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Properties

Label 20.20.4142461205...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{27}\cdot 11^{18}$
Root discriminant $76.01$
Ramified primes $5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5:C_4$ (as 20T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-33329, 127380, 551050, -556105, -2796615, -700216, 3851885, 2297075, -2174625, -1893595, 462661, 705485, 25635, -124135, -23405, 8844, 2655, -165, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 90*x^18 - 165*x^17 + 2655*x^16 + 8844*x^15 - 23405*x^14 - 124135*x^13 + 25635*x^12 + 705485*x^11 + 462661*x^10 - 1893595*x^9 - 2174625*x^8 + 2297075*x^7 + 3851885*x^6 - 700216*x^5 - 2796615*x^4 - 556105*x^3 + 551050*x^2 + 127380*x - 33329)
 
gp: K = bnfinit(x^20 - 90*x^18 - 165*x^17 + 2655*x^16 + 8844*x^15 - 23405*x^14 - 124135*x^13 + 25635*x^12 + 705485*x^11 + 462661*x^10 - 1893595*x^9 - 2174625*x^8 + 2297075*x^7 + 3851885*x^6 - 700216*x^5 - 2796615*x^4 - 556105*x^3 + 551050*x^2 + 127380*x - 33329, 1)
 

Normalized defining polynomial

\( x^{20} - 90 x^{18} - 165 x^{17} + 2655 x^{16} + 8844 x^{15} - 23405 x^{14} - 124135 x^{13} + 25635 x^{12} + 705485 x^{11} + 462661 x^{10} - 1893595 x^{9} - 2174625 x^{8} + 2297075 x^{7} + 3851885 x^{6} - 700216 x^{5} - 2796615 x^{4} - 556105 x^{3} + 551050 x^{2} + 127380 x - 33329 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41424612056406076855957508087158203125=5^{27}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.01$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{12} a^{16} + \frac{1}{6} a^{15} + \frac{1}{12} a^{14} - \frac{5}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{4} a^{11} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{5}{12} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{5}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{5}{12}$, $\frac{1}{12} a^{17} - \frac{1}{4} a^{15} - \frac{1}{12} a^{14} + \frac{5}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{5}{12} a^{10} + \frac{1}{12} a^{9} - \frac{5}{12} a^{8} - \frac{5}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{12} a^{5} + \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{24} a^{18} - \frac{1}{24} a^{17} + \frac{1}{12} a^{15} - \frac{1}{8} a^{14} - \frac{7}{24} a^{13} - \frac{3}{8} a^{12} - \frac{5}{24} a^{11} - \frac{1}{24} a^{10} + \frac{3}{8} a^{9} + \frac{1}{8} a^{8} + \frac{11}{24} a^{7} - \frac{1}{24} a^{6} + \frac{7}{24} a^{5} + \frac{5}{24} a^{4} + \frac{5}{12} a^{3} + \frac{1}{6} a^{2} - \frac{11}{24} a + \frac{1}{24}$, $\frac{1}{4124392105563923353987372502249835405567384} a^{19} + \frac{13570441720466114045365160537944908927733}{1374797368521307784662457500749945135189128} a^{18} - \frac{21776235661690453797903575553592157700367}{687398684260653892331228750374972567594564} a^{17} - \frac{3354455925970633864096251802098034676509}{2062196052781961676993686251124917702783692} a^{16} + \frac{891817981771103138009235638339926637265}{1374797368521307784662457500749945135189128} a^{15} + \frac{227471419013371880004233202067326985501713}{1374797368521307784662457500749945135189128} a^{14} - \frac{463569950461014197649046558045195583749809}{1374797368521307784662457500749945135189128} a^{13} + \frac{169475037489789520405422573170123892583033}{4124392105563923353987372502249835405567384} a^{12} - \frac{1837983379970606259241098145997436826447091}{4124392105563923353987372502249835405567384} a^{11} - \frac{508013447622110600772234798872712131931603}{1374797368521307784662457500749945135189128} a^{10} - \frac{410740566239301321310780277651307363884253}{1374797368521307784662457500749945135189128} a^{9} - \frac{588441251221339439614082913585874169619905}{1374797368521307784662457500749945135189128} a^{8} + \frac{373594616688454975634918165878597259877553}{4124392105563923353987372502249835405567384} a^{7} - \frac{462119213682002572211724485993220540359225}{1374797368521307784662457500749945135189128} a^{6} + \frac{57198814431370530394607217174028547833055}{4124392105563923353987372502249835405567384} a^{5} - \frac{167335591922119572630838488795578041689023}{1031098026390980838496843125562458851391846} a^{4} - \frac{298527291170613095603487797911781206536643}{687398684260653892331228750374972567594564} a^{3} - \frac{1642890115014794032248456222589196130701315}{4124392105563923353987372502249835405567384} a^{2} - \frac{182774061561554054320427005822163742209321}{4124392105563923353987372502249835405567384} a - \frac{4193732915536902975804271371744706411975}{515549013195490419248421562781229425695923}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $19$
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:  \( 547793455064 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:C_4$ (as 20T2):

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 $C_5:C_4$
Character table for $C_5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 5.5.228765625.1 x5, 10.10.261668555908203125.1

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

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$