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Properties

Label 20.4.37906719768...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{31}\cdot 13^{10}$
Root discriminant $75.67$
Ramified primes $3, 5, 13$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![521177724016, -73949086720, 91075080080, 9898849230, -55565363835, -176450583, 7876207515, -919172770, -275565135, 150274870, -35238496, -1234795, 1752770, -305645, 35340, -3602, -1085, 500, -35, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 35*x^18 + 500*x^17 - 1085*x^16 - 3602*x^15 + 35340*x^14 - 305645*x^13 + 1752770*x^12 - 1234795*x^11 - 35238496*x^10 + 150274870*x^9 - 275565135*x^8 - 919172770*x^7 + 7876207515*x^6 - 176450583*x^5 - 55565363835*x^4 + 9898849230*x^3 + 91075080080*x^2 - 73949086720*x + 521177724016)
 
gp: K = bnfinit(x^20 - 5*x^19 - 35*x^18 + 500*x^17 - 1085*x^16 - 3602*x^15 + 35340*x^14 - 305645*x^13 + 1752770*x^12 - 1234795*x^11 - 35238496*x^10 + 150274870*x^9 - 275565135*x^8 - 919172770*x^7 + 7876207515*x^6 - 176450583*x^5 - 55565363835*x^4 + 9898849230*x^3 + 91075080080*x^2 - 73949086720*x + 521177724016, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 35 x^{18} + 500 x^{17} - 1085 x^{16} - 3602 x^{15} + 35340 x^{14} - 305645 x^{13} + 1752770 x^{12} - 1234795 x^{11} - 35238496 x^{10} + 150274870 x^{9} - 275565135 x^{8} - 919172770 x^{7} + 7876207515 x^{6} - 176450583 x^{5} - 55565363835 x^{4} + 9898849230 x^{3} + 91075080080 x^{2} - 73949086720 x + 521177724016 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37906719768380750901997089385986328125=3^{10}\cdot 5^{31}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.67$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{19} - \frac{60171369321515932925554966686503711575523799150845538580876502478239473004908658025407792102480455615970493}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{18} - \frac{1243004170877276630285360462480877126276466905042352247525766497887552956085717836861299809207734184091558197}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{17} - \frac{495637003645739771259464777942139776475513492656836399915493758313789472707475685555368025491591528355939065}{5962653868090895567150167120990228475971960718845976917487342712130338397930739466174089331791546665168448116} a^{16} - \frac{167251148545779759635989152619644380593899718642848325724368788423474660263645075664210511945446655243075993}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{15} + \frac{189971156867272113968072979261503345068731938598459552171513809294372554526833738576117785625598756924043505}{2981326934045447783575083560495114237985980359422988458743671356065169198965369733087044665895773332584224058} a^{14} + \frac{1221968729574180882816603284608261456619388554763299334783178394870091249145420495073218368469279419987442547}{5962653868090895567150167120990228475971960718845976917487342712130338397930739466174089331791546665168448116} a^{13} + \frac{1726922445848399634224103015787188705475632121420923200985851835064319437696071413842684029347960939179153953}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{12} + \frac{55484513503494867000177334960336112662913100881505712316624108320989002468757533219181746073678802839131727}{1490663467022723891787541780247557118992990179711494229371835678032584599482684866543522332947886666292112029} a^{11} + \frac{5091860026473591258729525838566996390253277379679278250120395042748547116043790249429353940023864833403426277}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{10} - \frac{1457978015002878027479409287708520028742644598103104661113542855767959336970151063017131251708794616848105213}{5962653868090895567150167120990228475971960718845976917487342712130338397930739466174089331791546665168448116} a^{9} - \frac{92045512090308048059618038816145174556507587375882673636785614084673821899636778985999453290723334146139956}{1490663467022723891787541780247557118992990179711494229371835678032584599482684866543522332947886666292112029} a^{8} + \frac{1627504278173675688340961894154752289108775039303262069857908802185831151045297879642833230599435444603801675}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{7} + \frac{663816360001000673444284855189750078137839431250498028716804818824539183759109437338375432303077609845708234}{1490663467022723891787541780247557118992990179711494229371835678032584599482684866543522332947886666292112029} a^{6} - \frac{680874888547739233150939206334203603931410685607397957392951311944348492612807495055342506432891602841913445}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{5} - \frac{1826257548282044033154116709821819239460266821558006804395335503639491236621118752284899350421432184896086277}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{4} + \frac{4652996752515875438531178005170021597019836678580554695279066935285382566861397036582462813797430386206106595}{11925307736181791134300334241980456951943921437691953834974685424260676795861478932348178663583093330336896232} a^{3} - \frac{757610746152952550889512634852583189891450914615169491099680676903257770034737975519160409199597486867223391}{5962653868090895567150167120990228475971960718845976917487342712130338397930739466174089331791546665168448116} a^{2} - \frac{412938606001829610458668186668824106250361859577046069520723945871558537382430807971380418844318397039065658}{1490663467022723891787541780247557118992990179711494229371835678032584599482684866543522332947886666292112029} a - \frac{506124794664446410321831785933581094602461660996434807243656512424836142649674278996880874155402937432048171}{1490663467022723891787541780247557118992990179711494229371835678032584599482684866543522332947886666292112029}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.190125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$