↑

Properties

Label 28.28.1455848000...5625.1
Degree $28$
Signature $[28, 0]$
Discriminant $5^{21}\cdot 29^{27}$
Root discriminant $85.98$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_{28}$ (as 28T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -146, 2582, -7831, -39932, 201056, 67136, -1290327, 602998, 3724759, -2718459, -5646852, 4711051, 4803704, -4269698, -2405667, 2232740, 743298, -713109, -145523, 142913, 18037, -17950, -1364, 1364, 57, -57, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1364*x^24 - 1364*x^23 - 17950*x^22 + 18037*x^21 + 142913*x^20 - 145523*x^19 - 713109*x^18 + 743298*x^17 + 2232740*x^16 - 2405667*x^15 - 4269698*x^14 + 4803704*x^13 + 4711051*x^12 - 5646852*x^11 - 2718459*x^10 + 3724759*x^9 + 602998*x^8 - 1290327*x^7 + 67136*x^6 + 201056*x^5 - 39932*x^4 - 7831*x^3 + 2582*x^2 - 146*x + 1)
 
gp: K = bnfinit(x^28 - x^27 - 57*x^26 + 57*x^25 + 1364*x^24 - 1364*x^23 - 17950*x^22 + 18037*x^21 + 142913*x^20 - 145523*x^19 - 713109*x^18 + 743298*x^17 + 2232740*x^16 - 2405667*x^15 - 4269698*x^14 + 4803704*x^13 + 4711051*x^12 - 5646852*x^11 - 2718459*x^10 + 3724759*x^9 + 602998*x^8 - 1290327*x^7 + 67136*x^6 + 201056*x^5 - 39932*x^4 - 7831*x^3 + 2582*x^2 - 146*x + 1, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 57 x^{26} + 57 x^{25} + 1364 x^{24} - 1364 x^{23} - 17950 x^{22} + 18037 x^{21} + 142913 x^{20} - 145523 x^{19} - 713109 x^{18} + 743298 x^{17} + 2232740 x^{16} - 2405667 x^{15} - 4269698 x^{14} + 4803704 x^{13} + 4711051 x^{12} - 5646852 x^{11} - 2718459 x^{10} + 3724759 x^{9} + 602998 x^{8} - 1290327 x^{7} + 67136 x^{6} + 201056 x^{5} - 39932 x^{4} - 7831 x^{3} + 2582 x^{2} - 146 x + 1 \)

