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Properties

Label 28.28.8649266893...7777.1
Degree $28$
Signature $[28, 0]$
Discriminant $17^{21}\cdot 29^{24}$
Root discriminant $150.08$
Ramified primes $17, 29$
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![-43367, -804372, -4146895, -617338, 37970686, 39437768, -141255320, -164124215, 283623445, 294247369, -332007770, -281263173, 238211940, 158408220, -108806380, -55419172, 32407795, 12347113, -6351581, -1759929, 814194, 158075, -66698, -8554, 3319, 251, -90, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 90*x^26 + 251*x^25 + 3319*x^24 - 8554*x^23 - 66698*x^22 + 158075*x^21 + 814194*x^20 - 1759929*x^19 - 6351581*x^18 + 12347113*x^17 + 32407795*x^16 - 55419172*x^15 - 108806380*x^14 + 158408220*x^13 + 238211940*x^12 - 281263173*x^11 - 332007770*x^10 + 294247369*x^9 + 283623445*x^8 - 164124215*x^7 - 141255320*x^6 + 39437768*x^5 + 37970686*x^4 - 617338*x^3 - 4146895*x^2 - 804372*x - 43367)
 
gp: K = bnfinit(x^28 - 3*x^27 - 90*x^26 + 251*x^25 + 3319*x^24 - 8554*x^23 - 66698*x^22 + 158075*x^21 + 814194*x^20 - 1759929*x^19 - 6351581*x^18 + 12347113*x^17 + 32407795*x^16 - 55419172*x^15 - 108806380*x^14 + 158408220*x^13 + 238211940*x^12 - 281263173*x^11 - 332007770*x^10 + 294247369*x^9 + 283623445*x^8 - 164124215*x^7 - 141255320*x^6 + 39437768*x^5 + 37970686*x^4 - 617338*x^3 - 4146895*x^2 - 804372*x - 43367, 1)
 

Normalized defining polynomial

\( x^{28} - 3 x^{27} - 90 x^{26} + 251 x^{25} + 3319 x^{24} - 8554 x^{23} - 66698 x^{22} + 158075 x^{21} + 814194 x^{20} - 1759929 x^{19} - 6351581 x^{18} + 12347113 x^{17} + 32407795 x^{16} - 55419172 x^{15} - 108806380 x^{14} + 158408220 x^{13} + 238211940 x^{12} - 281263173 x^{11} - 332007770 x^{10} + 294247369 x^{9} + 283623445 x^{8} - 164124215 x^{7} - 141255320 x^{6} + 39437768 x^{5} + 37970686 x^{4} - 617338 x^{3} - 4146895 x^{2} - 804372 x - 43367 \)

