Properties

Label 28.0.141...481.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.412\times 10^{49}$
Root discriminant $56.93$
Ramified primes $3, 43$
Class number $203$ (GRH)
Class group $[203]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241)
 
gp: K = bnfinit(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, 13272, 9659, 53384, 57132, -58937, -52559, -105029, -13057, 49398, 87041, 45122, -3964, -71623, -15266, -15421, 34135, 3755, 5482, -7986, -566, -740, 923, 41, 45, -50, -1, -1, 1]);
 

\(x^{28} - x^{27} - x^{26} - 50 x^{25} + 45 x^{24} + 41 x^{23} + 923 x^{22} - 740 x^{21} - 566 x^{20} - 7986 x^{19} + 5482 x^{18} + 3755 x^{17} + 34135 x^{16} - 15421 x^{15} - 15266 x^{14} - 71623 x^{13} - 3964 x^{12} + 45122 x^{11} + 87041 x^{10} + 49398 x^{9} - 13057 x^{8} - 105029 x^{7} - 52559 x^{6} - 58937 x^{5} + 57132 x^{4} + 53384 x^{3} + 9659 x^{2} + 13272 x + 6241\)  Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(14121388821225670988853483488774192350843817726481\)\(\medspace = 3^{14}\cdot 43^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.93$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 43$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(70,·)$, $\chi_{129}(65,·)$, $\chi_{129}(8,·)$, $\chi_{129}(11,·)$, $\chi_{129}(64,·)$, $\chi_{129}(16,·)$, $\chi_{129}(82,·)$, $\chi_{129}(85,·)$, $\chi_{129}(22,·)$, $\chi_{129}(88,·)$, $\chi_{129}(94,·)$, $\chi_{129}(32,·)$, $\chi_{129}(97,·)$, $\chi_{129}(35,·)$, $\chi_{129}(41,·)$, $\chi_{129}(107,·)$, $\chi_{129}(44,·)$, $\chi_{129}(47,·)$, $\chi_{129}(113,·)$, $\chi_{129}(118,·)$, $\chi_{129}(121,·)$, $\chi_{129}(59,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{79} a^{23} + \frac{4}{79} a^{22} + \frac{30}{79} a^{21} - \frac{33}{79} a^{20} + \frac{15}{79} a^{19} - \frac{29}{79} a^{18} - \frac{25}{79} a^{17} + \frac{9}{79} a^{16} + \frac{24}{79} a^{15} + \frac{30}{79} a^{14} + \frac{10}{79} a^{13} + \frac{1}{79} a^{12} + \frac{14}{79} a^{11} + \frac{18}{79} a^{10} - \frac{36}{79} a^{9} - \frac{21}{79} a^{8} + \frac{5}{79} a^{7} + \frac{8}{79} a^{6} + \frac{21}{79} a^{5} + \frac{13}{79} a^{4} + \frac{1}{79} a^{3} + \frac{23}{79} a^{2} - \frac{37}{79} a$, $\frac{1}{553} a^{24} + \frac{3}{553} a^{23} + \frac{26}{553} a^{22} - \frac{142}{553} a^{21} - \frac{27}{79} a^{20} + \frac{114}{553} a^{19} - \frac{75}{553} a^{18} + \frac{113}{553} a^{17} + \frac{15}{553} a^{16} - \frac{73}{553} a^{15} + \frac{138}{553} a^{14} - \frac{9}{553} a^{13} - \frac{66}{553} a^{12} - \frac{22}{79} a^{11} - \frac{19}{79} a^{10} - \frac{64}{553} a^{9} + \frac{184}{553} a^{8} + \frac{23}{79} a^{7} + \frac{92}{553} a^{6} - \frac{166}{553} a^{5} + \frac{67}{553} a^{4} + \frac{37}{79} a^{3} + \frac{256}{553} a^{2} + \frac{116}{553} a - \frac{1}{7}$, $\frac{1}{169771} a^{25} - \frac{150}{169771} a^{24} + \frac{463}{169771} a^{23} - \frac{48094}{169771} a^{22} - \frac{72137}{169771} a^{21} + \frac{38726}{169771} a^{20} + \frac{66707}{169771} a^{19} + \frac{19337}{169771} a^{18} - \frac{12024}{169771} a^{17} + \frac{22839}{169771} a^{16} - \frac{6452}{169771} a^{15} + \frac{26771}{169771} a^{14} + \frac{83267}{169771} a^{13} + \frac{30748}{169771} a^{12} + \frac{1979}{24253} a^{11} - \frac{33265}{169771} a^{10} + \frac{31361}{169771} a^{9} + \frac{72641}{169771} a^{8} + \frac{75608}{169771} a^{7} - \frac{65139}{169771} a^{6} + \frac{6677}{169771} a^{5} - \frac{67469}{169771} a^{4} + \frac{79314}{169771} a^{3} + \frac{10312}{169771} a^{2} - \frac{37154}{169771} a + \frac{132}{2149}$, $\frac{1}{12208639947162743} a^{26} + \frac{25500050657}{12208639947162743} a^{25} + \frac{8172513473303}{12208639947162743} a^{24} - \frac{35401708760840}{12208639947162743} a^{23} - \frac{5407447383265908}{12208639947162743} a^{22} - \frac{45333011933414}{154539746166617} a^{21} - \frac{545176098296287}{12208639947162743} a^{20} - \frac{2138725957501061}{12208639947162743} a^{19} - \frac{5019996399970187}{12208639947162743} a^{18} + \frac{5668652282921482}{12208639947162743} a^{17} - \frac{585010836262925}{12208639947162743} a^{16} + \frac{426512719512321}{12208639947162743} a^{15} - \frac{4969460893359876}{12208639947162743} a^{14} + \frac{5102925782841738}{12208639947162743} a^{13} - \frac{36109037429781}{154539746166617} a^{12} - \frac{3536979576394748}{12208639947162743} a^{11} + \frac{4925818989105456}{12208639947162743} a^{10} - \frac{4364147001568068}{12208639947162743} a^{9} + \frac{440703115311966}{1744091421023249} a^{8} + \frac{2634527333178975}{12208639947162743} a^{7} - \frac{2229323157341243}{12208639947162743} a^{6} + \frac{3972994343637945}{12208639947162743} a^{5} - \frac{419010392445237}{12208639947162743} a^{4} - \frac{5821203586736733}{12208639947162743} a^{3} - \frac{477653372780291}{1744091421023249} a^{2} - \frac{2592150416563299}{12208639947162743} a + \frac{4722382228694}{154539746166617}$, $\frac{1}{977266965746288165860398234830366371819969467671} a^{27} - \frac{30846139009003708975946021034127}{977266965746288165860398234830366371819969467671} a^{26} + \frac{2121989472560265635463194394989613534077095}{977266965746288165860398234830366371819969467671} a^{25} - \frac{71048950744452308648952472455486367450250909}{977266965746288165860398234830366371819969467671} a^{24} - \frac{436356011282060622716582308812646563395770044}{139609566535184023694342604975766624545709923953} a^{23} + \frac{229611879369248765554585990404797035607737216320}{977266965746288165860398234830366371819969467671} a^{22} - \frac{48159924404725312081367271415598730351323939176}{977266965746288165860398234830366371819969467671} a^{21} + \frac{60172833658222373354789502012594458582494903173}{977266965746288165860398234830366371819969467671} a^{20} - \frac{445880919370254647571341244394323165862103665708}{977266965746288165860398234830366371819969467671} a^{19} - \frac{238358654659680748286774563588803350620263181120}{977266965746288165860398234830366371819969467671} a^{18} + \frac{286783369160233155763788827228130556173276085402}{977266965746288165860398234830366371819969467671} a^{17} - \frac{363878234670053392591664007681657010722434970431}{977266965746288165860398234830366371819969467671} a^{16} - \frac{60802971052613989432133912224481113884496608599}{139609566535184023694342604975766624545709923953} a^{15} + \frac{35420276677628489989615639104946505125945801330}{139609566535184023694342604975766624545709923953} a^{14} + \frac{431050587766517611363395969915298796164651338346}{977266965746288165860398234830366371819969467671} a^{13} - \frac{1438436398287349417323503827467815562336500798}{12370467920839090707093648542156536352151512249} a^{12} - \frac{76423689522391091327880149340362031380360106502}{977266965746288165860398234830366371819969467671} a^{11} - \frac{248220019113118974255328454708349982989803302171}{977266965746288165860398234830366371819969467671} a^{10} - \frac{201511280551677822678638657276672468568891867343}{977266965746288165860398234830366371819969467671} a^{9} + \frac{63139250117831176157936029989368928080019628866}{139609566535184023694342604975766624545709923953} a^{8} + \frac{51333456632848115061638144062866695835784294034}{977266965746288165860398234830366371819969467671} a^{7} + \frac{352284344298243290693918221548216134200175024839}{977266965746288165860398234830366371819969467671} a^{6} + \frac{211112911282388500011932333737753473007734515434}{977266965746288165860398234830366371819969467671} a^{5} - \frac{237706156506733876010392200060760546269482368546}{977266965746288165860398234830366371819969467671} a^{4} - \frac{70607364059626640624378645578214214907690192642}{977266965746288165860398234830366371819969467671} a^{3} + \frac{74631477567767577683853253249493399805252826046}{977266965746288165860398234830366371819969467671} a^{2} + \frac{234313017712533076446629571685348528760060321571}{977266965746288165860398234830366371819969467671} a + \frac{253056952745284426672317330804822151785035918}{12370467920839090707093648542156536352151512249}$  Toggle raw display

