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Properties

Label 28.28.3641753875...8125.1
Degree $28$
Signature $[28, 0]$
Discriminant $3^{14}\cdot 5^{21}\cdot 43^{24}$
Root discriminant $145.51$
Ramified primes $3, 5, 43$
Class number $1$ (GRH)
Class group Trivial (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![811, 43812, 491878, 572142, -10823713, -24893187, 70371961, 199817979, -118191520, -586030579, -164293830, 566561234, 329592453, -248307765, -204434963, 54518578, 65218506, -5368181, -12199271, -36832, 1402407, 59618, -99700, -5523, 4256, 212, -100, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 100*x^26 + 212*x^25 + 4256*x^24 - 5523*x^23 - 99700*x^22 + 59618*x^21 + 1402407*x^20 - 36832*x^19 - 12199271*x^18 - 5368181*x^17 + 65218506*x^16 + 54518578*x^15 - 204434963*x^14 - 248307765*x^13 + 329592453*x^12 + 566561234*x^11 - 164293830*x^10 - 586030579*x^9 - 118191520*x^8 + 199817979*x^7 + 70371961*x^6 - 24893187*x^5 - 10823713*x^4 + 572142*x^3 + 491878*x^2 + 43812*x + 811)
 
gp: K = bnfinit(x^28 - 3*x^27 - 100*x^26 + 212*x^25 + 4256*x^24 - 5523*x^23 - 99700*x^22 + 59618*x^21 + 1402407*x^20 - 36832*x^19 - 12199271*x^18 - 5368181*x^17 + 65218506*x^16 + 54518578*x^15 - 204434963*x^14 - 248307765*x^13 + 329592453*x^12 + 566561234*x^11 - 164293830*x^10 - 586030579*x^9 - 118191520*x^8 + 199817979*x^7 + 70371961*x^6 - 24893187*x^5 - 10823713*x^4 + 572142*x^3 + 491878*x^2 + 43812*x + 811, 1)
 