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:  \(8649266893998089480085334923884882761653761146262454088937777=17^{21}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.08$
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:  $17, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(493=17\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{493}(256,·)$, $\chi_{493}(1,·)$, $\chi_{493}(132,·)$, $\chi_{493}(455,·)$, $\chi_{493}(140,·)$, $\chi_{493}(458,·)$, $\chi_{493}(268,·)$, $\chi_{493}(16,·)$, $\chi_{493}(81,·)$, $\chi_{493}(339,·)$, $\chi_{493}(480,·)$, $\chi_{493}(407,·)$, $\chi_{493}(152,·)$, $\chi_{493}(285,·)$, $\chi_{493}(30,·)$, $\chi_{493}(344,·)$, $\chi_{493}(103,·)$, $\chi_{493}(169,·)$, $\chi_{493}(426,·)$, $\chi_{493}(429,·)$, $\chi_{493}(239,·)$, $\chi_{493}(52,·)$, $\chi_{493}(373,·)$, $\chi_{493}(310,·)$, $\chi_{493}(489,·)$, $\chi_{493}(378,·)$, $\chi_{493}(123,·)$, $\chi_{493}(460,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34} a^{24} + \frac{3}{34} a^{23} + \frac{4}{17} a^{22} + \frac{9}{34} a^{20} - \frac{3}{17} a^{19} + \frac{8}{17} a^{18} - \frac{7}{34} a^{17} + \frac{6}{17} a^{16} + \frac{3}{34} a^{15} - \frac{6}{17} a^{14} - \frac{3}{34} a^{13} + \frac{7}{34} a^{12} - \frac{3}{34} a^{11} - \frac{4}{17} a^{10} + \frac{7}{34} a^{9} - \frac{11}{34} a^{8} + \frac{15}{34} a^{7} + \frac{11}{34} a^{6} + \frac{3}{17} a^{5} - \frac{1}{34} a^{4} - \frac{1}{2} a$, $\frac{1}{578} a^{25} - \frac{7}{578} a^{24} - \frac{73}{578} a^{23} + \frac{73}{578} a^{22} - \frac{38}{289} a^{21} + \frac{159}{578} a^{20} + \frac{89}{289} a^{19} + \frac{44}{289} a^{18} - \frac{78}{289} a^{17} + \frac{223}{578} a^{16} - \frac{106}{289} a^{15} - \frac{52}{289} a^{14} + \frac{275}{578} a^{13} - \frac{209}{578} a^{12} - \frac{63}{578} a^{11} + \frac{137}{289} a^{10} + \frac{87}{289} a^{9} - \frac{79}{578} a^{8} - \frac{241}{578} a^{7} + \frac{185}{578} a^{6} + \frac{80}{289} a^{5} + \frac{163}{578} a^{4} + \frac{3}{17} a^{3} + \frac{7}{17} a^{2} - \frac{4}{17} a - \frac{3}{17}$, $\frac{1}{302294} a^{26} - \frac{209}{302294} a^{25} - \frac{1973}{151147} a^{24} - \frac{1099}{151147} a^{23} + \frac{6751}{302294} a^{22} + \frac{67531}{302294} a^{21} - \frac{65073}{302294} a^{20} - \frac{35597}{151147} a^{19} + \frac{148617}{302294} a^{18} + \frac{32041}{302294} a^{17} - \frac{128065}{302294} a^{16} - \frac{36143}{302294} a^{15} - \frac{87517}{302294} a^{14} + \frac{86395}{302294} a^{13} - \frac{29823}{302294} a^{12} - \frac{125465}{302294} a^{11} - \frac{24149}{302294} a^{10} - \frac{64440}{151147} a^{9} + \frac{45896}{151147} a^{8} + \frac{63487}{302294} a^{7} + \frac{7517}{302294} a^{6} - \frac{34107}{151147} a^{5} + \frac{37777}{302294} a^{4} - \frac{21}{8891} a^{3} - \frac{1017}{17782} a^{2} + \frac{2692}{8891} a - \frac{1151}{17782}$, $\frac{1}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{27} + \frac{15544463186023399912164436573378353496463955465032492591608271922378088451053}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{26} - \frac{18674750078143700758779713141261968847895024428308417797501688881719298198672599}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{25} + \frac{228616628641687855611072875870741856519961628454410243754738608594982077691503768}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{24} - \frac{5952638650864500329327647157919110403788036532237004765695396980246157193023375489}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{23} - \frac{4084668378305219256460258170764040481806484857136786327102941764700574495272580297}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{22} - \frac{11996939301499308547311834362640519536438850018169641067063400768663215459906684790}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{21} + \frac{6574629182947231636146444490648925873651587761941931624987987674854307068507138319}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{20} - \frac{10920106775666915797340164804281160255412652660135743724024127057730221180336018345}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{19} + \frac{2742231332799163222249756073739601180344121394503081485832525566660337940304329730}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{18} + \frac{29480961423929046748001820925948929530173708619161439242456556523152145426289224848}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{17} - \frac{43905438447084414032204003208793067363617626125573835487861537050646628979839869395}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{16} - \frac{18066832377764564469080305202012564661160511124708933551324561876409128830206918703}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{15} - \frac{34552824883735887676008645178506826488137006580254648694795280883570004371039675767}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{14} - \frac{18802249983340641797108863060975096747463054088323700429879333287176667802386968939}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{13} - \frac{21807110865030063675847460671601085611087222060887703645048789174857679911433785449}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{12} + \frac{52470654999120280055206476571542500075096918660651202276757802385457516059044224665}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{11} + \frac{13742868008125769708692635450466263138514292758095765657700810669992455775604791210}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{10} - \frac{12702517704950792287712567081349258911707582294764676241655956459168716190407590460}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{9} + \frac{29808710930443703613587707714114044737978511649276846771993039379410654748473874313}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{8} + \frac{38367083229116225104592529327463695967799092625590889116909888796164279590326696547}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{7} - \frac{26527386393490298394184480341360065956187209026433145436093669132434408224075121395}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{6} + \frac{9720270316842158863030235911961979581806521959933251085475256354411855150542370198}{61901585492784236910055642386334242480749391367348845765173444794479966740805975889} a^{5} + \frac{32200628722373368711940297771973284001711168383546105524802411054773734091894088685}{123803170985568473820111284772668484961498782734697691530346889588959933481611951778} a^{4} + \frac{328216095304381385271618315842521054386549149076718718525486221750047630180326916}{3641269734869660994709155434490249557691140668667579162657261458498821572988586817} a^{3} - \frac{15991021955751308094768667515492305309647018597489145325890427271045684354588369}{428384674690548352318724168763558771493075372784421077959677818646920185057480802} a^{2} - \frac{2570749000685819322656264728319544292504868119698303939318623036377488770867822655}{7282539469739321989418310868980499115382281337335158325314522916997643145977173634} a - \frac{438832651249292087837881936961470620678010842407212250581430163298067375459606}{1427389155182148567114525846526950042215264864236604924601043300077938680120967}$

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:  \( 4582667171549212600000 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, 7.7.594823321.1, 14.14.145183888628314852626522593.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ $28$ $28$ $28$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
29Data not computed