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Properties

Label 20.0.44639374932...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{14}\cdot 53^{15}$
Root discriminant $60.60$
Ramified primes $5, 53$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![473141, -1775774, 175064, 4177271, 939687, -540658, 118099, -486474, 59848, 95621, -31000, 13968, -7646, -2861, 2185, -436, 331, -124, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 124*x^17 + 331*x^16 - 436*x^15 + 2185*x^14 - 2861*x^13 - 7646*x^12 + 13968*x^11 - 31000*x^10 + 95621*x^9 + 59848*x^8 - 486474*x^7 + 118099*x^6 - 540658*x^5 + 939687*x^4 + 4177271*x^3 + 175064*x^2 - 1775774*x + 473141)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 124*x^17 + 331*x^16 - 436*x^15 + 2185*x^14 - 2861*x^13 - 7646*x^12 + 13968*x^11 - 31000*x^10 + 95621*x^9 + 59848*x^8 - 486474*x^7 + 118099*x^6 - 540658*x^5 + 939687*x^4 + 4177271*x^3 + 175064*x^2 - 1775774*x + 473141, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 124 x^{17} + 331 x^{16} - 436 x^{15} + 2185 x^{14} - 2861 x^{13} - 7646 x^{12} + 13968 x^{11} - 31000 x^{10} + 95621 x^{9} + 59848 x^{8} - 486474 x^{7} + 118099 x^{6} - 540658 x^{5} + 939687 x^{4} + 4177271 x^{3} + 175064 x^{2} - 1775774 x + 473141 \)

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:  \(446393749322684446560596661376953125=5^{14}\cdot 53^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.60$
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:  $5, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} + \frac{1}{8} a^{14} - \frac{1}{4} a^{12} + \frac{3}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{120} a^{18} - \frac{1}{60} a^{17} + \frac{11}{120} a^{16} + \frac{1}{120} a^{15} + \frac{23}{120} a^{14} - \frac{1}{15} a^{13} + \frac{1}{15} a^{12} + \frac{7}{40} a^{11} - \frac{1}{120} a^{10} - \frac{1}{20} a^{9} - \frac{19}{40} a^{8} - \frac{1}{30} a^{7} - \frac{47}{120} a^{6} - \frac{41}{120} a^{5} - \frac{19}{40} a^{4} + \frac{13}{30} a^{3} - \frac{1}{4} a^{2} + \frac{19}{120} a - \frac{19}{120}$, $\frac{1}{472043443338023363665628042792558421923132972542319595958328520} a^{19} - \frac{99866897475728750276541263992377419526051680503341210273673}{472043443338023363665628042792558421923132972542319595958328520} a^{18} - \frac{6595452838953454706496312456985825953274765683533471173065309}{157347814446007787888542680930852807307710990847439865319442840} a^{17} - \frac{170456241135607845808863702415396156829015001048537107599513}{15734781444600778788854268093085280730771099084743986531944284} a^{16} + \frac{2100869036074330496932346714753003786332751715879948855968748}{19668476805750973486067835116356600913463873855929983164930355} a^{15} - \frac{27690783956649489002485416876681624495770914556067648580662307}{157347814446007787888542680930852807307710990847439865319442840} a^{14} - \frac{9877838536579824704824615411247050265882344866294371878308419}{78673907223003893944271340465426403653855495423719932659721420} a^{13} - \frac{54388966559806005093994265905221932155223384766274884858794077}{472043443338023363665628042792558421923132972542319595958328520} a^{12} - \frac{24337496563185816862973795566302426208160083456217907631918871}{236021721669011681832814021396279210961566486271159797979164260} a^{11} - \frac{1329394841056190831271840968527500776399911304803025901966209}{94408688667604672733125608558511684384626594508463919191665704} a^{10} + \frac{78127708788469214510574725665315462276201493326598870719217223}{157347814446007787888542680930852807307710990847439865319442840} a^{9} - \frac{16139306457667682772807499495326141339335723721812956529211277}{472043443338023363665628042792558421923132972542319595958328520} a^{8} + \frac{42158168846833313814556334278938183318974014041973402190372419}{157347814446007787888542680930852807307710990847439865319442840} a^{7} + \frac{37254984769636899805406792455728158061441515079680725380741823}{236021721669011681832814021396279210961566486271159797979164260} a^{6} - \frac{79669502046280154362688480050565691597735432886562428294422193}{236021721669011681832814021396279210961566486271159797979164260} a^{5} + \frac{90621927863511766362388932472482340367319880899686422807848919}{472043443338023363665628042792558421923132972542319595958328520} a^{4} + \frac{12303371643751193075865562883297747391732402975776144269244846}{59005430417252920458203505349069802740391621567789949494791065} a^{3} - \frac{206812700522500446203958957123527459637931424600670964337383181}{472043443338023363665628042792558421923132972542319595958328520} a^{2} - \frac{26985123190845651779444830666678305828381741514004918979100923}{78673907223003893944271340465426403653855495423719932659721420} a + \frac{38540330804029730654902081943934152971067046479148737173744859}{472043443338023363665628042792558421923132972542319595958328520}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 745932630.7132764 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.3721925.1, 5.1.351125.1, 10.2.6534304578125.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
53Data not computed