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Properties

Label 20.0.64453982490...1801.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 127^{10}$
Root discriminant $19.52$
Ramified primes $3, 127$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -198, 1206, -407, -2392, 3679, -2523, 1133, -748, 849, -752, 358, -82, 84, -98, 38, -6, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 7*x^18 - 6*x^17 + 38*x^16 - 98*x^15 + 84*x^14 - 82*x^13 + 358*x^12 - 752*x^11 + 849*x^10 - 748*x^9 + 1133*x^8 - 2523*x^7 + 3679*x^6 - 2392*x^5 - 407*x^4 + 1206*x^3 - 198*x^2 + 81)
 
gp: K = bnfinit(x^20 - 5*x^19 + 7*x^18 - 6*x^17 + 38*x^16 - 98*x^15 + 84*x^14 - 82*x^13 + 358*x^12 - 752*x^11 + 849*x^10 - 748*x^9 + 1133*x^8 - 2523*x^7 + 3679*x^6 - 2392*x^5 - 407*x^4 + 1206*x^3 - 198*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 7 x^{18} - 6 x^{17} + 38 x^{16} - 98 x^{15} + 84 x^{14} - 82 x^{13} + 358 x^{12} - 752 x^{11} + 849 x^{10} - 748 x^{9} + 1133 x^{8} - 2523 x^{7} + 3679 x^{6} - 2392 x^{5} - 407 x^{4} + 1206 x^{3} - 198 x^{2} + 81 \)

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:  \(64453982490130814650341801=3^{10}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.52$
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:  $3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{63} a^{16} + \frac{1}{21} a^{15} - \frac{1}{21} a^{13} - \frac{10}{63} a^{12} + \frac{8}{63} a^{11} + \frac{2}{63} a^{10} - \frac{2}{63} a^{9} + \frac{10}{63} a^{8} - \frac{16}{63} a^{7} + \frac{5}{21} a^{6} + \frac{8}{21} a^{5} + \frac{5}{63} a^{4} - \frac{29}{63} a^{3} - \frac{8}{63} a^{2} + \frac{1}{21} a - \frac{3}{7}$, $\frac{1}{315} a^{17} - \frac{2}{315} a^{16} - \frac{8}{315} a^{15} - \frac{2}{63} a^{14} - \frac{1}{35} a^{13} + \frac{1}{35} a^{12} - \frac{38}{315} a^{11} - \frac{8}{63} a^{10} - \frac{22}{315} a^{9} + \frac{46}{315} a^{8} + \frac{19}{63} a^{7} + \frac{89}{315} a^{6} - \frac{1}{105} a^{5} + \frac{107}{315} a^{4} - \frac{23}{63} a^{3} - \frac{76}{315} a^{2} - \frac{1}{15} a + \frac{8}{35}$, $\frac{1}{945} a^{18} - \frac{1}{945} a^{17} + \frac{4}{315} a^{15} - \frac{19}{945} a^{14} - \frac{2}{63} a^{13} - \frac{8}{315} a^{12} + \frac{107}{945} a^{11} - \frac{7}{45} a^{10} - \frac{101}{945} a^{9} + \frac{346}{945} a^{8} + \frac{26}{105} a^{7} + \frac{26}{945} a^{6} - \frac{181}{945} a^{5} + \frac{2}{45} a^{4} + \frac{464}{945} a^{3} - \frac{8}{105} a^{2} - \frac{26}{105} a + \frac{16}{35}$, $\frac{1}{70682108655345705} a^{19} - \frac{2064802929673}{23560702885115235} a^{18} - \frac{21908368528754}{14136421731069141} a^{17} - \frac{26260321476943}{7853567628371745} a^{16} + \frac{1429470915863129}{70682108655345705} a^{15} - \frac{323190741632443}{70682108655345705} a^{14} - \frac{411921627173003}{7853567628371745} a^{13} - \frac{3576479976626038}{70682108655345705} a^{12} + \frac{7552902629117156}{70682108655345705} a^{11} + \frac{1674077835902603}{14136421731069141} a^{10} - \frac{2201128765517858}{14136421731069141} a^{9} + \frac{17569096364182213}{70682108655345705} a^{8} + \frac{2359918377682634}{70682108655345705} a^{7} - \frac{2974335169582121}{70682108655345705} a^{6} + \frac{7803847799447054}{70682108655345705} a^{5} + \frac{5287380068491817}{70682108655345705} a^{4} - \frac{29965780094638849}{70682108655345705} a^{3} + \frac{127431942612860}{523571175224783} a^{2} - \frac{570550820611489}{2617855876123915} a - \frac{535041520032891}{2617855876123915}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{20792896236083}{1121938232624535} a^{19} + \frac{316406947554743}{3365814697873605} a^{18} - \frac{493445238192283}{3365814697873605} a^{17} + \frac{106368932009524}{673162939574721} a^{16} - \frac{835330270712128}{1121938232624535} a^{15} + \frac{6479708874404764}{3365814697873605} a^{14} - \frac{6841796977136548}{3365814697873605} a^{13} + \frac{499593004405672}{224387646524907} a^{12} - \frac{24125278027072862}{3365814697873605} a^{11} + \frac{51609326879648291}{3365814697873605} a^{10} - \frac{13175111858410526}{673162939574721} a^{9} + \frac{22147923700244143}{1121938232624535} a^{8} - \frac{92052645347638613}{3365814697873605} a^{7} + \frac{181026481011228427}{3365814697873605} a^{6} - \frac{18178325426676785}{224387646524907} a^{5} + \frac{227350400067571229}{3365814697873605} a^{4} - \frac{60984261949487876}{3365814697873605} a^{3} - \frac{31891216649954833}{3365814697873605} a^{2} + \frac{355848576230272}{74795882174969} a - \frac{123058272896158}{74795882174969} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 191976.450688 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-127}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{381}) \), \(\Q(\sqrt{-3}, \sqrt{-127})\), 5.1.16129.1 x5, 10.0.33038369407.1, 10.0.63215147763.1 x5, 10.2.8028323765901.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$127$127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$