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Properties

Label 20.4.31525276730...8089.1
Degree $20$
Signature $[4, 8]$
Discriminant $47^{8}\cdot 163^{10}$
Root discriminant $59.56$
Ramified primes $47, 163$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 0, -94770, -23166, -45405, 13729, 38727, 16588, 10111, 3085, -4565, -195, 1759, 770, 675, 104, -10, 4, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 4*x^17 - 10*x^16 + 104*x^15 + 675*x^14 + 770*x^13 + 1759*x^12 - 195*x^11 - 4565*x^10 + 3085*x^9 + 10111*x^8 + 16588*x^7 + 38727*x^6 + 13729*x^5 - 45405*x^4 - 23166*x^3 - 94770*x^2 + 59049)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 4*x^17 - 10*x^16 + 104*x^15 + 675*x^14 + 770*x^13 + 1759*x^12 - 195*x^11 - 4565*x^10 + 3085*x^9 + 10111*x^8 + 16588*x^7 + 38727*x^6 + 13729*x^5 - 45405*x^4 - 23166*x^3 - 94770*x^2 + 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 4 x^{17} - 10 x^{16} + 104 x^{15} + 675 x^{14} + 770 x^{13} + 1759 x^{12} - 195 x^{11} - 4565 x^{10} + 3085 x^{9} + 10111 x^{8} + 16588 x^{7} + 38727 x^{6} + 13729 x^{5} - 45405 x^{4} - 23166 x^{3} - 94770 x^{2} + 59049 \)

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:  \(315252767308030713643991421826778089=47^{8}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.56$
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:  $47, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{45} a^{16} + \frac{16}{45} a^{15} - \frac{7}{45} a^{14} + \frac{13}{45} a^{13} - \frac{1}{45} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{14}{45} a^{4} + \frac{1}{45} a^{3} + \frac{2}{5} a^{2} + \frac{13}{45} a + \frac{1}{5}$, $\frac{1}{405} a^{17} - \frac{2}{405} a^{16} - \frac{32}{81} a^{15} + \frac{4}{405} a^{14} - \frac{2}{81} a^{13} + \frac{23}{405} a^{12} - \frac{1}{3} a^{11} - \frac{8}{81} a^{10} - \frac{37}{81} a^{9} - \frac{13}{27} a^{8} - \frac{22}{81} a^{7} - \frac{31}{81} a^{6} - \frac{14}{405} a^{5} - \frac{17}{405} a^{4} + \frac{2}{9} a^{3} - \frac{41}{405} a^{2} - \frac{1}{9} a - \frac{2}{5}$, $\frac{1}{3645} a^{18} - \frac{2}{3645} a^{17} + \frac{2}{3645} a^{16} - \frac{1454}{3645} a^{15} - \frac{739}{3645} a^{14} + \frac{104}{3645} a^{13} - \frac{56}{135} a^{12} + \frac{154}{729} a^{11} + \frac{206}{729} a^{10} - \frac{13}{243} a^{9} - \frac{184}{729} a^{8} - \frac{112}{729} a^{7} - \frac{824}{3645} a^{6} - \frac{1637}{3645} a^{5} - \frac{152}{405} a^{4} + \frac{1336}{3645} a^{3} + \frac{139}{405} a^{2} - \frac{16}{45} a + \frac{2}{5}$, $\frac{1}{85220652806464550541561825502955561139878502345} a^{19} + \frac{2903715819464735839564928053555833316152283}{85220652806464550541561825502955561139878502345} a^{18} - \frac{1935458166659264848526431903745350519057639}{6555434831266503887812448115611966241529115565} a^{17} - \frac{530897560526298628105510980768261823677881552}{85220652806464550541561825502955561139878502345} a^{16} - \frac{6713699762212998744274686933994778183246901818}{17044130561292910108312365100591112227975700469} a^{15} - \frac{2339973743650061320477043609836866935785778258}{17044130561292910108312365100591112227975700469} a^{14} + \frac{464057100718686571838765004822513777158520526}{1893792284588101123145818344510123580886188941} a^{13} - \frac{5596378602137879376469919376300675343283424271}{85220652806464550541561825502955561139878502345} a^{12} - \frac{4451699214260288368383263544817506794113877440}{17044130561292910108312365100591112227975700469} a^{11} + \frac{1869779162918534652215754149019172461474309302}{5681376853764303369437455033530370742658566823} a^{10} - \frac{305716515462095693804586544935284951612589478}{1311086966253300777562489623122393248305823113} a^{9} + \frac{3075441666584264656156838531477399356008580217}{17044130561292910108312365100591112227975700469} a^{8} - \frac{21295318496872797859541066356094254567001853824}{85220652806464550541561825502955561139878502345} a^{7} + \frac{38683043645307401296472987853839582899288555933}{85220652806464550541561825502955561139878502345} a^{6} - \frac{4183973899276261215629072675289753620380986013}{9468961422940505615729091722550617904430944705} a^{5} + \frac{3118785795149847312820709345792345427814732081}{6555434831266503887812448115611966241529115565} a^{4} - \frac{140702768423874658486202941750045637792741075}{1893792284588101123145818344510123580886188941} a^{3} + \frac{38822894696406772219676108488474954021271230}{210421364954233458127313149390013731209576549} a^{2} + \frac{1855711408588016133488229460085481601226411}{7793383887193831782493079607037545600354687} a - \frac{378396797110765936917145479058289983065901}{12988973145323052970821799345062576000591145}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  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:  \( 8004059626.39 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.4.26569.1, 5.1.2209.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
163Data not computed