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Properties

Label 20.0.58299958569...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{35}\cdot 11^{18}$
Root discriminant $973.38$
Ramified primes $2, 5, 11$
Class number $96731380360$ (GRH)
Class group $[2, 48365690180]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3596332960, 0, 9488880790400, 0, 446208997828000, 0, 103761119545600, 0, 7831272677800, 0, 278566111120, 0, 5249103200, 0, 54245400, 0, 306240, 0, 880, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 880*x^18 + 306240*x^16 + 54245400*x^14 + 5249103200*x^12 + 278566111120*x^10 + 7831272677800*x^8 + 103761119545600*x^6 + 446208997828000*x^4 + 9488880790400*x^2 + 3596332960)
 
gp: K = bnfinit(x^20 + 880*x^18 + 306240*x^16 + 54245400*x^14 + 5249103200*x^12 + 278566111120*x^10 + 7831272677800*x^8 + 103761119545600*x^6 + 446208997828000*x^4 + 9488880790400*x^2 + 3596332960, 1)
 

Normalized defining polynomial

\( x^{20} + 880 x^{18} + 306240 x^{16} + 54245400 x^{14} + 5249103200 x^{12} + 278566111120 x^{10} + 7831272677800 x^{8} + 103761119545600 x^{6} + 446208997828000 x^{4} + 9488880790400 x^{2} + 3596332960 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(582999585691243011742105600000000000000000000000000000000000=2^{55}\cdot 5^{35}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $973.38$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4400=2^{4}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4400}(1547,·)$, $\chi_{4400}(1,·)$, $\chi_{4400}(963,·)$, $\chi_{4400}(3041,·)$, $\chi_{4400}(2323,·)$, $\chi_{4400}(1929,·)$, $\chi_{4400}(1867,·)$, $\chi_{4400}(403,·)$, $\chi_{4400}(3281,·)$, $\chi_{4400}(2243,·)$, $\chi_{4400}(1849,·)$, $\chi_{4400}(2507,·)$, $\chi_{4400}(2721,·)$, $\chi_{4400}(4009,·)$, $\chi_{4400}(2987,·)$, $\chi_{4400}(2483,·)$, $\chi_{4400}(2561,·)$, $\chi_{4400}(3369,·)$, $\chi_{4400}(889,·)$, $\chi_{4400}(827,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{44} a^{10}$, $\frac{1}{44} a^{11}$, $\frac{1}{88} a^{12}$, $\frac{1}{88} a^{13}$, $\frac{1}{88} a^{14}$, $\frac{1}{88} a^{15}$, $\frac{1}{16657168} a^{16} + \frac{779}{757144} a^{14} + \frac{347}{8328584} a^{12} + \frac{26903}{4164292} a^{10} - \frac{2187}{378572} a^{8} + \frac{495}{6106} a^{6} - \frac{2654}{94643} a^{4} - \frac{13093}{94643} a^{2} - \frac{1861}{94643}$, $\frac{1}{16657168} a^{17} + \frac{779}{757144} a^{15} + \frac{347}{8328584} a^{13} + \frac{26903}{4164292} a^{11} - \frac{2187}{378572} a^{9} + \frac{495}{6106} a^{7} - \frac{2654}{94643} a^{5} - \frac{13093}{94643} a^{3} - \frac{1861}{94643} a$, $\frac{1}{2832169794875071370145200545873642336669776565973140016} a^{18} + \frac{1511481613908060989969291072481387232129051881}{2832169794875071370145200545873642336669776565973140016} a^{16} - \frac{221154292536792038918370763894207226048042830176294}{177010612179691960634075034117102646041861035373321251} a^{14} + \frac{31220501103228274964373308166359957005077504522546}{5710019747731998730131452713454924065866485012042621} a^{12} - \frac{412945898705044657488459451921841079312576062951335}{64367495338069803866936376042582780378858558317571364} a^{10} - \frac{3278077859738620952084598656119470305615788455823663}{32183747669034901933468188021291390189429279158785682} a^{8} + \frac{132831060609363752608900308434695572563800470480332}{16091873834517450966734094010645695094714639579392841} a^{6} - \frac{4685841087077497659168858109366118397781417184201873}{32183747669034901933468188021291390189429279158785682} a^{4} + \frac{398856994857551294321629553072059113070865627321098}{16091873834517450966734094010645695094714639579392841} a^{2} + \frac{384704148078015409723470147660372831146204229218730}{16091873834517450966734094010645695094714639579392841}$, $\frac{1}{1220665181591155760532581435271539847104673699934423346896} a^{19} - \frac{1392002327937222626096153870188242396080407026301}{76291573849447235033286339704471240444042106245901459181} a^{17} + \frac{23461190819461780819017672501033554151089587932859727}{4298116836588576621593596603068802278537583450473321644} a^{15} + \frac{683911519399378229123156602849427039781797808389478517}{305166295397788940133145358817884961776168424983605836724} a^{13} - \frac{961650670380983522013609573898299389653881896871322519}{152583147698894470066572679408942480888084212491802918362} a^{11} + \frac{8299715573571619746852558846688974469388569544908417}{390737894235325147417599691188072934412507586406665604} a^{9} - \frac{1532320606873584962388123933151036545836494281332645035}{13871195245354042733324789037176589171644019317436628942} a^{7} - \frac{1914033947584817252290835790566363918914081055250624163}{13871195245354042733324789037176589171644019317436628942} a^{5} + \frac{1933397561619321341639371968524939480647086675527412701}{6935597622677021366662394518588294585822009658718314471} a^{3} - \frac{2548955909025289366418324255076885543970581576272339554}{6935597622677021366662394518588294585822009658718314471} a$

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

Class group and class number

$C_{2}\times C_{48365690180}$, which has order $96731380360$ (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:  $9$
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:  \( 86035906530.77673 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{10}) \), 4.0.30976000.2, 5.5.5719140625.1, 10.10.5358972025000000000000000.3

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R $20$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed