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Properties

Label 28.28.2710927942...2417.1
Degree $28$
Signature $[28, 0]$
Discriminant $113^{27}$
Root discriminant $95.45$
Ramified prime $113$
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![1, -106, 1761, -8671, -2990, 123876, -185425, -526380, 1156658, 1130459, -3009246, -1539463, 4180450, 1402578, -3415294, -830930, 1732121, 313492, -560240, -74822, 116343, 11161, -15311, -1001, 1225, 49, -54, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 54*x^26 + 49*x^25 + 1225*x^24 - 1001*x^23 - 15311*x^22 + 11161*x^21 + 116343*x^20 - 74822*x^19 - 560240*x^18 + 313492*x^17 + 1732121*x^16 - 830930*x^15 - 3415294*x^14 + 1402578*x^13 + 4180450*x^12 - 1539463*x^11 - 3009246*x^10 + 1130459*x^9 + 1156658*x^8 - 526380*x^7 - 185425*x^6 + 123876*x^5 - 2990*x^4 - 8671*x^3 + 1761*x^2 - 106*x + 1)
 
gp: K = bnfinit(x^28 - x^27 - 54*x^26 + 49*x^25 + 1225*x^24 - 1001*x^23 - 15311*x^22 + 11161*x^21 + 116343*x^20 - 74822*x^19 - 560240*x^18 + 313492*x^17 + 1732121*x^16 - 830930*x^15 - 3415294*x^14 + 1402578*x^13 + 4180450*x^12 - 1539463*x^11 - 3009246*x^10 + 1130459*x^9 + 1156658*x^8 - 526380*x^7 - 185425*x^6 + 123876*x^5 - 2990*x^4 - 8671*x^3 + 1761*x^2 - 106*x + 1, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 54 x^{26} + 49 x^{25} + 1225 x^{24} - 1001 x^{23} - 15311 x^{22} + 11161 x^{21} + 116343 x^{20} - 74822 x^{19} - 560240 x^{18} + 313492 x^{17} + 1732121 x^{16} - 830930 x^{15} - 3415294 x^{14} + 1402578 x^{13} + 4180450 x^{12} - 1539463 x^{11} - 3009246 x^{10} + 1130459 x^{9} + 1156658 x^{8} - 526380 x^{7} - 185425 x^{6} + 123876 x^{5} - 2990 x^{4} - 8671 x^{3} + 1761 x^{2} - 106 x + 1 \)

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:  \(27109279426973463946497256695670162843066176000405662417=113^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.45$
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:  $113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(113\)
Dirichlet character group:    $\lbrace$$\chi_{113}(64,·)$, $\chi_{113}(1,·)$, $\chi_{113}(2,·)$, $\chi_{113}(4,·)$, $\chi_{113}(7,·)$, $\chi_{113}(8,·)$, $\chi_{113}(14,·)$, $\chi_{113}(15,·)$, $\chi_{113}(16,·)$, $\chi_{113}(81,·)$, $\chi_{113}(83,·)$, $\chi_{113}(85,·)$, $\chi_{113}(28,·)$, $\chi_{113}(30,·)$, $\chi_{113}(32,·)$, $\chi_{113}(97,·)$, $\chi_{113}(98,·)$, $\chi_{113}(99,·)$, $\chi_{113}(105,·)$, $\chi_{113}(106,·)$, $\chi_{113}(109,·)$, $\chi_{113}(111,·)$, $\chi_{113}(112,·)$, $\chi_{113}(49,·)$, $\chi_{113}(53,·)$, $\chi_{113}(56,·)$, $\chi_{113}(57,·)$, $\chi_{113}(60,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{211} a^{25} + \frac{105}{211} a^{24} - \frac{33}{211} a^{23} + \frac{92}{211} a^{22} - \frac{6}{211} a^{21} - \frac{18}{211} a^{20} + \frac{77}{211} a^{19} - \frac{101}{211} a^{18} + \frac{89}{211} a^{17} + \frac{84}{211} a^{16} - \frac{94}{211} a^{15} + \frac{39}{211} a^{14} + \frac{31}{211} a^{13} + \frac{43}{211} a^{12} - \frac{88}{211} a^{11} - \frac{34}{211} a^{10} - \frac{46}{211} a^{9} - \frac{66}{211} a^{8} + \frac{70}{211} a^{7} - \frac{83}{211} a^{6} + \frac{97}{211} a^{5} - \frac{12}{211} a^{4} - \frac{55}{211} a^{3} + \frac{24}{211} a^{2} + \frac{23}{211} a + \frac{47}{211}$, $\frac{1}{229795669} a^{26} + \frac{65570}{229795669} a^{25} + \frac{28122969}{229795669} a^{24} - \frac{8356690}{229795669} a^{23} + \frac{68773330}{229795669} a^{22} - \frac{16890898}{229795669} a^{21} - \frac{43681711}{229795669} a^{20} - \frac{92388335}{229795669} a^{19} + \frac{50579674}{229795669} a^{18} - \frac{98764543}{229795669} a^{17} + \frac{109218548}{229795669} a^{16} + \frac{18910175}{229795669} a^{15} + \frac{110295885}{229795669} a^{14} - \frac{3477431}{229795669} a^{13} - \frac{8855925}{229795669} a^{12} + \frac{22298459}{229795669} a^{11} + \frac{107292850}{229795669} a^{10} - \frac{83180062}{229795669} a^{9} - \frac{30785717}{229795669} a^{8} + \frac{50052123}{229795669} a^{7} - \frac{46513299}{229795669} a^{6} - \frac{75216599}{229795669} a^{5} - \frac{108667825}{229795669} a^{4} + \frac{78746630}{229795669} a^{3} - \frac{95971163}{229795669} a^{2} + \frac{40728532}{229795669} a + \frac{97964188}{229795669}$, $\frac{1}{47770253739616358179743954550686958346681} a^{27} - \frac{42429198729298159776726980766491}{47770253739616358179743954550686958346681} a^{26} - \frac{40212003079964902309617637521648326105}{47770253739616358179743954550686958346681} a^{25} + \frac{20492712004607359179398961757608396377615}{47770253739616358179743954550686958346681} a^{24} - \frac{5191540613224324052351723626855694428966}{47770253739616358179743954550686958346681} a^{23} - \frac{19269918370936786334501385505804597648842}{47770253739616358179743954550686958346681} a^{22} - \frac{20447551809743011722227223357571801596424}{47770253739616358179743954550686958346681} a^{21} - \frac{16197893674855708852684347803699338328513}{47770253739616358179743954550686958346681} a^{20} - \frac{17867128419978114243834150247194696541476}{47770253739616358179743954550686958346681} a^{19} - \frac{6160941216907722440288453228663691649025}{47770253739616358179743954550686958346681} a^{18} - \frac{10464747426078014393383802770725394269395}{47770253739616358179743954550686958346681} a^{17} + \frac{10623246077090522573929162301832606377878}{47770253739616358179743954550686958346681} a^{16} + \frac{12170593852874892434088684083478891792798}{47770253739616358179743954550686958346681} a^{15} + \frac{12647577305549979395351908333860046391}{304269132099467249552509264654057059533} a^{14} + \frac{11343225552792761280692101329556815935956}{47770253739616358179743954550686958346681} a^{13} - \frac{12732574566640281258707532073665697071199}{47770253739616358179743954550686958346681} a^{12} + \frac{7278403867231363267005502039540492425436}{47770253739616358179743954550686958346681} a^{11} - \frac{3044330112486089843263871601605018679398}{47770253739616358179743954550686958346681} a^{10} + \frac{8397503181616661030044451743343489081173}{47770253739616358179743954550686958346681} a^{9} - \frac{17168220472104340951862224817137704029164}{47770253739616358179743954550686958346681} a^{8} - \frac{14054607525657526985825109612262215893178}{47770253739616358179743954550686958346681} a^{7} + \frac{11503849448599064657065665813055986190067}{47770253739616358179743954550686958346681} a^{6} - \frac{3214699391061281717933901350691469471528}{47770253739616358179743954550686958346681} a^{5} - \frac{20043651879194821636938787776125284605310}{47770253739616358179743954550686958346681} a^{4} + \frac{9472451457054946111854354297456664054079}{47770253739616358179743954550686958346681} a^{3} - \frac{19917194040586578278651619395046088277834}{47770253739616358179743954550686958346681} a^{2} - \frac{1422868921612626891919356025728383450990}{47770253739616358179743954550686958346681} a + \frac{13066506138000212660289699154579655760937}{47770253739616358179743954550686958346681}$

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:  \( 5378761397781380000 \) (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

\(\Q(\sqrt{113}) \), 4.4.1442897.1, 7.7.2081951752609.1, 14.14.489801110321660601428677553.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ $28$ $28$ $28$ $28$ ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ $28$

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
113Data not computed