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Properties

Label 20.0.77671748484...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{31}\cdot 7^{10}$
Root discriminant $55.53$
Ramified primes $3, 5, 7$
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![1410941641, 2827663210, 2942848715, 1874737995, 734798300, 71914741, -100137290, -79891605, -28263385, -5300355, 1072566, 799150, 317315, 40785, 7885, -1319, 430, -50, 30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 30*x^18 - 50*x^17 + 430*x^16 - 1319*x^15 + 7885*x^14 + 40785*x^13 + 317315*x^12 + 799150*x^11 + 1072566*x^10 - 5300355*x^9 - 28263385*x^8 - 79891605*x^7 - 100137290*x^6 + 71914741*x^5 + 734798300*x^4 + 1874737995*x^3 + 2942848715*x^2 + 2827663210*x + 1410941641)
 
gp: K = bnfinit(x^20 - 5*x^19 + 30*x^18 - 50*x^17 + 430*x^16 - 1319*x^15 + 7885*x^14 + 40785*x^13 + 317315*x^12 + 799150*x^11 + 1072566*x^10 - 5300355*x^9 - 28263385*x^8 - 79891605*x^7 - 100137290*x^6 + 71914741*x^5 + 734798300*x^4 + 1874737995*x^3 + 2942848715*x^2 + 2827663210*x + 1410941641, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 30 x^{18} - 50 x^{17} + 430 x^{16} - 1319 x^{15} + 7885 x^{14} + 40785 x^{13} + 317315 x^{12} + 799150 x^{11} + 1072566 x^{10} - 5300355 x^{9} - 28263385 x^{8} - 79891605 x^{7} - 100137290 x^{6} + 71914741 x^{5} + 734798300 x^{4} + 1874737995 x^{3} + 2942848715 x^{2} + 2827663210 x + 1410941641 \)

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:  \(77671748484489507973194122314453125=3^{10}\cdot 5^{31}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.53$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{16} - \frac{1}{12} a^{15} + \frac{1}{36} a^{14} + \frac{1}{36} a^{13} - \frac{1}{4} a^{11} - \frac{1}{18} a^{10} - \frac{17}{36} a^{9} + \frac{5}{36} a^{8} + \frac{1}{9} a^{7} - \frac{17}{36} a^{6} + \frac{1}{9} a^{5} - \frac{11}{36} a^{4} + \frac{5}{12} a^{3} - \frac{5}{18} a^{2} + \frac{1}{18} a + \frac{13}{36}$, $\frac{1}{36} a^{17} - \frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{5}{36} a^{11} + \frac{1}{36} a^{10} - \frac{4}{9} a^{9} + \frac{1}{36} a^{8} + \frac{13}{36} a^{7} - \frac{11}{36} a^{6} + \frac{13}{36} a^{5} + \frac{1}{3} a^{4} - \frac{7}{36} a^{3} - \frac{5}{18} a^{2} - \frac{5}{36} a - \frac{1}{12}$, $\frac{1}{72} a^{18} - \frac{1}{72} a^{16} + \frac{1}{72} a^{15} - \frac{1}{36} a^{14} + \frac{1}{18} a^{13} - \frac{5}{72} a^{12} - \frac{1}{9} a^{11} + \frac{1}{12} a^{10} - \frac{2}{9} a^{9} - \frac{1}{2} a^{8} + \frac{29}{72} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{4} a^{4} + \frac{23}{72} a^{3} - \frac{3}{8} a^{2} - \frac{25}{72} a + \frac{19}{72}$, $\frac{1}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a^{19} + \frac{3967216947502594194929589006660025854080885423610454390754827181447785973}{637959195904890961125624920768651297067276956430176093503603229248559704149} a^{18} + \frac{15875565205635071859724465518659491295376411962592116654268170021694133429}{1701224522413042563001666455383070125512738550480469582676275277996159211064} a^{17} - \frac{4610537680169556399312761128528284955733505723378148540436362445451186523}{567074840804347521000555485127690041837579516826823194225425092665386403688} a^{16} + \frac{275982945823550889927606433015467544655684621759681106689730056391972717}{3589081270913591905066806867896772416693541245739387305224209447249281036} a^{15} - \frac{3354025841047424114391473425809339455838461574688053186904642278558934931}{212653065301630320375208306922883765689092318810058697834534409749519901383} a^{14} - \frac{163109226478360400067699733535936058842040078089056203627927528991346292637}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a^{13} + \frac{6817997259893747879700539140603046679766787571448632434330199497932081048}{637959195904890961125624920768651297067276956430176093503603229248559704149} a^{12} - \frac{83406670282564143473419045874742705147317964426215125062763271698335027905}{2551836783619563844502499683074605188269107825720704374014412916994238816596} a^{11} - \frac{59042789768764260948302098129446022240030871758556941542487751796555785879}{425306130603260640750416613845767531378184637620117395669068819499039802766} a^{10} - \frac{189396058500384659852336826635753443649049873845080633513967403041768627116}{637959195904890961125624920768651297067276956430176093503603229248559704149} a^{9} + \frac{70207334151229112134132364653723468503060534987216233834877441382065579405}{567074840804347521000555485127690041837579516826823194225425092665386403688} a^{8} - \frac{150661003142282626294284952322200687457962480447756620130495254505449991313}{637959195904890961125624920768651297067276956430176093503603229248559704149} a^{7} - \frac{48176644155966995816102197895145271935339402355849555756498023363567277063}{637959195904890961125624920768651297067276956430176093503603229248559704149} a^{6} - \frac{1121931803563895998860537556665650702578602436426563822667044079977990904661}{2551836783619563844502499683074605188269107825720704374014412916994238816596} a^{5} - \frac{1531934066011900388290014464120121689488795919666312217681737011749387551053}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a^{4} + \frac{1584473114451027202283204134550892245521271037930292480606730795909233995925}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a^{3} + \frac{665995113348895889861067653766104917821748822209979163758907609364364182655}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a^{2} + \frac{1875259087990221152735554648385070234850135427853465757653024791008470937807}{5103673567239127689004999366149210376538215651441408748028825833988477633192} a - \frac{364234824894064426147369387653110049169762948920884125056985729143786440795}{1275918391809781922251249841537302594134553912860352187007206458497119408298}$

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:  $9$
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:  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

\(\Q(\sqrt{5}) \), 4.0.55125.1, 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 R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$$[\ ]_{2}^{4}$
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$