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Properties

Label 28.0.20510294149...0000.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 5^{14}\cdot 29^{24}$
Root discriminant $80.17$
Ramified primes $2, 5, 29$
Class number $90944$ (GRH)
Class group $[4, 28, 812]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, 0, 92266, 0, 1320817, 0, 8079074, 0, 24218096, 0, 38013350, 0, 34627149, 0, 19717109, 0, 7322517, 0, 1804843, 0, 294090, 0, 30865, 0, 1974, 0, 69, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681)
 
gp: K = bnfinit(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681, 1)
 

Normalized defining polynomial

\( x^{28} + 69 x^{26} + 1974 x^{24} + 30865 x^{22} + 294090 x^{20} + 1804843 x^{18} + 7322517 x^{16} + 19717109 x^{14} + 34627149 x^{12} + 38013350 x^{10} + 24218096 x^{8} + 8079074 x^{6} + 1320817 x^{4} + 92266 x^{2} + 1681 \)

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:  \(205102941494858641164670947835018741350400000000000000=2^{28}\cdot 5^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.17$
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$$\chi_{580}(1,·)$, $\chi_{580}(451,·)$, $\chi_{580}(199,·)$, $\chi_{580}(139,·)$, $\chi_{580}(141,·)$, $\chi_{580}(59,·)$, $\chi_{580}(401,·)$, $\chi_{580}(339,·)$, $\chi_{580}(529,·)$, $\chi_{580}(471,·)$, $\chi_{580}(281,·)$, $\chi_{580}(431,·)$, $\chi_{580}(349,·)$, $\chi_{580}(459,·)$, $\chi_{580}(161,·)$, $\chi_{580}(291,·)$, $\chi_{580}(81,·)$, $\chi_{580}(169,·)$, $\chi_{580}(429,·)$, $\chi_{580}(239,·)$, $\chi_{580}(49,·)$, $\chi_{580}(371,·)$, $\chi_{580}(181,·)$, $\chi_{580}(219,·)$, $\chi_{580}(489,·)$, $\chi_{580}(111,·)$, $\chi_{580}(571,·)$, $\chi_{580}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{41} a^{19} + \frac{5}{41} a^{17} + \frac{16}{41} a^{15} - \frac{15}{41} a^{13} - \frac{4}{41} a^{11} - \frac{3}{41} a^{9} - \frac{14}{41} a^{7} - \frac{3}{41} a^{5} + \frac{1}{41} a^{3} + \frac{16}{41} a$, $\frac{1}{41} a^{20} + \frac{5}{41} a^{18} + \frac{16}{41} a^{16} - \frac{15}{41} a^{14} - \frac{4}{41} a^{12} - \frac{3}{41} a^{10} - \frac{14}{41} a^{8} - \frac{3}{41} a^{6} + \frac{1}{41} a^{4} + \frac{16}{41} a^{2}$, $\frac{1}{41} a^{21} - \frac{9}{41} a^{17} - \frac{13}{41} a^{15} - \frac{11}{41} a^{13} + \frac{17}{41} a^{11} + \frac{1}{41} a^{9} - \frac{15}{41} a^{7} + \frac{16}{41} a^{5} + \frac{11}{41} a^{3} + \frac{2}{41} a$, $\frac{1}{697} a^{22} + \frac{6}{697} a^{20} + \frac{103}{697} a^{18} + \frac{329}{697} a^{16} - \frac{224}{697} a^{14} + \frac{116}{697} a^{12} - \frac{181}{697} a^{10} - \frac{222}{697} a^{8} - \frac{125}{697} a^{6} - \frac{188}{697} a^{4} - \frac{66}{697} a^{2} - \frac{1}{17}$, $\frac{1}{697} a^{23} + \frac{6}{697} a^{21} + \frac{1}{697} a^{19} - \frac{181}{697} a^{17} + \frac{235}{697} a^{15} + \frac{252}{697} a^{13} + \frac{227}{697} a^{11} + \frac{84}{697} a^{9} - \frac{91}{697} a^{7} + \frac{118}{697} a^{5} - \frac{168}{697} a^{3} - \frac{279}{697} a$, $\frac{1}{4980118669} a^{24} + \frac{2783324}{4980118669} a^{22} - \frac{59097396}{4980118669} a^{20} + \frac{260142159}{4980118669} a^{18} - \frac{799373620}{4980118669} a^{16} + \frac{1385484718}{4980118669} a^{14} - \frac{1906305581}{4980118669} a^{12} - \frac{1815298344}{4980118669} a^{10} - \frac{2191529635}{4980118669} a^{8} + \frac{1939393871}{4980118669} a^{6} + \frac{1967003488}{4980118669} a^{4} - \frac{1175056885}{4980118669} a^{2} + \frac{43383412}{121466309}$, $\frac{1}{4980118669} a^{25} + \frac{2783324}{4980118669} a^{23} - \frac{59097396}{4980118669} a^{21} + \frac{17209541}{4980118669} a^{19} - \frac{2014036710}{4980118669} a^{17} + \frac{2478681499}{4980118669} a^{15} + \frac{42382529}{121466309} a^{13} - \frac{843567872}{4980118669} a^{11} - \frac{1462731781}{4980118669} a^{9} + \frac{360331854}{4980118669} a^{7} - \frac{2284317327}{4980118669} a^{5} - \frac{1417989503}{4980118669} a^{3} - \frac{2108201996}{4980118669} a$, $\frac{1}{48726178279089778669} a^{26} + \frac{2581602224}{48726178279089778669} a^{24} - \frac{32927740787461464}{48726178279089778669} a^{22} - \frac{404349695046022644}{48726178279089778669} a^{20} - \frac{677986517887625026}{2866245781122928157} a^{18} - \frac{1439335712819665506}{48726178279089778669} a^{16} + \frac{23088731755815567971}{48726178279089778669} a^{14} + \frac{9369540585218245119}{48726178279089778669} a^{12} + \frac{12343186782872122929}{48726178279089778669} a^{10} - \frac{6691554883289951982}{48726178279089778669} a^{8} + \frac{20259423277446911726}{48726178279089778669} a^{6} - \frac{8869523208522272700}{48726178279089778669} a^{4} - \frac{20404191615545548828}{48726178279089778669} a^{2} + \frac{3308501989891473}{28986423723432349}$, $\frac{1}{48726178279089778669} a^{27} + \frac{2581602224}{48726178279089778669} a^{25} - \frac{32927740787461464}{48726178279089778669} a^{23} - \frac{404349695046022644}{48726178279089778669} a^{21} + \frac{21097818971625744}{2866245781122928157} a^{19} + \frac{9256654641126871275}{48726178279089778669} a^{17} + \frac{18334958265172662735}{48726178279089778669} a^{15} - \frac{22718430476621365224}{48726178279089778669} a^{13} + \frac{13531630155532849238}{48726178279089778669} a^{11} + \frac{6381322215978037417}{48726178279089778669} a^{9} + \frac{55885942214564473}{48726178279089778669} a^{7} + \frac{4203353890745716699}{48726178279089778669} a^{5} - \frac{8519757888938285738}{48726178279089778669} a^{3} + \frac{19702886691820997}{1188443372660726309} a$

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

Class group and class number

$C_{4}\times C_{28}\times C_{812}$, which has order $90944$ (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:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{76312508254}{1018332217582181} a^{27} + \frac{4887465691706}{1018332217582181} a^{25} + \frac{125531913681362}{1018332217582181} a^{23} + \frac{1673688288942610}{1018332217582181} a^{21} + \frac{12513121902505317}{1018332217582181} a^{19} + \frac{3033452798317255}{59901895151893} a^{17} + \frac{89691424738823298}{1018332217582181} a^{15} - \frac{132263574798686820}{1018332217582181} a^{13} - \frac{1006293990451061196}{1018332217582181} a^{11} - \frac{2148704059783968070}{1018332217582181} a^{9} - \frac{2239360689609567932}{1018332217582181} a^{7} - \frac{1125749116331006403}{1018332217582181} a^{5} - \frac{219195043656905800}{1018332217582181} a^{3} - \frac{276729386944036}{24837371160541} a \) (order $4$)
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:  \( 34681517373.86067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, 14.0.452882922503000372480000000.1, 14.0.5796901408038404767744.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
2Data 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}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$