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

Class group and class number

$C_{203}$, which has order $203$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1469565974509958327034868919014095}{13589686546556770354122015967062022987} a^{27} - \frac{2185707542180747637832708906726247}{13589686546556770354122015967062022987} a^{26} - \frac{7423076916778607575505668606219}{172021348690592029799012860342557253} a^{25} - \frac{73439535746083352207161767503807445}{13589686546556770354122015967062022987} a^{24} + \frac{102056386983756908318367238772596210}{13589686546556770354122015967062022987} a^{23} + \frac{20112019365816358240629209106930190}{13589686546556770354122015967062022987} a^{22} + \frac{1360145432505719268638469568936869224}{13589686546556770354122015967062022987} a^{21} - \frac{1753206778039274686307086151665928270}{13589686546556770354122015967062022987} a^{20} - \frac{163205604854518903203388760690261033}{13589686546556770354122015967062022987} a^{19} - \frac{11924696640706567412378496798402125083}{13589686546556770354122015967062022987} a^{18} + \frac{13848196158548853725689842851966575139}{13589686546556770354122015967062022987} a^{17} + \frac{417485373565694113786508502403038620}{13589686546556770354122015967062022987} a^{16} + \frac{52384474480375976058742825612964545713}{13589686546556770354122015967062022987} a^{15} - \frac{47527746331131048620434744262972068572}{13589686546556770354122015967062022987} a^{14} - \frac{6234873556739011773170419315120310751}{13589686546556770354122015967062022987} a^{13} - \frac{113173553420117357450659172523095744476}{13589686546556770354122015967062022987} a^{12} + \frac{44128646302373508697263147156693373974}{13589686546556770354122015967062022987} a^{11} + \frac{57441928696070197496832368274180043024}{13589686546556770354122015967062022987} a^{10} + \frac{17870280564843121166570103024525552756}{1941383792365252907731716566723146141} a^{9} + \frac{29156360210991818865782673380619600489}{13589686546556770354122015967062022987} a^{8} - \frac{40840584820498036407055156919440751331}{13589686546556770354122015967062022987} a^{7} - \frac{164990259650080922193773411795086730354}{13589686546556770354122015967062022987} a^{6} - \frac{30186355712350454417212570511258960916}{13589686546556770354122015967062022987} a^{5} - \frac{91841198497284455494610331339211512711}{13589686546556770354122015967062022987} a^{4} + \frac{19657519881538952482738490037871420732}{1941383792365252907731716566723146141} a^{3} + \frac{40280007276170701259769319385155426104}{13589686546556770354122015967062022987} a^{2} + \frac{22718577442160810286994597934338804912}{13589686546556770354122015967062022987} a + \frac{50996908933488818958189033790849093}{24574478384370289971287551477508179} \) (order $6$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 158620092291.7724 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 158620092291.7724 \cdot 203}{6\sqrt{14121388821225670988853483488774192350843817726481}}\approx 0.213443255955932$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{129}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-43})\), 7.7.6321363049.1, 14.14.3757843639805369947326441.1, 14.0.1718264124282290785243.1, 14.0.87391712553613254588987.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$
43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$