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Properties

Label 20.0.38877457874...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{26}\cdot 41^{15}$
Root discriminant $301.64$
Ramified primes $2, 5, 41$
Class number $170689280$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2667020]$ (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![107326429960, 512305253600, 821957983120, -51110077080, 217327929960, -31559673260, 48159593480, -1620898960, 9051016800, 579321460, 1009324309, 52805040, 62913495, 1155600, 2376790, -23576, 47525, -410, 390, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 390*x^18 - 410*x^17 + 47525*x^16 - 23576*x^15 + 2376790*x^14 + 1155600*x^13 + 62913495*x^12 + 52805040*x^11 + 1009324309*x^10 + 579321460*x^9 + 9051016800*x^8 - 1620898960*x^7 + 48159593480*x^6 - 31559673260*x^5 + 217327929960*x^4 - 51110077080*x^3 + 821957983120*x^2 + 512305253600*x + 107326429960)
 
gp: K = bnfinit(x^20 + 390*x^18 - 410*x^17 + 47525*x^16 - 23576*x^15 + 2376790*x^14 + 1155600*x^13 + 62913495*x^12 + 52805040*x^11 + 1009324309*x^10 + 579321460*x^9 + 9051016800*x^8 - 1620898960*x^7 + 48159593480*x^6 - 31559673260*x^5 + 217327929960*x^4 - 51110077080*x^3 + 821957983120*x^2 + 512305253600*x + 107326429960, 1)
 

Normalized defining polynomial

\( x^{20} + 390 x^{18} - 410 x^{17} + 47525 x^{16} - 23576 x^{15} + 2376790 x^{14} + 1155600 x^{13} + 62913495 x^{12} + 52805040 x^{11} + 1009324309 x^{10} + 579321460 x^{9} + 9051016800 x^{8} - 1620898960 x^{7} + 48159593480 x^{6} - 31559673260 x^{5} + 217327929960 x^{4} - 51110077080 x^{3} + 821957983120 x^{2} + 512305253600 x + 107326429960 \)

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:  \(38877457874788447772215025000000000000000000000000=2^{24}\cdot 5^{26}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $301.64$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{19} - \frac{367829978229205310247405994164175650318085163074815660258384475509252574053777470068892921202698919440672598965415164789}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{18} + \frac{436657950142369937747161465909495463668125699436797122636461383407074027029776738956946662992695682960781292225986560739}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{17} + \frac{445154211809258310599475720256503541680772312592763524368929006956239310665084430306618669985485315881867146367342125391}{2802385063307381309341086475665056735739365249226188227533155588648218272682427473084602048527323722970841023159833005302} a^{16} - \frac{257527834540200093250648108172841638061610707710898897391283762791281737075384672369884931912216398662510015791277815791}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{15} - \frac{465629660011048605483732371399739885275168459884030815001450116384343740952034277512289103713727641516519167481260619257}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{14} - \frac{1307029209500272551615634569202668228721359401527172647967941331497389302349028705484203698292244293930782965023389339649}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{13} - \frac{572335610426348128406814519011959857399621454497293370132052075498595657409516295058862011926487623037930082575467682065}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{12} - \frac{2729430276507594071712368820057249305195386096141095443657913807847317896082064579061252881223281643751428568696943256003}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{11} - \frac{379504364039757261588459224587253082588856432119470578637097562260843117587032322873895162907984234191463107578240112581}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{10} - \frac{460182539715821976165642820457402205742386268550957656752455481487164167919754528529778062922679684105990693539039682418}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{9} - \frac{1108285786526616017592673535548222404430586404116368166368408815209492031951187309540968046985395539672492089437331482571}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{8} + \frac{1352800527535222331943145799499130511281112866834788423354072853261597588944131812253250471574695178668654541836492141401}{5604770126614762618682172951330113471478730498452376455066311177296436545364854946169204097054647445941682046319666010604} a^{7} + \frac{463377854064538603509106322404088590131429643779837066584743232341550712807622928566462256650362269326565245519099221178}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{6} - \frac{193337624050239018319800859702544015726525289644779265162962205223017938627104331136324983458095647959814107911330161323}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{5} + \frac{498219683752872186291113346569405708023044702057977899656599415227094658696271044590527807306214116237368382788921161107}{2802385063307381309341086475665056735739365249226188227533155588648218272682427473084602048527323722970841023159833005302} a^{4} - \frac{590210978227706304191725963980577087959771107442563022543351073131557638270810544732354645968010173766043247091221934586}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{3} - \frac{670431495772099685177230206055533534375085554459111110237955803648924536256980149950391359409174073846833461872458477970}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a^{2} - \frac{518865091234865580967438859999224393797589608872338191509264875576246061808111564560635839688362363795972935642822498963}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651} a - \frac{258666104688940668827406847402143829789096657264613886295906295914108118477838905809217345052633467649696183631417288852}{1401192531653690654670543237832528367869682624613094113766577794324109136341213736542301024263661861485420511579916502651}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2667020}$, which has order $170689280$ (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:  \( 824768892.422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_5.Q_8$ (as 20T105):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_5:D_5.Q_8$
Character table for $C_5:D_5.Q_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.27568400.4, 10.10.452563285156250000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$5$5.10.15.16$x^{10} - 10 x^{6} + 10$$10$$1$$15$$F_{5}\times C_2$$[7/4]_{4}^{2}$
5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
41Data not computed