Properties

Label 28.0.186...125.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.863\times 10^{49}$
Root discriminant $57.50$
Ramified primes $5, 29$
Class number $24224$ (GRH)
Class group $[2, 2, 2, 2, 1514]$ (GRH)
Galois group $C_{28}$ (as 28T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851)
 
gp: K = bnfinit(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1149851, -1149822, 1149822, -1148807, 1148807, -1138251, 1138251, -1086979, 1086979, -945981, 945981, -702439, 702439, -421429, 421429, -196621, 196621, -69340, 69340, -17981, 17981, -3307, 3307, -407, 407, -30, 30, -1, 1]);
 

\( x^{28} - x^{27} + 30 x^{26} - 30 x^{25} + 407 x^{24} - 407 x^{23} + 3307 x^{22} - 3307 x^{21} + 17981 x^{20} - 17981 x^{19} + 69340 x^{18} - 69340 x^{17} + 196621 x^{16} - 196621 x^{15} + 421429 x^{14} - 421429 x^{13} + 702439 x^{12} - 702439 x^{11} + 945981 x^{10} - 945981 x^{9} + 1086979 x^{8} - 1086979 x^{7} + 1138251 x^{6} - 1138251 x^{5} + 1148807 x^{4} - 1148807 x^{3} + 1149822 x^{2} - 1149822 x + 1149851 \)

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:  \(18634854406558377293367533932433545897882080078125\)\(\medspace = 5^{14}\cdot 29^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $57.50$
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:  $5, 29$
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:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(69,·)$, $\chi_{145}(6,·)$, $\chi_{145}(71,·)$, $\chi_{145}(136,·)$, $\chi_{145}(141,·)$, $\chi_{145}(14,·)$, $\chi_{145}(79,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(19,·)$, $\chi_{145}(84,·)$, $\chi_{145}(86,·)$, $\chi_{145}(89,·)$, $\chi_{145}(91,·)$, $\chi_{145}(96,·)$, $\chi_{145}(99,·)$, $\chi_{145}(36,·)$, $\chi_{145}(134,·)$, $\chi_{145}(39,·)$, $\chi_{145}(104,·)$, $\chi_{145}(44,·)$, $\chi_{145}(111,·)$, $\chi_{145}(114,·)$, $\chi_{145}(51,·)$, $\chi_{145}(119,·)$, $\chi_{145}(121,·)$, $\chi_{145}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{514229} a^{15} - \frac{196418}{514229} a^{14} + \frac{15}{514229} a^{13} - \frac{178707}{514229} a^{12} + \frac{90}{514229} a^{11} - \frac{211545}{514229} a^{10} + \frac{275}{514229} a^{9} - \frac{109460}{514229} a^{8} + \frac{450}{514229} a^{7} - \frac{153244}{514229} a^{6} + \frac{378}{514229} a^{5} + \frac{69247}{514229} a^{4} + \frac{140}{514229} a^{3} + \frac{145869}{514229} a^{2} + \frac{15}{514229} a + \frac{121393}{514229}$, $\frac{1}{514229} a^{16} + \frac{16}{514229} a^{14} + \frac{196418}{514229} a^{13} + \frac{104}{514229} a^{12} - \frac{17711}{514229} a^{11} + \frac{352}{514229} a^{10} - \frac{88555}{514229} a^{9} + \frac{660}{514229} a^{8} - \frac{212532}{514229} a^{7} + \frac{672}{514229} a^{6} - \frac{247954}{514229} a^{5} + \frac{336}{514229} a^{4} - \frac{123977}{514229} a^{3} + \frac{64}{514229} a^{2} - \frac{17711}{514229} a + \frac{2}{514229}$, $\frac{1}{514229} a^{17} + \frac{253732}{514229} a^{14} - \frac{136}{514229} a^{13} - \frac{243773}{514229} a^{12} - \frac{1088}{514229} a^{11} + \frac{210791}{514229} a^{10} - \frac{3740}{514229} a^{9} - \frac{3859}{514229} a^{8} - \frac{6528}{514229} a^{7} + \frac{147034}{514229} a^{6} - \frac{5712}{514229} a^{5} - \frac{203471}{514229} a^{4} - \frac{2176}{514229} a^{3} + \frac{219530}{514229} a^{2} - \frac{238}{514229} a + \frac{114628}{514229}$, $\frac{1}{514229} a^{18} - \frac{153}{514229} a^{14} + \frac{64079}{514229} a^{13} - \frac{1326}{514229} a^{12} + \frac{987}{514229} a^{11} - \frac{5049}{514229} a^{10} + \frac{154985}{514229} a^{9} - \frac{10098}{514229} a^{8} + \frac{126472}{514229} a^{7} - \frac{10710}{514229} a^{6} + \frac{46656}{514229} a^{5} - \frac{5508}{514229} a^{4} + \frac{178851}{514229} a^{3} - \frac{1071}{514229} a^{2} - \frac{91749}{514229} a - \frac{34}{514229}$, $\frac{1}{514229} a^{19} - \frac{162593}{514229} a^{14} + \frac{969}{514229} a^{13} - \frac{87047}{514229} a^{12} + \frac{8721}{514229} a^{11} + \frac{185027}{514229} a^{10} + \frac{31977}{514229} a^{9} - \frac{165580}{514229} a^{8} + \frac{58140}{514229} a^{7} + \frac{254858}{514229} a^{6} + \frac{52326}{514229} a^{5} - \frac{25167}{514229} a^{4} + \frac{20349}{514229} a^{3} + \frac{114361}{514229} a^{2} + \frac{2261}{514229} a + \frac{60885}{514229}$, $\frac{1}{514229} a^{20} + \frac{1140}{514229} a^{14} - \frac{219297}{514229} a^{13} + \frac{11115}{514229} a^{12} - \frac{94244}{514229} a^{11} + \frac{45144}{514229} a^{10} - \frac{190428}{514229} a^{9} + \frac{94050}{514229} a^{8} - \frac{113039}{514229} a^{7} + \frac{102600}{514229} a^{6} + \frac{241736}{514229} a^{5} + \frac{53865}{514229} a^{4} + \frac{251305}{514229} a^{3} + \frac{10640}{514229} a^{2} - \frac{71365}{514229} a + \frac{342}{514229}$, $\frac{1}{514229} a^{21} + \frac{7608}{514229} a^{14} - \frac{5985}{514229} a^{13} - \frac{2948}{514229} a^{12} - \frac{57456}{514229} a^{11} - \frac{202529}{514229} a^{10} - \frac{219450}{514229} a^{9} + \frac{227943}{514229} a^{8} + \frac{103829}{514229} a^{7} + \frac{102036}{514229} a^{6} + \frac{137174}{514229} a^{5} - \frac{13238}{514229} a^{4} - \frac{148960}{514229} a^{3} + \frac{248171}{514229} a^{2} - \frac{16758}{514229} a - \frac{60419}{514229}$, $\frac{1}{514229} a^{22} - \frac{7315}{514229} a^{14} - \frac{117068}{514229} a^{13} - \frac{76076}{514229} a^{12} + \frac{141209}{514229} a^{11} + \frac{192369}{514229} a^{10} + \frac{192659}{514229} a^{9} - \frac{175471}{514229} a^{8} - \frac{236190}{514229} a^{7} - \frac{253846}{514229} a^{6} + \frac{196312}{514229} a^{5} + \frac{104589}{514229} a^{4} + \frac{211509}{514229} a^{3} - \frac{81928}{514229} a^{2} - \frac{174539}{514229} a - \frac{2660}{514229}$, $\frac{1}{514229} a^{23} - \frac{158912}{514229} a^{14} + \frac{33649}{514229} a^{13} + \frac{69622}{514229} a^{12} - \frac{177739}{514229} a^{11} + \frac{56045}{514229} a^{10} - \frac{220762}{514229} a^{9} + \frac{232692}{514229} a^{8} - \frac{47470}{514229} a^{7} + \frac{235672}{514229} a^{6} - \frac{215715}{514229} a^{5} + \frac{237749}{514229} a^{4} - \frac{86286}{514229} a^{3} - \frac{167979}{514229} a^{2} + \frac{107065}{514229} a - \frac{83688}{514229}$, $\frac{1}{514229} a^{24} + \frac{42504}{514229} a^{14} - \frac{117843}{514229} a^{13} - \frac{53769}{514229} a^{12} - \frac{40287}{514229} a^{11} - \frac{53156}{514229} a^{10} + \frac{224027}{514229} a^{9} - \frac{244836}{514229} a^{8} - \frac{245988}{514229} a^{7} - \frac{183490}{514229} a^{6} + \frac{141692}{514229} a^{5} + \frac{106607}{514229} a^{4} - \frac{32146}{514229} a^{3} + \frac{26731}{514229} a^{2} + \frac{243076}{514229} a + \frac{17710}{514229}$, $\frac{1}{514229} a^{25} - \frac{74986}{514229} a^{14} - \frac{177100}{514229} a^{13} + \frac{45482}{514229} a^{12} + \frac{235316}{514229} a^{11} - \frac{75587}{514229} a^{10} - \frac{106169}{514229} a^{9} + \frac{12089}{514229} a^{8} + \frac{230412}{514229} a^{7} - \frac{114075}{514229} a^{6} - \frac{18806}{514229} a^{5} + \frac{140162}{514229} a^{4} + \frac{246919}{514229} a^{3} - \frac{228076}{514229} a^{2} - \frac{105621}{514229} a + \frac{85714}{514229}$, $\frac{1}{514229} a^{26} - \frac{230230}{514229} a^{14} + \frac{141814}{514229} a^{13} + \frac{5725}{514229} a^{12} - \frac{11824}{514229} a^{11} - \frac{83347}{514229} a^{10} + \frac{64079}{514229} a^{9} - \frac{128079}{514229} a^{8} + \frac{204740}{514229} a^{7} - \frac{212156}{514229} a^{6} + \frac{202275}{514229} a^{5} + \frac{118019}{514229} a^{4} - \frac{14616}{514229} a^{3} - \frac{137846}{514229} a^{2} + \frac{182046}{514229} a - \frac{106260}{514229}$, $\frac{1}{514229} a^{27} + \frac{123934}{514229} a^{14} - \frac{140428}{514229} a^{13} + \frac{252085}{514229} a^{12} + \frac{68193}{514229} a^{11} + \frac{230006}{514229} a^{10} - \frac{64996}{514229} a^{9} + \frac{49543}{514229} a^{8} + \frac{31315}{514229} a^{7} + \frac{87845}{514229} a^{6} + \frac{240258}{514229} a^{5} + \frac{80507}{514229} a^{4} + \frac{212156}{514229} a^{3} - \frac{179845}{514229} a^{2} - \frac{252413}{514229} a - \frac{35760}{514229}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1514}$, which has order $24224$ (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:  \( -1 \) (order $2$)
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:  \( 487075979.1876791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 24224}{2\sqrt{18634854406558377293367533932433545897882080078125}}\approx 0.204252663933886$ (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.609725.2, 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 $28$ $28$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed