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Properties

Label 28.28.1891740735...1232.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{77}\cdot 29^{24}$
Root discriminant $120.59$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (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![-23953, 211268, 46562, -3152012, 1821397, 17598492, -10970104, -51247828, 26775430, 86510324, -35706314, -88343908, 30251654, 56540840, -17269984, -23204920, 6666823, 6176908, -1718136, -1063528, 290042, 115948, -31106, -7620, 2007, 272, -70, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 4*x^27 - 70*x^26 + 272*x^25 + 2007*x^24 - 7620*x^23 - 31106*x^22 + 115948*x^21 + 290042*x^20 - 1063528*x^19 - 1718136*x^18 + 6176908*x^17 + 6666823*x^16 - 23204920*x^15 - 17269984*x^14 + 56540840*x^13 + 30251654*x^12 - 88343908*x^11 - 35706314*x^10 + 86510324*x^9 + 26775430*x^8 - 51247828*x^7 - 10970104*x^6 + 17598492*x^5 + 1821397*x^4 - 3152012*x^3 + 46562*x^2 + 211268*x - 23953)
 
gp: K = bnfinit(x^28 - 4*x^27 - 70*x^26 + 272*x^25 + 2007*x^24 - 7620*x^23 - 31106*x^22 + 115948*x^21 + 290042*x^20 - 1063528*x^19 - 1718136*x^18 + 6176908*x^17 + 6666823*x^16 - 23204920*x^15 - 17269984*x^14 + 56540840*x^13 + 30251654*x^12 - 88343908*x^11 - 35706314*x^10 + 86510324*x^9 + 26775430*x^8 - 51247828*x^7 - 10970104*x^6 + 17598492*x^5 + 1821397*x^4 - 3152012*x^3 + 46562*x^2 + 211268*x - 23953, 1)
 

Normalized defining polynomial

\( x^{28} - 4 x^{27} - 70 x^{26} + 272 x^{25} + 2007 x^{24} - 7620 x^{23} - 31106 x^{22} + 115948 x^{21} + 290042 x^{20} - 1063528 x^{19} - 1718136 x^{18} + 6176908 x^{17} + 6666823 x^{16} - 23204920 x^{15} - 17269984 x^{14} + 56540840 x^{13} + 30251654 x^{12} - 88343908 x^{11} - 35706314 x^{10} + 86510324 x^{9} + 26775430 x^{8} - 51247828 x^{7} - 10970104 x^{6} + 17598492 x^{5} + 1821397 x^{4} - 3152012 x^{3} + 46562 x^{2} + 211268 x - 23953 \)

