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Properties

Label 28.0.51408015904...5461.1
Degree $28$
Signature $[0, 14]$
Discriminant $17^{14}\cdot 29^{27}$
Root discriminant $106.02$
Ramified primes $17, 29$
Class number Not computed
Class group Not computed
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![702418373381, -694633745157, 694633745157, -626518248197, 626518248197, -449417956101, 449417956101, -234367601413, 234367601413, -86520482565, 86520482565, -22677408517, 22677408517, -4261137157, 4261137157, -577882885, 577882885, -56539909, 56539909, -3948293, 3948293, -191749, 191749, -6149, 6149, -117, 117, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 117*x^26 - 117*x^25 + 6149*x^24 - 6149*x^23 + 191749*x^22 - 191749*x^21 + 3948293*x^20 - 3948293*x^19 + 56539909*x^18 - 56539909*x^17 + 577882885*x^16 - 577882885*x^15 + 4261137157*x^14 - 4261137157*x^13 + 22677408517*x^12 - 22677408517*x^11 + 86520482565*x^10 - 86520482565*x^9 + 234367601413*x^8 - 234367601413*x^7 + 449417956101*x^6 - 449417956101*x^5 + 626518248197*x^4 - 626518248197*x^3 + 694633745157*x^2 - 694633745157*x + 702418373381)
 
gp: K = bnfinit(x^28 - x^27 + 117*x^26 - 117*x^25 + 6149*x^24 - 6149*x^23 + 191749*x^22 - 191749*x^21 + 3948293*x^20 - 3948293*x^19 + 56539909*x^18 - 56539909*x^17 + 577882885*x^16 - 577882885*x^15 + 4261137157*x^14 - 4261137157*x^13 + 22677408517*x^12 - 22677408517*x^11 + 86520482565*x^10 - 86520482565*x^9 + 234367601413*x^8 - 234367601413*x^7 + 449417956101*x^6 - 449417956101*x^5 + 626518248197*x^4 - 626518248197*x^3 + 694633745157*x^2 - 694633745157*x + 702418373381, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 117 x^{26} - 117 x^{25} + 6149 x^{24} - 6149 x^{23} + 191749 x^{22} - 191749 x^{21} + 3948293 x^{20} - 3948293 x^{19} + 56539909 x^{18} - 56539909 x^{17} + 577882885 x^{16} - 577882885 x^{15} + 4261137157 x^{14} - 4261137157 x^{13} + 22677408517 x^{12} - 22677408517 x^{11} + 86520482565 x^{10} - 86520482565 x^{9} + 234367601413 x^{8} - 234367601413 x^{7} + 449417956101 x^{6} - 449417956101 x^{5} + 626518248197 x^{4} - 626518248197 x^{3} + 694633745157 x^{2} - 694633745157 x + 702418373381 \)

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:  \(514080159043940282786363725114031369092948206215491155461=17^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.02$
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$$\chi_{493}(256,·)$, $\chi_{493}(1,·)$, $\chi_{493}(322,·)$, $\chi_{493}(324,·)$, $\chi_{493}(135,·)$, $\chi_{493}(460,·)$, $\chi_{493}(271,·)$, $\chi_{493}(84,·)$, $\chi_{493}(341,·)$, $\chi_{493}(86,·)$, $\chi_{493}(154,·)$, $\chi_{493}(475,·)$, $\chi_{493}(220,·)$, $\chi_{493}(477,·)$, $\chi_{493}(288,·)$, $\chi_{493}(35,·)$, $\chi_{493}(356,·)$, $\chi_{493}(101,·)$, $\chi_{493}(103,·)$, $\chi_{493}(424,·)$, $\chi_{493}(426,·)$, $\chi_{493}(239,·)$, $\chi_{493}(305,·)$, $\chi_{493}(50,·)$, $\chi_{493}(52,·)$, $\chi_{493}(118,·)$, $\chi_{493}(120,·)$, $\chi_{493}(186,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{170361678269} a^{15} + \frac{64818294450}{170361678269} a^{14} + \frac{60}{170361678269} a^{13} + \frac{52229245551}{170361678269} a^{12} + \frac{1440}{170361678269} a^{11} - \frac{43488345761}{170361678269} a^{10} + \frac{17600}{170361678269} a^{9} - \frac{71745259666}{170361678269} a^{8} + \frac{115200}{170361678269} a^{7} + \frac{7094573716}{170361678269} a^{6} + \frac{387072}{170361678269} a^{5} - \frac{37868362847}{170361678269} a^{4} + \frac{573440}{170361678269} a^{3} - \frac{37868362847}{170361678269} a^{2} + \frac{245760}{170361678269} a + \frac{66829557977}{170361678269}$, $\frac{1}{170361678269} a^{16} + \frac{64}{170361678269} a^{14} + \frac{81450178738}{170361678269} a^{13} + \frac{1664}{170361678269} a^{12} - \frac{23632662349}{170361678269} a^{11} + \frac{22528}{170361678269} a^{10} + \frac{38431787827}{170361678269} a^{9} + \frac{168960}{170361678269} a^{8} + \frac{62294142255}{170361678269} a^{7} + \frac{688128}{170361678269} a^{6} - \frac{50017359348}{170361678269} a^{5} + \frac{1376256}{170361678269} a^{4} + \frac{70326959573}{170361678269} a^{3} + \frac{1048576}{170361678269} a^{2} - \frac{8487931178}{170361678269} a + \frac{131072}{170361678269}$, $\frac{1}{170361678269} a^{17} + \frac{21759612394}{170361678269} a^{14} - \frac{2176}{170361678269} a^{13} + \frac{40929187767}{170361678269} a^{12} - \frac{69632}{170361678269} a^{11} - \frac{74462614042}{170361678269} a^{10} - \frac{957440}{170361678269} a^{9} + \frac{54225447616}{170361678269} a^{8} - \frac{6684672}{170361678269} a^{7} + \frac{7014957635}{170361678269} a^{6} - \frac{23396352}{170361678269} a^{5} - \frac{61522992254}{170361678269} a^{4} - \frac{35651584}{170361678269} a^{3} + \frac{30023795264}{170361678269} a^{2} - \frac{15597568}{170361678269} a - \frac{18049753803}{170361678269}$, $\frac{1}{170361678269} a^{18} - \frac{2448}{170361678269} a^{14} - \frac{72115807990}{170361678269} a^{13} - \frac{84864}{170361678269} a^{12} - \frac{61755659906}{170361678269} a^{11} - \frac{1292544}{170361678269} a^{10} + \frac{58100061928}{170361678269} a^{9} - \frac{10340352}{170361678269} a^{8} + \frac{1401218901}{170361678269} a^{7} - \frac{43868160}{170361678269} a^{6} + \frac{83162056738}{170361678269} a^{5} - \frac{90243072}{170361678269} a^{4} - \frac{1705963729}{170361678269} a^{3} - \frac{70189056}{170361678269} a^{2} - \frac{7310839333}{170361678269} a - \frac{8912896}{170361678269}$, $\frac{1}{170361678269} a^{19} - \frac{3653462829}{170361678269} a^{14} + \frac{62016}{170361678269} a^{13} + \frac{24178747192}{170361678269} a^{12} + \frac{2232576}{170361678269} a^{11} + \frac{74678557125}{170361678269} a^{10} + \frac{32744448}{170361678269} a^{9} + \frac{11895851872}{170361678269} a^{8} + \frac{238141440}{170361678269} a^{7} + \frac{73787330068}{170361678269} a^{6} + \frac{857309184}{170361678269} a^{5} - \frac{26705234849}{170361678269} a^{4} + \frac{1333592064}{170361678269} a^{3} - \frac{32310110453}{170361678269} a^{2} + \frac{592707584}{170361678269} a + \frac{51546789456}{170361678269}$, $\frac{1}{170361678269} a^{20} + \frac{72960}{170361678269} a^{14} + \frac{73024838663}{170361678269} a^{13} + \frac{2845440}{170361678269} a^{12} + \frac{54453004546}{170361678269} a^{11} + \frac{46227456}{170361678269} a^{10} - \frac{83872743410}{170361678269} a^{9} + \frac{385228800}{170361678269} a^{8} - \frac{11001771831}{170361678269} a^{7} + \frac{1680998400}{170361678269} a^{6} - \frac{45832399130}{170361678269} a^{5} + \frac{3530096640}{170361678269} a^{4} + \frac{71856877414}{170361678269} a^{3} + \frac{2789212160}{170361678269} a^{2} - \frac{49834511403}{170361678269} a + \frac{358612992}{170361678269}$, $\frac{1}{170361678269} a^{21} + \frac{88835834}{170361678269} a^{14} - \frac{1532160}{170361678269} a^{13} + \frac{58717124578}{170361678269} a^{12} - \frac{58834944}{170361678269} a^{11} + \frac{9937897294}{170361678269} a^{10} - \frac{898867200}{170361678269} a^{9} - \frac{9783033765}{170361678269} a^{8} - \frac{6723993600}{170361678269} a^{7} + \frac{63209541001}{170361678269} a^{6} - \frac{24710676480}{170361678269} a^{5} + \frac{21912027892}{170361678269} a^{4} - \frac{39048970240}{170361678269} a^{3} + \frac{70582317344}{170361678269} a^{2} - \frac{17572036608}{170361678269} a + \frac{37043735129}{170361678269}$, $\frac{1}{170361678269} a^{22} - \frac{1872640}{170361678269} a^{14} + \frac{53386974538}{170361678269} a^{13} - \frac{77901824}{170361678269} a^{12} + \frac{52375974603}{170361678269} a^{11} - \frac{1318338560}{170361678269} a^{10} - \frac{40038607744}{170361678269} a^{9} - \frac{11300044800}{170361678269} a^{8} + \frac{51022160341}{170361678269} a^{7} - \frac{50336563200}{170361678269} a^{6} + \frac{49107100182}{170361678269} a^{5} + \frac{62977010109}{170361678269} a^{4} + \frac{66703470815}{170361678269} a^{3} + \frac{84453943741}{170361678269} a^{2} + \frac{11043989721}{170361678269} a - \frac{11156848640}{170361678269}$, $\frac{1}{170361678269} a^{23} + \frac{51432586190}{170361678269} a^{14} + \frac{34456576}{170361678269} a^{13} - \frac{57069772885}{170361678269} a^{12} + \frac{1378263040}{170361678269} a^{11} - \frac{62781556714}{170361678269} a^{10} + \frac{21658419200}{170361678269} a^{9} + \frac{19741216647}{170361678269} a^{8} - \frac{4970113469}{170361678269} a^{7} - \frac{23849177543}{170361678269} a^{6} - \frac{63984871156}{170361678269} a^{5} - \frac{14272150939}{170361678269} a^{4} - \frac{34231122542}{170361678269} a^{3} - \frac{69931632033}{170361678269} a^{2} - \frac{62021877047}{170361678269} a + \frac{14593641880}{170361678269}$, $\frac{1}{170361678269} a^{24} + \frac{43524096}{170361678269} a^{14} - \frac{76514735443}{170361678269} a^{13} + \frac{1886044160}{170361678269} a^{12} - \frac{18375623299}{170361678269} a^{11} + \frac{32829603840}{170361678269} a^{10} - \frac{62179084156}{170361678269} a^{9} - \frac{53464322938}{170361678269} a^{8} - \frac{48969747992}{170361678269} a^{7} - \frac{62973758952}{170361678269} a^{6} - \frac{3274727817}{170361678269} a^{5} + \frac{82039628848}{170361678269} a^{4} - \frac{47329461546}{170361678269} a^{3} + \frac{54248874343}{170361678269} a^{2} - \frac{73069244065}{170361678269} a - \frac{43598861178}{170361678269}$, $\frac{1}{170361678269} a^{25} - \frac{41850320677}{170361678269} a^{14} - \frac{725401600}{170361678269} a^{13} + \frac{48853142638}{170361678269} a^{12} - \frac{29845094400}{170361678269} a^{11} + \frac{53960340306}{170361678269} a^{10} + \frac{32319978807}{170361678269} a^{9} + \frac{71936319602}{170361678269} a^{8} + \frac{33900729918}{170361678269} a^{7} + \frac{60497208941}{170361678269} a^{6} - \frac{69474787702}{170361678269} a^{5} + \frac{9154584296}{170361678269} a^{4} - \frac{31403708623}{170361678269} a^{3} - \frac{16585198223}{170361678269} a^{2} - \frac{7294963191}{170361678269} a + \frac{82234032865}{170361678269}$, $\frac{1}{170361678269} a^{26} - \frac{943022080}{170361678269} a^{14} + \frac{4447209223}{170361678269} a^{13} - \frac{42031841280}{170361678269} a^{12} + \frac{10388007960}{170361678269} a^{11} - \frac{65426774284}{170361678269} a^{10} - \frac{6316600354}{170361678269} a^{9} + \frac{5230009291}{170361678269} a^{8} - \frac{18055813359}{170361678269} a^{7} + \frac{76620197671}{170361678269} a^{6} - \frac{84421892363}{170361678269} a^{5} + \frac{74246788590}{170361678269} a^{4} - \frac{47952255104}{170361678269} a^{3} - \frac{72006144247}{170361678269} a^{2} - \frac{28558521952}{170361678269} a + \frac{24202598658}{170361678269}$, $\frac{1}{170361678269} a^{27} - \frac{65168397631}{170361678269} a^{14} + \frac{14549483520}{170361678269} a^{13} + \frac{30232624323}{170361678269} a^{12} - \frac{70368405236}{170361678269} a^{11} - \frac{80820075175}{170361678269} a^{10} + \frac{77335825198}{170361678269} a^{9} - \frac{25474877771}{170361678269} a^{8} + \frac{22013078049}{170361678269} a^{7} + \frac{10320689545}{170361678269} a^{6} + \frac{6612807883}{170361678269} a^{5} + \frac{34332951715}{170361678269} a^{4} - \frac{33391414853}{170361678269} a^{3} + \frac{53726684867}{170361678269} a^{2} - \frac{80935144651}{170361678269} a - \frac{21333369962}{170361678269}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{29}) \), 4.0.7048421.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ R $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29Data not computed