Normalized defining polynomial

\( x^{28} - 3 x^{27} - 100 x^{26} + 212 x^{25} + 4256 x^{24} - 5523 x^{23} - 99700 x^{22} + 59618 x^{21} + 1402407 x^{20} - 36832 x^{19} - 12199271 x^{18} - 5368181 x^{17} + 65218506 x^{16} + 54518578 x^{15} - 204434963 x^{14} - 248307765 x^{13} + 329592453 x^{12} + 566561234 x^{11} - 164293830 x^{10} - 586030579 x^{9} - 118191520 x^{8} + 199817979 x^{7} + 70371961 x^{6} - 24893187 x^{5} - 10823713 x^{4} + 572142 x^{3} + 491878 x^{2} + 43812 x + 811 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3641753875281030900637863258408801848555415234088897705078125=3^{14}\cdot 5^{21}\cdot 43^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.51$
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:  $3, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(645=3\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{645}(256,·)$, $\chi_{645}(1,·)$, $\chi_{645}(514,·)$, $\chi_{645}(259,·)$, $\chi_{645}(4,·)$, $\chi_{645}(391,·)$, $\chi_{645}(64,·)$, $\chi_{645}(398,·)$, $\chi_{645}(557,·)$, $\chi_{645}(16,·)$, $\chi_{645}(274,·)$, $\chi_{645}(451,·)$, $\chi_{645}(527,·)$, $\chi_{645}(226,·)$, $\chi_{645}(484,·)$, $\chi_{645}(293,·)$, $\chi_{645}(422,·)$, $\chi_{645}(623,·)$, $\chi_{645}(107,·)$, $\chi_{645}(428,·)$, $\chi_{645}(173,·)$, $\chi_{645}(302,·)$, $\chi_{645}(47,·)$, $\chi_{645}(563,·)$, $\chi_{645}(121,·)$, $\chi_{645}(379,·)$, $\chi_{645}(188,·)$, $\chi_{645}(317,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7} a^{20} + \frac{3}{7} a^{19} + \frac{3}{7} a^{18} - \frac{1}{7} a^{17} + \frac{2}{7} a^{16} + \frac{2}{7} a^{15} + \frac{1}{7} a^{14} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{21} + \frac{1}{7} a^{19} - \frac{3}{7} a^{18} - \frac{2}{7} a^{17} + \frac{3}{7} a^{16} + \frac{2}{7} a^{15} - \frac{3}{7} a^{14} - \frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{19} + \frac{2}{7} a^{18} - \frac{3}{7} a^{17} + \frac{2}{7} a^{15} - \frac{1}{7} a^{14} - \frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{23} - \frac{1}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{24} - \frac{3}{7} a^{19} - \frac{3}{7} a^{18} - \frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{25} - \frac{1}{7} a^{19} + \frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{16} - \frac{2}{7} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{26} - \frac{3}{7} a^{19} - \frac{1}{7} a^{18} - \frac{2}{7} a^{17} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{27} + \frac{4506810127072755501435282141641578268477042591830079013176724166036959451170655944729706352708}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{26} - \frac{204524033663932352707266484275594178271761830718793130805104297044986697413616981348325882440}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{25} + \frac{7021813503332415007766398264771641589304763577489595390304927645784250041059332302545087918962}{102619549977903248884768615055298585618367619340300367045106502217115298587923874250396036441689} a^{24} - \frac{11956105477737152439956723048389012331182416544658013469053556570905319663767181460495473811544}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{23} - \frac{33905468198312588052182841424721905579271968071794229726589103698129751578971453310634872760881}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{22} + \frac{8107095248334374551891408680196768359375813488244832877463289582056533732221726218362195243769}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{21} + \frac{5537878627816607856910713174945919265763906885533150185338885848692628613619201150743161004040}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{20} + \frac{221893343444841887047553056459017346979365191528820622282433076867173401692302863034192316019023}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{19} + \frac{21829604622438743647444999237301303931275596778760739418336615437339385027250776594326033243272}{102619549977903248884768615055298585618367619340300367045106502217115298587923874250396036441689} a^{18} - \frac{218740760798418252958381568150396801597922043892373369061017103273066808414708103129505742349676}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{17} - \frac{131013328474350256845905415746688578339798371045104102109264783125134850551021658294681283302455}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{16} - \frac{331116787148803361620256838773881876870441141680314425100532746122723133466852347870095467888049}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{15} + \frac{51421963818875014912329543241347101461507968195697085922159960885414080564005568372361921867675}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{14} - \frac{69772199808427404600233246016381726456405792574072631337120693272057781689476505324622156934686}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{13} - \frac{315278676053055227059864472371596572369817285282473990998223566952023350296607791511739874824955}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{12} - \frac{216132099546180111631656448444611428780707148397684895042714500360278970730739031721921337824127}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{11} + \frac{29059057796684568736441800522413908037425714840876987278975597071774461609039827499210357947742}{102619549977903248884768615055298585618367619340300367045106502217115298587923874250396036441689} a^{10} + \frac{298212984614441642829198519151211482250298708843620869326436878168923582960339176364875224494965}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{9} + \frac{223952422284490035861645373473666569712967173243157961782170510914083295987789827417014173044693}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{8} - \frac{229650824436898799581108506787383731972285769119870733134839438138373689448858630631038433004611}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{7} - \frac{99583291337056261949813279201787483206885812458724597232281576608662488078078137672776927571731}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{6} + \frac{357500736440944248969218296524171442680002612594590794784935336341726710574839460784913830878063}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{5} + \frac{22305210939247377458870982290491430885543693687346744639769904363341018997553538294167165143566}{102619549977903248884768615055298585618367619340300367045106502217115298587923874250396036441689} a^{4} - \frac{146913435078372132114819349866382176973221507463015724555927578419793091663645097261707850128885}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a^{3} + \frac{45699725234537255719926396138565704708469601544960589044588067960219622210115422793024721370946}{102619549977903248884768615055298585618367619340300367045106502217115298587923874250396036441689} a^{2} - \frac{130399107916145931167602218951992563599676762155382079628082993912729707207931817968816045569265}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823} a - \frac{313348555288553887514798788162867147061015792660274056030804530209516095845990348172093508603646}{718336849845322742193380305387090099328573335382102569315745515519807090115467119752772255091823}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $27$
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:  \( 1634421464198991000000 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 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 $28$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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
3Data not computed
5Data not computed
43Data not computed