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:  \(18917407352603402612306290142504402709970002327473311711232=2^{77}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.59$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(464=2^{4}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{464}(1,·)$, $\chi_{464}(197,·)$, $\chi_{464}(257,·)$, $\chi_{464}(393,·)$, $\chi_{464}(141,·)$, $\chi_{464}(45,·)$, $\chi_{464}(401,·)$, $\chi_{464}(297,·)$, $\chi_{464}(277,·)$, $\chi_{464}(25,·)$, $\chi_{464}(397,·)$, $\chi_{464}(281,·)$, $\chi_{464}(285,·)$, $\chi_{464}(161,·)$, $\chi_{464}(373,·)$, $\chi_{464}(165,·)$, $\chi_{464}(65,·)$, $\chi_{464}(81,·)$, $\chi_{464}(169,·)$, $\chi_{464}(429,·)$, $\chi_{464}(413,·)$, $\chi_{464}(49,·)$, $\chi_{464}(181,·)$, $\chi_{464}(349,·)$, $\chi_{464}(233,·)$, $\chi_{464}(313,·)$, $\chi_{464}(117,·)$, $\chi_{464}(53,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} + \frac{6}{17} a^{14} + \frac{8}{17} a^{13} - \frac{1}{17} a^{12} + \frac{8}{17} a^{11} + \frac{8}{17} a^{10} - \frac{6}{17} a^{9} - \frac{5}{17} a^{8} + \frac{2}{17} a^{7} - \frac{1}{17} a^{6} + \frac{6}{17} a^{5} - \frac{4}{17} a^{4} - \frac{5}{17} a^{3} - \frac{3}{17} a^{2} + \frac{3}{17} a$, $\frac{1}{17} a^{16} + \frac{6}{17} a^{14} + \frac{2}{17} a^{13} - \frac{3}{17} a^{12} - \frac{6}{17} a^{11} - \frac{3}{17} a^{10} - \frac{3}{17} a^{9} - \frac{2}{17} a^{8} + \frac{4}{17} a^{7} - \frac{5}{17} a^{6} - \frac{6}{17} a^{5} + \frac{2}{17} a^{4} - \frac{7}{17} a^{3} + \frac{4}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} - \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} - \frac{1}{17} a^{7}$, $\frac{1}{289} a^{24} - \frac{7}{289} a^{23} - \frac{6}{289} a^{22} + \frac{4}{289} a^{21} + \frac{6}{289} a^{20} + \frac{1}{289} a^{19} - \frac{6}{289} a^{18} - \frac{4}{289} a^{17} - \frac{3}{289} a^{16} + \frac{67}{289} a^{14} + \frac{96}{289} a^{13} + \frac{43}{289} a^{12} - \frac{33}{289} a^{11} - \frac{76}{289} a^{10} - \frac{76}{289} a^{9} + \frac{90}{289} a^{8} + \frac{80}{289} a^{7} + \frac{38}{289} a^{6} - \frac{37}{289} a^{5} - \frac{29}{289} a^{4} + \frac{122}{289} a^{3} + \frac{45}{289} a^{2} + \frac{75}{289} a - \frac{6}{17}$, $\frac{1}{289} a^{25} - \frac{4}{289} a^{23} - \frac{4}{289} a^{22} - \frac{8}{289} a^{20} + \frac{1}{289} a^{19} + \frac{5}{289} a^{18} + \frac{3}{289} a^{17} - \frac{4}{289} a^{16} - \frac{1}{289} a^{15} - \frac{30}{289} a^{14} - \frac{84}{289} a^{13} - \frac{4}{289} a^{12} - \frac{86}{289} a^{11} - \frac{47}{289} a^{10} - \frac{5}{17} a^{9} - \frac{140}{289} a^{8} - \frac{99}{289} a^{7} - \frac{111}{289} a^{6} + \frac{103}{289} a^{5} - \frac{13}{289} a^{4} - \frac{36}{289} a^{3} + \frac{33}{289} a^{2} - \frac{121}{289} a - \frac{8}{17}$, $\frac{1}{15698139221542769281} a^{26} + \frac{6108405607370287}{15698139221542769281} a^{25} - \frac{23517317032195676}{15698139221542769281} a^{24} + \frac{23069043629753715}{15698139221542769281} a^{23} - \frac{382238707362947693}{15698139221542769281} a^{22} - \frac{139716445921718211}{15698139221542769281} a^{21} - \frac{119956891304452291}{15698139221542769281} a^{20} + \frac{270156437789745809}{15698139221542769281} a^{19} + \frac{143484733476097478}{15698139221542769281} a^{18} - \frac{299141511581112184}{15698139221542769281} a^{17} - \frac{216127259526829208}{15698139221542769281} a^{16} - \frac{449115500779830690}{15698139221542769281} a^{15} - \frac{3061829901535367553}{15698139221542769281} a^{14} - \frac{71587092340123892}{15698139221542769281} a^{13} - \frac{4782703866820071824}{15698139221542769281} a^{12} + \frac{5274422981369140058}{15698139221542769281} a^{11} - \frac{6335204769763236030}{15698139221542769281} a^{10} + \frac{5091977876658823112}{15698139221542769281} a^{9} + \frac{1098030436986027959}{15698139221542769281} a^{8} - \frac{4487147056039712413}{15698139221542769281} a^{7} + \frac{5549475272892342842}{15698139221542769281} a^{6} - \frac{2237129696050613971}{15698139221542769281} a^{5} + \frac{4951641786020613409}{15698139221542769281} a^{4} - \frac{4120381262656247977}{15698139221542769281} a^{3} - \frac{6585772650167318001}{15698139221542769281} a^{2} - \frac{6237952836505258215}{15698139221542769281} a - \frac{134139842736508928}{923419954208398193}$, $\frac{1}{57644075786493336122536356085426288267313} a^{27} - \frac{1541539404109973477592}{57644075786493336122536356085426288267313} a^{26} - \frac{52922917094727059692883374969333433981}{57644075786493336122536356085426288267313} a^{25} - \frac{47792919482891096651010813882706687647}{57644075786493336122536356085426288267313} a^{24} - \frac{82531058679032858205993333564544731965}{3390827987440784477796256240319193427489} a^{23} + \frac{1518678486663700958917586570218682636202}{57644075786493336122536356085426288267313} a^{22} - \frac{1357789706675813478997702919347759569420}{57644075786493336122536356085426288267313} a^{21} + \frac{1061828565675958555672168858640487709411}{57644075786493336122536356085426288267313} a^{20} + \frac{987073919399265675219483996276260421291}{57644075786493336122536356085426288267313} a^{19} - \frac{199978868401865831499316797038839987732}{57644075786493336122536356085426288267313} a^{18} + \frac{672121144284601366208143320403495089754}{57644075786493336122536356085426288267313} a^{17} + \frac{575187030509192025895725277862335534765}{57644075786493336122536356085426288267313} a^{16} - \frac{97432958861262851036933892830171899022}{3390827987440784477796256240319193427489} a^{15} + \frac{2526908846955223728749597473868290653137}{57644075786493336122536356085426288267313} a^{14} + \frac{13183955946472295171839178641544117490315}{57644075786493336122536356085426288267313} a^{13} + \frac{6345297608232758347372450861073326062919}{57644075786493336122536356085426288267313} a^{12} - \frac{14862737998695272290050120285614810093011}{57644075786493336122536356085426288267313} a^{11} - \frac{6996640325294377871276975050806052211081}{57644075786493336122536356085426288267313} a^{10} + \frac{9416493662196438516243398639524977980779}{57644075786493336122536356085426288267313} a^{9} + \frac{27293487170840781404607231840971674670510}{57644075786493336122536356085426288267313} a^{8} + \frac{19343711322673296713886625314625163090503}{57644075786493336122536356085426288267313} a^{7} + \frac{6738859161087985776649932290665766263657}{57644075786493336122536356085426288267313} a^{6} + \frac{19875403544611030534118895974053328423023}{57644075786493336122536356085426288267313} a^{5} - \frac{18647354548424607450397215899254185015137}{57644075786493336122536356085426288267313} a^{4} - \frac{8054977676423595290213374440892108092373}{57644075786493336122536356085426288267313} a^{3} + \frac{5431163694316093566645191037627710707082}{57644075786493336122536356085426288267313} a^{2} + \frac{17414627903025071849819813179974548555397}{57644075786493336122536356085426288267313} a - \frac{1240022595818377081933372719823739415166}{3390827987440784477796256240319193427489}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 614883944308310100000 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 7.7.594823321.1, 14.14.742003380228915810271232.1

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 $28$ $28$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{7}$

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
29Data not computed