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Properties

Label 28.28.1347590420...0000.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{28}\cdot 5^{21}\cdot 29^{26}$
Root discriminant $152.47$
Ramified primes $2, 5, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![65703125, 0, -854140625, 0, 4349546875, 0, -11274656250, 0, 16449434375, 0, -14139312500, 0, 7347186250, 0, -2365022500, 0, 480954125, 0, -62582000, 0, 5210575, 0, -272600, 0, 8555, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 145*x^26 + 8555*x^24 - 272600*x^22 + 5210575*x^20 - 62582000*x^18 + 480954125*x^16 - 2365022500*x^14 + 7347186250*x^12 - 14139312500*x^10 + 16449434375*x^8 - 11274656250*x^6 + 4349546875*x^4 - 854140625*x^2 + 65703125)
 
gp: K = bnfinit(x^28 - 145*x^26 + 8555*x^24 - 272600*x^22 + 5210575*x^20 - 62582000*x^18 + 480954125*x^16 - 2365022500*x^14 + 7347186250*x^12 - 14139312500*x^10 + 16449434375*x^8 - 11274656250*x^6 + 4349546875*x^4 - 854140625*x^2 + 65703125, 1)
 

Normalized defining polynomial

\( x^{28} - 145 x^{26} + 8555 x^{24} - 272600 x^{22} + 5210575 x^{20} - 62582000 x^{18} + 480954125 x^{16} - 2365022500 x^{14} + 7347186250 x^{12} - 14139312500 x^{10} + 16449434375 x^{8} - 11274656250 x^{6} + 4349546875 x^{4} - 854140625 x^{2} + 65703125 \)

