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Properties

Label 20.0.19379770369...1712.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 17^{14}\cdot 37^{8}$
Root discriminant $51.80$
Ramified primes $2, 17, 37$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20591, -18028, 54617, -30932, 99762, -87721, 52825, -7812, -13469, 3940, 7777, -5757, -1781, 2882, 105, -700, 56, 83, -11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 11*x^18 + 83*x^17 + 56*x^16 - 700*x^15 + 105*x^14 + 2882*x^13 - 1781*x^12 - 5757*x^11 + 7777*x^10 + 3940*x^9 - 13469*x^8 - 7812*x^7 + 52825*x^6 - 87721*x^5 + 99762*x^4 - 30932*x^3 + 54617*x^2 - 18028*x + 20591)
 
gp: K = bnfinit(x^20 - 5*x^19 - 11*x^18 + 83*x^17 + 56*x^16 - 700*x^15 + 105*x^14 + 2882*x^13 - 1781*x^12 - 5757*x^11 + 7777*x^10 + 3940*x^9 - 13469*x^8 - 7812*x^7 + 52825*x^6 - 87721*x^5 + 99762*x^4 - 30932*x^3 + 54617*x^2 - 18028*x + 20591, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 11 x^{18} + 83 x^{17} + 56 x^{16} - 700 x^{15} + 105 x^{14} + 2882 x^{13} - 1781 x^{12} - 5757 x^{11} + 7777 x^{10} + 3940 x^{9} - 13469 x^{8} - 7812 x^{7} + 52825 x^{6} - 87721 x^{5} + 99762 x^{4} - 30932 x^{3} + 54617 x^{2} - 18028 x + 20591 \)

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:  \(19379770369172092530288888794611712=2^{15}\cdot 17^{14}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.80$
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:  $2, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{26} a^{16} - \frac{2}{13} a^{15} + \frac{5}{13} a^{14} + \frac{2}{13} a^{13} + \frac{4}{13} a^{12} - \frac{9}{26} a^{11} - \frac{2}{13} a^{10} - \frac{7}{26} a^{9} + \frac{5}{26} a^{8} - \frac{9}{26} a^{7} - \frac{5}{26} a^{6} + \frac{7}{26} a^{5} - \frac{1}{2} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} - \frac{1}{26} a - \frac{3}{13}$, $\frac{1}{26} a^{17} - \frac{3}{13} a^{15} - \frac{4}{13} a^{14} - \frac{1}{13} a^{13} - \frac{3}{26} a^{12} + \frac{6}{13} a^{11} + \frac{3}{26} a^{10} + \frac{3}{26} a^{9} + \frac{11}{26} a^{8} + \frac{11}{26} a^{7} - \frac{1}{2} a^{6} - \frac{11}{26} a^{5} + \frac{6}{13} a^{4} - \frac{5}{13} a^{3} + \frac{1}{26} a^{2} - \frac{5}{13} a + \frac{1}{13}$, $\frac{1}{551784748918} a^{18} - \frac{2184537548}{275892374459} a^{17} + \frac{766134484}{275892374459} a^{16} - \frac{117845632919}{551784748918} a^{15} + \frac{123989432690}{275892374459} a^{14} + \frac{178099452723}{551784748918} a^{13} + \frac{14600633575}{275892374459} a^{12} - \frac{233691995283}{551784748918} a^{11} - \frac{100082467280}{275892374459} a^{10} + \frac{226772844203}{551784748918} a^{9} + \frac{1096974657}{21222490343} a^{8} - \frac{7386193432}{21222490343} a^{7} - \frac{104922317699}{275892374459} a^{6} - \frac{197045969411}{551784748918} a^{5} - \frac{244466914719}{551784748918} a^{4} - \frac{81266209610}{275892374459} a^{3} - \frac{97284749610}{275892374459} a^{2} + \frac{12339256312}{275892374459} a - \frac{264658574431}{551784748918}$, $\frac{1}{6701372192474074599802774952767540594} a^{19} - \frac{156632550109553101249637}{257745084325925946146260575106443869} a^{18} + \frac{18169816396434669390439546789536991}{3350686096237037299901387476383770297} a^{17} - \frac{111095958606723417846651540738334993}{6701372192474074599802774952767540594} a^{16} - \frac{525048138351332500858695988377625783}{6701372192474074599802774952767540594} a^{15} + \frac{650916125719494286049455211344641463}{6701372192474074599802774952767540594} a^{14} + \frac{893193506666288233546415686930975045}{3350686096237037299901387476383770297} a^{13} - \frac{1109813369550137537051755518558807687}{6701372192474074599802774952767540594} a^{12} - \frac{382555364811739055762733269512508506}{3350686096237037299901387476383770297} a^{11} + \frac{178932683286519532899723522023089135}{3350686096237037299901387476383770297} a^{10} + \frac{1440609326051312277484999640364559886}{3350686096237037299901387476383770297} a^{9} - \frac{2693320681383649462743144200936788839}{6701372192474074599802774952767540594} a^{8} - \frac{97387380173077178846825445342147627}{515490168651851892292521150212887738} a^{7} + \frac{1487777424623309725613458619775129582}{3350686096237037299901387476383770297} a^{6} + \frac{1541109384152428944949648218674793488}{3350686096237037299901387476383770297} a^{5} + \frac{1616702080879064870392799407430596969}{6701372192474074599802774952767540594} a^{4} + \frac{922276805675277749292591597982802125}{6701372192474074599802774952767540594} a^{3} - \frac{1267519248208603918153657945336442571}{3350686096237037299901387476383770297} a^{2} - \frac{2667060823756809100445012259245325815}{6701372192474074599802774952767540594} a + \frac{1531075082718463130065531434997004389}{6701372192474074599802774952767540594}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( 7343104.2357 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.2312.1, 5.5.6725897.1, 10.10.769040737728353.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$