↑

Properties

Label 28.28.2439474914...4989.1
Degree $28$
Signature $[28, 0]$
Discriminant $19^{14}\cdot 29^{27}$
Root discriminant $112.08$
Ramified primes $19, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6930211781, 170071741344, -170071741344, -1068941930531, 1068941930531, 1508206506969, -1508206506969, -995309118031, 995309118031, 381624475719, -381624475719, -94043493031, 94043493031, 15726038219, -15726038219, -1837086781, 1837086781, 151678844, -151678844, -8818031, 8818031, 353219, -353219, -9281, 9281, 144, -144, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 144*x^26 + 144*x^25 + 9281*x^24 - 9281*x^23 - 353219*x^22 + 353219*x^21 + 8818031*x^20 - 8818031*x^19 - 151678844*x^18 + 151678844*x^17 + 1837086781*x^16 - 1837086781*x^15 - 15726038219*x^14 + 15726038219*x^13 + 94043493031*x^12 - 94043493031*x^11 - 381624475719*x^10 + 381624475719*x^9 + 995309118031*x^8 - 995309118031*x^7 - 1508206506969*x^6 + 1508206506969*x^5 + 1068941930531*x^4 - 1068941930531*x^3 - 170071741344*x^2 + 170071741344*x + 6930211781)
 
gp: K = bnfinit(x^28 - x^27 - 144*x^26 + 144*x^25 + 9281*x^24 - 9281*x^23 - 353219*x^22 + 353219*x^21 + 8818031*x^20 - 8818031*x^19 - 151678844*x^18 + 151678844*x^17 + 1837086781*x^16 - 1837086781*x^15 - 15726038219*x^14 + 15726038219*x^13 + 94043493031*x^12 - 94043493031*x^11 - 381624475719*x^10 + 381624475719*x^9 + 995309118031*x^8 - 995309118031*x^7 - 1508206506969*x^6 + 1508206506969*x^5 + 1068941930531*x^4 - 1068941930531*x^3 - 170071741344*x^2 + 170071741344*x + 6930211781, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 144 x^{26} + 144 x^{25} + 9281 x^{24} - 9281 x^{23} - 353219 x^{22} + 353219 x^{21} + 8818031 x^{20} - 8818031 x^{19} - 151678844 x^{18} + 151678844 x^{17} + 1837086781 x^{16} - 1837086781 x^{15} - 15726038219 x^{14} + 15726038219 x^{13} + 94043493031 x^{12} - 94043493031 x^{11} - 381624475719 x^{10} + 381624475719 x^{9} + 995309118031 x^{8} - 995309118031 x^{7} - 1508206506969 x^{6} + 1508206506969 x^{5} + 1068941930531 x^{4} - 1068941930531 x^{3} - 170071741344 x^{2} + 170071741344 x + 6930211781 \)

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:  \(2439474914825139472333913721484598054632328281342568904989=19^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.08$
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:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(551=19\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{551}(512,·)$, $\chi_{551}(1,·)$, $\chi_{551}(322,·)$, $\chi_{551}(75,·)$, $\chi_{551}(324,·)$, $\chi_{551}(457,·)$, $\chi_{551}(267,·)$, $\chi_{551}(398,·)$, $\chi_{551}(400,·)$, $\chi_{551}(248,·)$, $\chi_{551}(18,·)$, $\chi_{551}(20,·)$, $\chi_{551}(343,·)$, $\chi_{551}(474,·)$, $\chi_{551}(286,·)$, $\chi_{551}(96,·)$, $\chi_{551}(417,·)$, $\chi_{551}(419,·)$, $\chi_{551}(37,·)$, $\chi_{551}(360,·)$, $\chi_{551}(210,·)$, $\chi_{551}(381,·)$, $\chi_{551}(113,·)$, $\chi_{551}(115,·)$, $\chi_{551}(246,·)$, $\chi_{551}(56,·)$, $\chi_{551}(379,·)$, $\chi_{551}(189,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6056880491} a^{15} + \frac{2871329889}{6056880491} a^{14} - \frac{75}{6056880491} a^{13} - \frac{1116036027}{6056880491} a^{12} + \frac{2250}{6056880491} a^{11} - \frac{2621851958}{6056880491} a^{10} - \frac{34375}{6056880491} a^{9} - \frac{588756246}{6056880491} a^{8} + \frac{281250}{6056880491} a^{7} - \frac{1935586769}{6056880491} a^{6} - \frac{1181250}{6056880491} a^{5} - \frac{1623884758}{6056880491} a^{4} + \frac{2187500}{6056880491} a^{3} - \frac{998584298}{6056880491} a^{2} - \frac{1171875}{6056880491} a - \frac{43426898}{6056880491}$, $\frac{1}{6056880491} a^{16} - \frac{80}{6056880491} a^{14} + \frac{2242888463}{6056880491} a^{13} + \frac{2600}{6056880491} a^{12} - \frac{422618311}{6056880491} a^{11} - \frac{44000}{6056880491} a^{10} - \frac{1548303207}{6056880491} a^{9} + \frac{412500}{6056880491} a^{8} + \frac{408997011}{6056880491} a^{7} - \frac{2100000}{6056880491} a^{6} + \frac{642624348}{6056880491} a^{5} + \frac{5250000}{6056880491} a^{4} - \frac{1606560870}{6056880491} a^{3} - \frac{5000000}{6056880491} a^{2} + \frac{282274837}{6056880491} a + \frac{781250}{6056880491}$, $\frac{1}{6056880491} a^{17} + \frac{1787820925}{6056880491} a^{14} - \frac{3400}{6056880491} a^{13} + \frac{1147706894}{6056880491} a^{12} + \frac{136000}{6056880491} a^{11} + \frac{694357338}{6056880491} a^{10} - \frac{2337500}{6056880491} a^{9} + \frac{1763541259}{6056880491} a^{8} + \frac{20400000}{6056880491} a^{7} - \frac{2782304897}{6056880491} a^{6} - \frac{89250000}{6056880491} a^{5} + \frac{1734029292}{6056880491} a^{4} + \frac{170000000}{6056880491} a^{3} - \frac{865022620}{6056880491} a^{2} - \frac{92968750}{6056880491} a + \frac{2582728651}{6056880491}$, $\frac{1}{6056880491} a^{18} - \frac{3825}{6056880491} a^{14} + \frac{1982905467}{6056880491} a^{13} + \frac{165750}{6056880491} a^{12} - \frac{134077888}{6056880491} a^{11} - \frac{3155625}{6056880491} a^{10} - \frac{1058504043}{6056880491} a^{9} + \frac{31556250}{6056880491} a^{8} + \frac{2687140691}{6056880491} a^{7} - \frac{167343750}{6056880491} a^{6} + \frac{567127590}{6056880491} a^{5} + \frac{430312500}{6056880491} a^{4} + \frac{1968893179}{6056880491} a^{3} - \frac{418359375}{6056880491} a^{2} - \frac{17026329}{6056880491} a + \frac{66406250}{6056880491}$, $\frac{1}{6056880491} a^{19} - \frac{2361479782}{6056880491} a^{14} - \frac{121125}{6056880491} a^{13} + \frac{1128864992}{6056880491} a^{12} + \frac{5450625}{6056880491} a^{11} + \frac{551849703}{6056880491} a^{10} - \frac{99928125}{6056880491} a^{9} - \frac{2202838098}{6056880491} a^{8} + \frac{908437500}{6056880491} a^{7} - \frac{1544303833}{6056880491} a^{6} + \frac{1968911741}{6056880491} a^{5} - \frac{1087802896}{6056880491} a^{4} + \frac{1891947634}{6056880491} a^{3} + \frac{2289623642}{6056880491} a^{2} + \frac{1640864866}{6056880491} a - \frac{2572111593}{6056880491}$, $\frac{1}{6056880491} a^{20} - \frac{142500}{6056880491} a^{14} - \frac{332584419}{6056880491} a^{13} + \frac{6946875}{6056880491} a^{12} + \frac{1997168596}{6056880491} a^{11} - \frac{141075000}{6056880491} a^{10} + \frac{2298876525}{6056880491} a^{9} + \frac{1469531250}{6056880491} a^{8} - \frac{2585856938}{6056880491} a^{7} - \frac{1958744509}{6056880491} a^{6} - \frac{2770160346}{6056880491} a^{5} + \frac{2870374152}{6056880491} a^{4} + \frac{1591508981}{6056880491} a^{3} - \frac{2610608527}{6056880491} a^{2} + \frac{2777173093}{6056880491} a - \frac{2717036741}{6056880491}$, $\frac{1}{6056880491} a^{21} - \frac{2328090933}{6056880491} a^{14} - \frac{3740625}{6056880491} a^{13} + \frac{2374373283}{6056880491} a^{12} + \frac{179550000}{6056880491} a^{11} + \frac{1011068369}{6056880491} a^{10} + \frac{2627974241}{6056880491} a^{9} - \frac{442350606}{6056880491} a^{8} + \frac{1778097545}{6056880491} a^{7} + \frac{395936803}{6056880491} a^{6} - \frac{1921977591}{6056880491} a^{5} + \frac{1132652636}{6056880491} a^{4} + \frac{207236432}{6056880491} a^{3} - \frac{1191916844}{6056880491} a^{2} - \frac{116570493}{6056880491} a + \frac{1798896802}{6056880491}$, $\frac{1}{6056880491} a^{22} - \frac{4571875}{6056880491} a^{14} - \frac{2639792944}{6056880491} a^{13} + \frac{237737500}{6056880491} a^{12} + \frac{14042904}{6056880491} a^{11} + \frac{1027817991}{6056880491} a^{10} + \frac{993755102}{6056880491} a^{9} - \frac{629111919}{6056880491} a^{8} + \frac{2962243989}{6056880491} a^{7} + \frac{2814727675}{6056880491} a^{6} - \frac{2377580956}{6056880491} a^{5} + \frac{569900188}{6056880491} a^{4} - \frac{75378036}{6056880491} a^{3} - \frac{569900188}{6056880491} a^{2} + \frac{1200751012}{6056880491} a + \frac{2688322189}{6056880491}$, $\frac{1}{6056880491} a^{23} + \frac{2991832045}{6056880491} a^{14} - \frac{105153125}{6056880491} a^{13} - \frac{502474411}{6056880491} a^{12} - \frac{799224241}{6056880491} a^{11} + \frac{55780037}{6056880491} a^{10} - \frac{308422278}{6056880491} a^{9} - \frac{2968453915}{6056880491} a^{8} - \frac{1460973158}{6056880491} a^{7} + \frac{2796021426}{6056880491} a^{6} + \frac{2779954410}{6056880491} a^{5} - \frac{1169038000}{6056880491} a^{4} + \frac{496971671}{6056880491} a^{3} + \frac{2567825967}{6056880491} a^{2} - \frac{695339392}{6056880491} a + \frac{2193201230}{6056880491}$, $\frac{1}{6056880491} a^{24} - \frac{132825000}{6056880491} a^{14} - \frac{219649203}{6056880491} a^{13} + \frac{1137807009}{6056880491} a^{12} - \frac{2372095712}{6056880491} a^{11} + \frac{935142766}{6056880491} a^{10} + \frac{1484236271}{6056880491} a^{9} - \frac{1899913328}{6056880491} a^{8} + \frac{2155577351}{6056880491} a^{7} - \frac{204344891}{6056880491} a^{6} + \frac{1520827115}{6056880491} a^{5} + \frac{2368795353}{6056880491} a^{4} - \frac{1067432285}{6056880491} a^{3} - \frac{6678749}{6056880491} a^{2} - \frac{185682200}{6056880491} a - \frac{882201824}{6056880491}$, $\frac{1}{6056880491} a^{25} - \frac{1098246015}{6056880491} a^{14} - \frac{2767187500}{6056880491} a^{13} - \frac{1439123782}{6056880491} a^{12} + \frac{3004248707}{6056880491} a^{11} + \frac{55030160}{6056880491} a^{10} - \frac{871398114}{6056880491} a^{9} + \frac{720982623}{6056880491} a^{8} - \frac{2011963379}{6056880491} a^{7} - \frac{550770771}{6056880491} a^{6} + \frac{269784217}{6056880491} a^{5} - \frac{2237946653}{6056880491} a^{4} + \frac{66787490}{6056880491} a^{3} + \frac{1277460760}{6056880491} a^{2} + \frac{592661385}{6056880491} a + \frac{1555546485}{6056880491}$, $\frac{1}{6056880491} a^{26} + \frac{2459536741}{6056880491} a^{14} + \frac{988751967}{6056880491} a^{13} + \frac{546381297}{6056880491} a^{12} - \frac{98676418}{6056880491} a^{11} + \frac{94270260}{6056880491} a^{10} + \frac{1050317401}{6056880491} a^{9} + \frac{2990472994}{6056880491} a^{8} - \frac{1593451548}{6056880491} a^{7} + \frac{1594012552}{6056880491} a^{6} - \frac{281439586}{6056880491} a^{5} + \frac{1709573531}{6056880491} a^{4} + \frac{1243562038}{6056880491} a^{3} - \frac{2975041289}{6056880491} a^{2} + \frac{2871609477}{6056880491} a - \frac{2238532756}{6056880491}$, $\frac{1}{6056880491} a^{27} + \frac{1420943606}{6056880491} a^{14} - \frac{2751658349}{6056880491} a^{13} - \frac{67688693}{6056880491} a^{12} + \frac{2125371784}{6056880491} a^{11} - \frac{2830285282}{6056880491} a^{10} + \frac{1571171000}{6056880491} a^{9} - \frac{2086815655}{6056880491} a^{8} + \frac{1092722430}{6056880491} a^{7} - \frac{2884771328}{6056880491} a^{6} + \frac{1392239847}{6056880491} a^{5} - \frac{173466307}{6056880491} a^{4} + \frac{2547849637}{6056880491} a^{3} + \frac{100234540}{6056880491} a^{2} - \frac{2168784078}{6056880491} a - \frac{1782999591}{6056880491}$

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:  $27$
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:  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

\(\Q(\sqrt{29}) \), 4.4.8804429.1, 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 $28$ $28$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
19Data not computed
29Data not computed