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:  \(13475904202904384157772520869472715740288000000000000000000000=2^{28}\cdot 5^{21}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.47$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(323,·)$, $\chi_{580}(81,·)$, $\chi_{580}(267,·)$, $\chi_{580}(141,·)$, $\chi_{580}(207,·)$, $\chi_{580}(527,·)$, $\chi_{580}(401,·)$, $\chi_{580}(67,·)$, $\chi_{580}(529,·)$, $\chi_{580}(463,·)$, $\chi_{580}(281,·)$, $\chi_{580}(283,·)$, $\chi_{580}(349,·)$, $\chi_{580}(161,·)$, $\chi_{580}(347,·)$, $\chi_{580}(167,·)$, $\chi_{580}(169,·)$, $\chi_{580}(429,·)$, $\chi_{580}(303,·)$, $\chi_{580}(489,·)$, $\chi_{580}(49,·)$, $\chi_{580}(383,·)$, $\chi_{580}(181,·)$, $\chi_{580}(183,·)$, $\chi_{580}(187,·)$, $\chi_{580}(509,·)$, $\chi_{580}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{3625} a^{14}$, $\frac{1}{3625} a^{15}$, $\frac{1}{18125} a^{16}$, $\frac{1}{18125} a^{17}$, $\frac{1}{743125} a^{18} - \frac{16}{743125} a^{16} - \frac{11}{148625} a^{14} - \frac{2}{5125} a^{12} + \frac{16}{1025} a^{10} - \frac{13}{1025} a^{8} + \frac{7}{205} a^{6} - \frac{12}{205} a^{4} + \frac{16}{41} a^{2} + \frac{5}{41}$, $\frac{1}{743125} a^{19} - \frac{16}{743125} a^{17} - \frac{11}{148625} a^{15} - \frac{2}{5125} a^{13} + \frac{16}{1025} a^{11} - \frac{13}{1025} a^{9} + \frac{7}{205} a^{7} - \frac{12}{205} a^{5} + \frac{16}{41} a^{3} + \frac{5}{41} a$, $\frac{1}{63165625} a^{20} + \frac{1}{2526625} a^{18} + \frac{9}{743125} a^{16} - \frac{28}{2526625} a^{14} + \frac{262}{87125} a^{12} - \frac{19}{17425} a^{10} - \frac{2}{1025} a^{8} + \frac{96}{3485} a^{6} + \frac{139}{3485} a^{4} + \frac{165}{697} a^{2} - \frac{2}{17}$, $\frac{1}{63165625} a^{21} + \frac{1}{2526625} a^{19} + \frac{9}{743125} a^{17} - \frac{28}{2526625} a^{15} + \frac{262}{87125} a^{13} - \frac{19}{17425} a^{11} - \frac{2}{1025} a^{9} + \frac{96}{3485} a^{7} + \frac{139}{3485} a^{5} + \frac{165}{697} a^{3} - \frac{2}{17} a$, $\frac{1}{63165625} a^{22} - \frac{6}{12633125} a^{18} + \frac{64}{12633125} a^{16} + \frac{308}{2526625} a^{14} - \frac{304}{87125} a^{12} - \frac{103}{17425} a^{10} - \frac{319}{17425} a^{8} + \frac{17}{205} a^{6} - \frac{151}{3485} a^{4} + \frac{128}{697} a^{2} - \frac{211}{697}$, $\frac{1}{63165625} a^{23} - \frac{6}{12633125} a^{19} + \frac{64}{12633125} a^{17} + \frac{308}{2526625} a^{15} - \frac{304}{87125} a^{13} - \frac{103}{17425} a^{11} - \frac{319}{17425} a^{9} + \frac{17}{205} a^{7} - \frac{151}{3485} a^{5} + \frac{128}{697} a^{3} - \frac{211}{697} a$, $\frac{1}{18633859375} a^{24} + \frac{16}{3726771875} a^{22} - \frac{18}{3726771875} a^{20} - \frac{218}{745354375} a^{18} + \frac{11566}{745354375} a^{16} + \frac{14221}{149070875} a^{14} - \frac{9216}{5140375} a^{12} + \frac{2921}{205615} a^{10} + \frac{7578}{1028075} a^{8} - \frac{288}{205615} a^{6} - \frac{131}{205615} a^{4} + \frac{6496}{41123} a^{2} + \frac{9780}{41123}$, $\frac{1}{18633859375} a^{25} + \frac{16}{3726771875} a^{23} - \frac{18}{3726771875} a^{21} - \frac{218}{745354375} a^{19} + \frac{11566}{745354375} a^{17} + \frac{14221}{149070875} a^{15} - \frac{9216}{5140375} a^{13} + \frac{2921}{205615} a^{11} + \frac{7578}{1028075} a^{9} - \frac{288}{205615} a^{7} - \frac{131}{205615} a^{5} + \frac{6496}{41123} a^{3} + \frac{9780}{41123} a$, $\frac{1}{32147860914265625} a^{26} + \frac{26321}{1108546928078125} a^{24} + \frac{32231}{13041728565625} a^{22} - \frac{2139}{751557239375} a^{20} + \frac{27720789}{44341877123125} a^{18} + \frac{18266907}{1081509198125} a^{16} + \frac{776646232}{8868375424625} a^{14} - \frac{26442842861}{8868375424625} a^{12} + \frac{1089502891}{61161209825} a^{10} - \frac{738564808}{61161209825} a^{8} + \frac{804084111}{12232241965} a^{6} + \frac{232028072}{12232241965} a^{4} - \frac{277708371}{2446448393} a^{2} - \frac{1195024387}{2446448393}$, $\frac{1}{32147860914265625} a^{27} + \frac{26321}{1108546928078125} a^{25} + \frac{32231}{13041728565625} a^{23} - \frac{2139}{751557239375} a^{21} + \frac{27720789}{44341877123125} a^{19} + \frac{18266907}{1081509198125} a^{17} + \frac{776646232}{8868375424625} a^{15} - \frac{26442842861}{8868375424625} a^{13} + \frac{1089502891}{61161209825} a^{11} - \frac{738564808}{61161209825} a^{9} + \frac{804084111}{12232241965} a^{7} + \frac{232028072}{12232241965} a^{5} - \frac{277708371}{2446448393} a^{3} - \frac{1195024387}{2446448393} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1362591492474554000000 \) (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

\(\Q(\sqrt{5}) \), 4.4.1682000.2, 7.7.594823321.1, 14.14.27641779937927268828125.1

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 R $28$ R $28$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
2Data not computed
5Data not computed
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$