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:  \(1455848000512373226044338588471370773272037506103515625=5^{21}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.98$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(128,·)$, $\chi_{145}(1,·)$, $\chi_{145}(2,·)$, $\chi_{145}(4,·)$, $\chi_{145}(129,·)$, $\chi_{145}(8,·)$, $\chi_{145}(9,·)$, $\chi_{145}(141,·)$, $\chi_{145}(77,·)$, $\chi_{145}(143,·)$, $\chi_{145}(144,·)$, $\chi_{145}(17,·)$, $\chi_{145}(18,·)$, $\chi_{145}(73,·)$, $\chi_{145}(68,·)$, $\chi_{145}(32,·)$, $\chi_{145}(16,·)$, $\chi_{145}(34,·)$, $\chi_{145}(36,·)$, $\chi_{145}(81,·)$, $\chi_{145}(64,·)$, $\chi_{145}(109,·)$, $\chi_{145}(111,·)$, $\chi_{145}(136,·)$, $\chi_{145}(113,·)$, $\chi_{145}(137,·)$, $\chi_{145}(72,·)$, $\chi_{145}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} - \frac{4}{17} a^{14} - \frac{1}{17} a^{13} + \frac{4}{17} a^{12} + \frac{1}{17} a^{11} - \frac{4}{17} a^{10} - \frac{1}{17} a^{9} + \frac{4}{17} a^{8} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} - \frac{1}{17} a^{5} + \frac{4}{17} a^{4} + \frac{1}{17} a^{3} - \frac{4}{17} a^{2} - \frac{1}{17} a + \frac{4}{17}$, $\frac{1}{17} a^{16} - \frac{1}{17}$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} - \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} - \frac{1}{17} a^{7}$, $\frac{1}{17} a^{24} - \frac{1}{17} a^{8}$, $\frac{1}{17} a^{25} - \frac{1}{17} a^{9}$, $\frac{1}{7172287267} a^{26} + \frac{64091731}{7172287267} a^{25} - \frac{97707799}{7172287267} a^{24} - \frac{116897231}{7172287267} a^{23} - \frac{33019805}{7172287267} a^{22} - \frac{180063001}{7172287267} a^{21} - \frac{15079777}{7172287267} a^{20} - \frac{24574813}{7172287267} a^{19} - \frac{90020743}{7172287267} a^{18} - \frac{113598916}{7172287267} a^{17} - \frac{981477}{421899251} a^{16} - \frac{20372831}{7172287267} a^{15} - \frac{2681809971}{7172287267} a^{14} - \frac{1821841655}{7172287267} a^{13} - \frac{1677735836}{7172287267} a^{12} + \frac{2759677799}{7172287267} a^{11} - \frac{2956307629}{7172287267} a^{10} + \frac{988886}{24817603} a^{9} - \frac{3030449065}{7172287267} a^{8} - \frac{2418772837}{7172287267} a^{7} - \frac{366577413}{7172287267} a^{6} - \frac{106067572}{7172287267} a^{5} + \frac{2412429615}{7172287267} a^{4} - \frac{3301315877}{7172287267} a^{3} + \frac{419769765}{7172287267} a^{2} - \frac{200924295}{421899251} a + \frac{1852279569}{7172287267}$, $\frac{1}{161345623225237524806681473170404851} a^{27} - \frac{4862415666708142183809934}{161345623225237524806681473170404851} a^{26} - \frac{402077186431944499868268310436034}{161345623225237524806681473170404851} a^{25} - \frac{3652368299166853773213153377690503}{161345623225237524806681473170404851} a^{24} - \frac{2100540096300082405296509757725425}{161345623225237524806681473170404851} a^{23} + \frac{156275088618060137706556436161688}{161345623225237524806681473170404851} a^{22} - \frac{4097555807375415744058283280008129}{161345623225237524806681473170404851} a^{21} - \frac{1744154022148413475398781550912303}{161345623225237524806681473170404851} a^{20} + \frac{5130474636918688903906573231238}{286581924023512477454141160160577} a^{19} - \frac{973851104633789161299510217487775}{161345623225237524806681473170404851} a^{18} - \frac{2346539817324016996088891877534374}{161345623225237524806681473170404851} a^{17} + \frac{1557251659717840491626505126542500}{161345623225237524806681473170404851} a^{16} - \frac{3481978727393119962208014490579053}{161345623225237524806681473170404851} a^{15} + \frac{57959375235097715065290446042417447}{161345623225237524806681473170404851} a^{14} + \frac{331415000375749583944776176559241}{161345623225237524806681473170404851} a^{13} - \frac{28817585912984963486271839187292625}{161345623225237524806681473170404851} a^{12} - \frac{71702984246428619660563495537165170}{161345623225237524806681473170404851} a^{11} + \frac{62333671080349615244659193293320659}{161345623225237524806681473170404851} a^{10} + \frac{45338988833254995078293112872206277}{161345623225237524806681473170404851} a^{9} + \frac{7678494824589975970272598341483107}{161345623225237524806681473170404851} a^{8} + \frac{44934333941157514049533437015680834}{161345623225237524806681473170404851} a^{7} + \frac{38036367592855670773765716465634312}{161345623225237524806681473170404851} a^{6} - \frac{59513586063415383071285864571629566}{161345623225237524806681473170404851} a^{5} - \frac{1950364284559126959398580184731961}{9490919013249266165098910186494403} a^{4} + \frac{54934998073080686566140133844903123}{161345623225237524806681473170404851} a^{3} + \frac{79509221829247151067680836426617511}{161345623225237524806681473170404851} a^{2} - \frac{20263374973159783921826129555315202}{161345623225237524806681473170404851} a + \frac{70360102618006064776886764755492216}{161345623225237524806681473170404851}$

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

Class group and class number

$C_{4}$, 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:  \( 1266343040764484400 \) (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

\(\Q(\sqrt{145}) \), 4.4.3048625.1, 7.7.594823321.1, 14.14.801611618199890796015625.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R $28$ $28$ $28$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ $28$ $28$ R $28$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
5Data not computed
29Data not computed