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Properties

Label 28.28.8912993094...3125.1
Degree $28$
Signature $[28, 0]$
Discriminant $3^{14}\cdot 5^{14}\cdot 29^{27}$
Root discriminant $99.59$
Ramified primes $3, 5, 29$
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![928458781, 6856169443, -6856169443, -61259327517, 61259327517, 115840964579, -115840964579, -99209390109, 99209390109, 48637728739, -48637728739, -15205345309, 15205345309, 3210926051, -3210926051, -472328221, 472328221, 49014755, -49014755, -3576861, 3576861, 179683, -179683, -5917, 5917, 115, -115, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781)
 
gp: K = bnfinit(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 115 x^{26} + 115 x^{25} + 5917 x^{24} - 5917 x^{23} - 179683 x^{22} + 179683 x^{21} + 3576861 x^{20} - 3576861 x^{19} - 49014755 x^{18} + 49014755 x^{17} + 472328221 x^{16} - 472328221 x^{15} - 3210926051 x^{14} + 3210926051 x^{13} + 15205345309 x^{12} - 15205345309 x^{11} - 48637728739 x^{10} + 48637728739 x^{9} + 99209390109 x^{8} - 99209390109 x^{7} - 115840964579 x^{6} + 115840964579 x^{5} + 61259327517 x^{4} - 61259327517 x^{3} - 6856169443 x^{2} + 6856169443 x + 928458781 \)

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:  \(89129930946082115284480820405277744589647154669189453125=3^{14}\cdot 5^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.59$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(196,·)$, $\chi_{435}(389,·)$, $\chi_{435}(134,·)$, $\chi_{435}(136,·)$, $\chi_{435}(329,·)$, $\chi_{435}(269,·)$, $\chi_{435}(14,·)$, $\chi_{435}(16,·)$, $\chi_{435}(404,·)$, $\chi_{435}(151,·)$, $\chi_{435}(89,·)$, $\chi_{435}(91,·)$, $\chi_{435}(286,·)$, $\chi_{435}(224,·)$, $\chi_{435}(226,·)$, $\chi_{435}(164,·)$, $\chi_{435}(359,·)$, $\chi_{435}(104,·)$, $\chi_{435}(361,·)$, $\chi_{435}(44,·)$, $\chi_{435}(241,·)$, $\chi_{435}(181,·)$, $\chi_{435}(374,·)$, $\chi_{435}(119,·)$, $\chi_{435}(376,·)$, $\chi_{435}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{139256549} a^{15} + \frac{9004948}{139256549} a^{14} - \frac{60}{139256549} a^{13} + \frac{52749108}{139256549} a^{12} + \frac{1440}{139256549} a^{11} - \frac{46427984}{139256549} a^{10} - \frac{17600}{139256549} a^{9} - \frac{12560039}{139256549} a^{8} + \frac{115200}{139256549} a^{7} - \frac{13217711}{139256549} a^{6} - \frac{387072}{139256549} a^{5} - \frac{57590470}{139256549} a^{4} + \frac{573440}{139256549} a^{3} + \frac{57590470}{139256549} a^{2} - \frac{245760}{139256549} a + \frac{10491267}{139256549}$, $\frac{1}{139256549} a^{16} - \frac{64}{139256549} a^{14} + \frac{36019792}{139256549} a^{13} + \frac{1664}{139256549} a^{12} - \frac{62694047}{139256549} a^{11} - \frac{22528}{139256549} a^{10} + \frac{571999}{139256549} a^{9} + \frac{168960}{139256549} a^{8} - \frac{61193810}{139256549} a^{7} - \frac{688128}{139256549} a^{6} + \frac{53476865}{139256549} a^{5} + \frac{1376256}{139256549} a^{4} + \frac{32302819}{139256549} a^{3} - \frac{1048576}{139256549} a^{2} + \frac{1435039}{139256549} a + \frac{131072}{139256549}$, $\frac{1}{139256549} a^{17} + \frac{55310268}{139256549} a^{14} - \frac{2176}{139256549} a^{13} - \frac{28908311}{139256549} a^{12} + \frac{69632}{139256549} a^{11} - \frac{46431448}{139256549} a^{10} - \frac{957440}{139256549} a^{9} - \frac{29497012}{139256549} a^{8} + \frac{6684672}{139256549} a^{7} + \frac{43082655}{139256549} a^{6} - \frac{23396352}{139256549} a^{5} - \frac{32816987}{139256549} a^{4} + \frac{35651584}{139256549} a^{3} + \frac{66554845}{139256549} a^{2} - \frac{15597568}{139256549} a - \frac{24841657}{139256549}$, $\frac{1}{139256549} a^{18} - \frac{2448}{139256549} a^{14} - \frac{52449407}{139256549} a^{13} + \frac{84864}{139256549} a^{12} - \frac{38471340}{139256549} a^{11} - \frac{1292544}{139256549} a^{10} + \frac{27942278}{139256549} a^{9} + \frac{10340352}{139256549} a^{8} - \frac{16391450}{139256549} a^{7} - \frac{43868160}{139256549} a^{6} - \frac{91853}{139256549} a^{5} - \frac{49013477}{139256549} a^{4} + \frac{18073165}{139256549} a^{3} + \frac{69067493}{139256549} a^{2} + \frac{55617584}{139256549} a + \frac{8912896}{139256549}$, $\frac{1}{139256549} a^{19} - \frac{10871445}{139256549} a^{14} - \frac{62016}{139256549} a^{13} + \frac{524121}{139256549} a^{12} + \frac{2232576}{139256549} a^{11} + \frac{5581430}{139256549} a^{10} - \frac{32744448}{139256549} a^{9} + \frac{12330407}{139256549} a^{8} - \frac{40371658}{139256549} a^{7} - \frac{49529013}{139256549} a^{6} - \frac{21769890}{139256549} a^{5} - \frac{35769807}{139256549} a^{4} - \frac{58973426}{139256549} a^{3} - \frac{29795993}{139256549} a^{2} - \frac{35681388}{139256549} a + \frac{59416600}{139256549}$, $\frac{1}{139256549} a^{20} - \frac{72960}{139256549} a^{14} + \frac{44520166}{139256549} a^{13} + \frac{2845440}{139256549} a^{12} + \frac{63728742}{139256549} a^{11} - \frac{46227456}{139256549} a^{10} + \frac{13396733}{139256549} a^{9} - \frac{32540847}{139256549} a^{8} + \frac{6789830}{139256549} a^{7} - \frac{9919812}{139256549} a^{6} - \frac{13331165}{139256549} a^{5} + \frac{48682915}{139256549} a^{4} - \frac{6304276}{139256549} a^{3} - \frac{4081180}{139256549} a^{2} + \frac{69242514}{139256549} a - \frac{59156655}{139256549}$, $\frac{1}{139256549} a^{21} + \frac{33128064}{139256549} a^{14} - \frac{1532160}{139256549} a^{13} + \frac{5403709}{139256549} a^{12} + \frac{58834944}{139256549} a^{11} + \frac{43238518}{139256549} a^{10} - \frac{63327906}{139256549} a^{9} - \frac{65563190}{139256549} a^{8} + \frac{39679248}{139256549} a^{7} - \frac{25923900}{139256549} a^{6} - \frac{62267307}{139256549} a^{5} - \frac{19142499}{139256549} a^{4} + \frac{57136520}{139256549} a^{3} - \frac{57175812}{139256549} a^{2} - \frac{25711434}{139256549} a - \frac{50409533}{139256549}$, $\frac{1}{139256549} a^{22} - \frac{1872640}{139256549} a^{14} + \frac{43495863}{139256549} a^{13} - \frac{61354725}{139256549} a^{12} - \frac{35433884}{139256549} a^{11} - \frac{65029619}{139256549} a^{10} + \frac{60449096}{139256549} a^{9} + \frac{20264331}{139256549} a^{8} - \frac{53171355}{139256549} a^{7} - \frac{64949011}{139256549} a^{6} + \frac{44557640}{139256549} a^{5} + \frac{17868881}{139256549} a^{4} - \frac{53551039}{139256549} a^{3} + \frac{13556205}{139256549} a^{2} + \frac{7718371}{139256549} a + \frac{16324720}{139256549}$, $\frac{1}{139256549} a^{23} - \frac{63226023}{139256549} a^{14} - \frac{34456576}{139256549} a^{13} - \frac{47039875}{139256549} a^{12} - \frac{14302450}{139256549} a^{11} + \frac{37267800}{139256549} a^{10} + \frac{65602444}{139256549} a^{9} - \frac{53478215}{139256549} a^{8} - \frac{45215412}{139256549} a^{7} + \frac{46276056}{139256549} a^{6} + \frac{1696346}{139256549} a^{5} - \frac{11714632}{139256549} a^{4} + \frac{52988466}{139256549} a^{3} - \frac{34118036}{139256549} a^{2} + \frac{39212765}{139256549} a + \frac{52301960}{139256549}$, $\frac{1}{139256549} a^{24} - \frac{43524096}{139256549} a^{14} + \frac{58582117}{139256549} a^{13} - \frac{63547526}{139256549} a^{12} + \frac{8957874}{139256549} a^{11} + \frac{34941724}{139256549} a^{10} - \frac{32399956}{139256549} a^{9} - \frac{27226987}{139256549} a^{8} + \frac{9586760}{139256549} a^{7} + \frac{40217715}{139256549} a^{6} + \frac{50288521}{139256549} a^{5} - \frac{3316335}{139256549} a^{4} + \frac{18439640}{139256549} a^{3} - \frac{43738983}{139256549} a^{2} + \frac{9883449}{139256549} a - \frac{48980206}{139256549}$, $\frac{1}{139256549} a^{25} + \frac{14397487}{139256549} a^{14} - \frac{29118855}{139256549} a^{13} - \frac{23320728}{139256549} a^{12} + \frac{44192914}{139256549} a^{11} - \frac{18553457}{139256549} a^{10} - \frac{1040538}{139256549} a^{9} + \frac{61089024}{139256549} a^{8} - \frac{55226379}{139256549} a^{7} - \frac{15313679}{139256549} a^{6} + \frac{16581675}{139256549} a^{5} + \frac{54122291}{139256549} a^{4} + \frac{19620183}{139256549} a^{3} - \frac{25799202}{139256549} a^{2} + \frac{43228622}{139256549} a - \frac{8384113}{139256549}$, $\frac{1}{139256549} a^{26} + \frac{31773763}{139256549} a^{14} + \frac{4989198}{139256549} a^{13} - \frac{23636518}{139256549} a^{12} - \frac{1708936}{139256549} a^{11} - \frac{40615073}{139256549} a^{10} + \frac{9941044}{139256549} a^{9} - \frac{41273826}{139256549} a^{8} - \frac{60317489}{139256549} a^{7} - \frac{33701312}{139256549} a^{6} + \frac{10375924}{139256549} a^{5} + \frac{60870449}{139256549} a^{4} - \frac{17723919}{139256549} a^{3} + \frac{1978356}{139256549} a^{2} - \frac{51632534}{139256549} a - \frac{61472552}{139256549}$, $\frac{1}{139256549} a^{27} + \frac{22451391}{139256549} a^{14} - \frac{66802424}{139256549} a^{13} - \frac{4795901}{139256549} a^{12} + \frac{20570828}{139256549} a^{11} - \frac{61402275}{139256549} a^{10} + \frac{61910739}{139256549} a^{9} + \frac{34350205}{139256549} a^{8} - \frac{12808447}{139256549} a^{7} - \frac{18439586}{139256549} a^{6} - \frac{65032197}{139256549} a^{5} + \frac{59117441}{139256549} a^{4} - \frac{17805204}{139256549} a^{3} + \frac{10782655}{139256549} a^{2} - \frac{13206298}{139256549} a + \frac{4019617}{139256549}$

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

\(\Q(\sqrt{29}) \), 4.4.5487525.2, 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 $28$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
3Data not computed
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed