{16} - 52161 x^{15} + 217434 x^{14} - 199373 x^{13} - 369218 x^{12} - 846630 x^{11} + 173519 x^{10} + 680690 x^{9} + 994183 x^{8} + 128151 x^{7} - 20587 x^{6} - 51405 x^{5} + 189716 x^{4} + 67288 x^{3} + 27471 x^{2} - 13272 x + 6241 \)

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:  \(18020212407199631553450678294049740884820855712890625=5^{14}\cdot 43^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.50$
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$$\chi_{215}(64,·)$, $\chi_{215}(1,·)$, $\chi_{215}(194,·)$, $\chi_{215}(131,·)$, $\chi_{215}(4,·)$, $\chi_{215}(199,·)$, $\chi_{215}(11,·)$, $\chi_{215}(204,·)$, $\chi_{215}(16,·)$, $\chi_{215}(211,·)$, $\chi_{215}(84,·)$, $\chi_{215}(21,·)$, $\chi_{215}(214,·)$, $\chi_{215}(151,·)$, $\chi_{215}(156,·)$, $\chi_{215}(94,·)$, $\chi_{215}(161,·)$, $\chi_{215}(164,·)$, $\chi_{215}(39,·)$, $\chi_{215}(41,·)$, $\chi_{215}(171,·)$, $\chi_{215}(44,·)$, $\chi_{215}(174,·)$, $\chi_{215}(176,·)$, $\chi_{215}(51,·)$, $\chi_{215}(54,·)$, $\chi_{215}(121,·)$, $\chi_{215}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{7} a^{23} - \frac{3}{7} a^{22} + \frac{3}{7} a^{21} + \frac{3}{7} a^{20} + \frac{1}{7} a^{19} - \frac{3}{7} a^{18} - \frac{1}{7} a^{17} + \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{553} a^{24} - \frac{18}{553} a^{23} - \frac{1}{553} a^{22} + \frac{9}{79} a^{21} - \frac{135}{553} a^{20} + \frac{276}{553} a^{19} + \frac{9}{553} a^{18} + \frac{232}{553} a^{17} - \frac{37}{79} a^{16} + \frac{13}{79} a^{15} + \frac{213}{553} a^{14} + \frac{258}{553} a^{13} - \frac{73}{553} a^{12} + \frac{113}{553} a^{11} - \frac{187}{553} a^{10} - \frac{18}{553} a^{9} + \frac{29}{553} a^{8} - \frac{184}{553} a^{7} - \frac{4}{553} a^{6} - \frac{141}{553} a^{5} + \frac{12}{553} a^{4} + \frac{271}{553} a^{3} + \frac{242}{553} a - \frac{3}{7}$, $\frac{1}{553} a^{25} - \frac{9}{553} a^{23} + \frac{29}{79} a^{22} - \frac{265}{553} a^{21} - \frac{100}{553} a^{20} - \frac{3}{7} a^{19} - \frac{1}{553} a^{18} - \frac{270}{553} a^{17} - \frac{21}{79} a^{16} + \frac{192}{553} a^{15} + \frac{221}{553} a^{14} - \frac{11}{553} a^{13} - \frac{16}{553} a^{12} - \frac{128}{553} a^{11} - \frac{66}{553} a^{10} - \frac{216}{553} a^{9} + \frac{37}{79} a^{8} + \frac{239}{553} a^{7} - \frac{55}{553} a^{6} + \frac{239}{553} a^{5} + \frac{171}{553} a^{4} + \frac{59}{553} a^{3} - \frac{232}{553} a^{2} - \frac{226}{553} a - \frac{2}{7}$, $\frac{1}{12208639947162743} a^{26} + \frac{5562754799298}{12208639947162743} a^{25} - \frac{2219957351104}{12208639947162743} a^{24} + \frac{348422260258456}{12208639947162743} a^{23} + \frac{122692935896474}{1744091421023249} a^{22} - \frac{3221572792559999}{12208639947162743} a^{21} - \frac{433570732839492}{12208639947162743} a^{20} - \frac{3085318733222289}{12208639947162743} a^{19} - \frac{5268641230823780}{12208639947162743} a^{18} - \frac{6006053287712190}{12208639947162743} a^{17} - \frac{144061424137355}{12208639947162743} a^{16} - \frac{1719833418254509}{12208639947162743} a^{15} - \frac{2933367870771301}{12208639947162743} a^{14} + \frac{1443138597646920}{12208639947162743} a^{13} + \frac{2708406483601923}{12208639947162743} a^{12} - \frac{4398651889001861}{12208639947162743} a^{11} - \frac{5905686164691942}{12208639947162743} a^{10} - \frac{810227257344289}{1744091421023249} a^{9} + \frac{495394206705322}{12208639947162743} a^{8} + \frac{4859119367880684}{12208639947162743} a^{7} - \frac{810005115075384}{1744091421023249} a^{6} - \frac{2592251544400146}{12208639947162743} a^{5} - \frac{328617019755}{1744091421023249} a^{4} - \frac{6093461301584345}{12208639947162743} a^{3} - \frac{151763227237913}{1744091421023249} a^{2} + \frac{5184384534907988}{12208639947162743} a - \frac{29172598001914}{154539746166617}$, $\frac{1}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{27} - \frac{81768822143005338740966703059653042167774441678744265}{2462805101666521351224269894818808496818156905647154129891597873557473} a^{26} + \frac{1228365873295932154893776779988430968166566400316293302225310781134244}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{25} + \frac{2375680634841015307952828057997242216778675838601248889484757030550189}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{24} + \frac{251884781834966423992453600737462717392880554634631730973943200302740534}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{23} + \frac{1090636039552508585649428514885344787209673196045446516773910139669053116}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{22} - \frac{117805488931095617263536905774747166832554215921800151969642800958474020}{829965319261617695362578954553938463427718877203090941773468483388868401} a^{21} - \frac{1160349066506418439266542024180392126704084626905482606490106349104945258}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{20} - \frac{1445785985313549597789686222744371314700961146643330966584384312000603707}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{19} + \frac{1836145515108300105824534418813130139398251628443777326584230927346017327}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{18} + \frac{2384590857043749492653613439280382059268635214820538408512802828898600424}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{17} - \frac{905956616805744473489119792724640138133234513188469504841888217794312718}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{16} - \frac{1054790800633494818496986381329198683505490203925267365349871506083139705}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{15} - \frac{2811151822900750001553577209352716004470313878682293031014857055612492966}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{14} + \frac{2463659307876019342745726935166955316084992091987373419954118620864859826}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{13} - \frac{268432667566322690577028535587986658649508137898048474074190193090939630}{829965319261617695362578954553938463427718877203090941773468483388868401} a^{12} + \frac{2188671446581699946871549306838589546858136548464619056773631857848850213}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{11} + \frac{1283649560229118346836752838186767770109003095353596483696987307924683194}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{10} + \frac{1129771513872701883153883139265320203687894036214557987454131767892798630}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{9} + \frac{782811876439481649399648458106656628420701323716360723896118894062888042}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{8} + \frac{1568417932604945464321264323302452556684335929309417185645033034343340428}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{7} - \frac{1696189217370613750927277928537533902346525781585510706794586358770910486}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{6} + \frac{572793458664809108551649490865814545059046972703529629556964990000736028}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{5} + \frac{710977151382545535622103004275716080968449214491321592888705924868084922}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{4} - \frac{1025976625815882371450044830789405406987292324098419104806620713721272777}{5809757234831323867538052681877569243994032140421636592414279383722078807} a^{3} + \frac{229954727213078335377660494387213285692427652217562073814764077151764444}{829965319261617695362578954553938463427718877203090941773468483388868401} a^{2} - \frac{2651299515489003733903174244929808464594890016552105064651983713052383120}{5809757234831323867538052681877569243994032140421636592414279383722078807} a - \frac{19814813532314646108669241703056121761103419525112488580770386665328791}{73541230820649669209342439011108471442962432157235906233092144097747833}$

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

Class group and class number

$C_{1421}$, which has order $1421$ (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:  \( -1 \) (order $2$)
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

$C_2\times C_{14}$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-215}) \), \(\Q(\sqrt{5}, \sqrt{-43})\), 7.7.6321363049.1, 14.14.3121846156036138781328125.1, 14.0.1718264124282290785243.1, 14.0.134239384709553967597109375.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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