↑

Properties

Label 20.0.33618693882...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number $31856$ (GRH)
Class group $[2, 2, 7964]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![175645009, -109373078, 183092351, -178742094, 176867685, -132713622, 96245877, -57031380, 31769904, -14901524, 6580762, -2466574, 871872, -259722, 73284, -16932, 3723, -624, 101, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 101*x^18 - 624*x^17 + 3723*x^16 - 16932*x^15 + 73284*x^14 - 259722*x^13 + 871872*x^12 - 2466574*x^11 + 6580762*x^10 - 14901524*x^9 + 31769904*x^8 - 57031380*x^7 + 96245877*x^6 - 132713622*x^5 + 176867685*x^4 - 178742094*x^3 + 183092351*x^2 - 109373078*x + 175645009)
 
gp: K = bnfinit(x^20 - 10*x^19 + 101*x^18 - 624*x^17 + 3723*x^16 - 16932*x^15 + 73284*x^14 - 259722*x^13 + 871872*x^12 - 2466574*x^11 + 6580762*x^10 - 14901524*x^9 + 31769904*x^8 - 57031380*x^7 + 96245877*x^6 - 132713622*x^5 + 176867685*x^4 - 178742094*x^3 + 183092351*x^2 - 109373078*x + 175645009, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 101 x^{18} - 624 x^{17} + 3723 x^{16} - 16932 x^{15} + 73284 x^{14} - 259722 x^{13} + 871872 x^{12} - 2466574 x^{11} + 6580762 x^{10} - 14901524 x^{9} + 31769904 x^{8} - 57031380 x^{7} + 96245877 x^{6} - 132713622 x^{5} + 176867685 x^{4} - 178742094 x^{3} + 183092351 x^{2} - 109373078 x + 175645009 \)

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:  \(3361869388230684433628866560000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(529,·)$, $\chi_{660}(659,·)$, $\chi_{660}(239,·)$, $\chi_{660}(491,·)$, $\chi_{660}(229,·)$, $\chi_{660}(289,·)$, $\chi_{660}(611,·)$, $\chi_{660}(421,·)$, $\chi_{660}(359,·)$, $\chi_{660}(169,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(431,·)$, $\chi_{660}(49,·)$, $\chi_{660}(371,·)$, $\chi_{660}(181,·)$, $\chi_{660}(361,·)$, $\chi_{660}(479,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{9} - \frac{4}{27} a^{6} + \frac{8}{27} a^{3} + \frac{10}{27}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{4}{27} a^{7} - \frac{1}{27} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{8}{27} a - \frac{1}{3}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{11} - \frac{1}{27} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{4}{27} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{11}{27} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{4} - \frac{7}{27} a^{3} + \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{4}{27}$, $\frac{1}{81} a^{16} + \frac{1}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{81} a^{12} - \frac{1}{81} a^{11} + \frac{4}{81} a^{10} + \frac{1}{27} a^{9} + \frac{4}{81} a^{8} + \frac{4}{27} a^{7} + \frac{7}{81} a^{6} - \frac{8}{81} a^{5} + \frac{11}{81} a^{4} + \frac{13}{27} a^{3} - \frac{10}{81} a^{2} + \frac{35}{81} a + \frac{34}{81}$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{4}{81} a^{11} - \frac{1}{81} a^{10} + \frac{4}{81} a^{9} - \frac{1}{81} a^{8} - \frac{5}{81} a^{7} + \frac{1}{27} a^{6} - \frac{8}{81} a^{5} + \frac{1}{81} a^{4} - \frac{34}{81} a^{3} + \frac{4}{9} a^{2} - \frac{1}{81} a - \frac{10}{81}$, $\frac{1}{3939001327911543391452321} a^{18} - \frac{1}{437666814212393710161369} a^{17} + \frac{4970392491037368970036}{3939001327911543391452321} a^{16} - \frac{39763139928298951760084}{3939001327911543391452321} a^{15} + \frac{39093529548556403821616}{3939001327911543391452321} a^{14} - \frac{5155524102352293703975}{1313000442637181130484107} a^{13} - \frac{69084412248368352814564}{3939001327911543391452321} a^{12} + \frac{119349489381528396745346}{3939001327911543391452321} a^{11} - \frac{54595423441687266352088}{3939001327911543391452321} a^{10} - \frac{171406921456817756541697}{3939001327911543391452321} a^{9} + \frac{60772184747595465717094}{3939001327911543391452321} a^{8} - \frac{161516928754180162946767}{1313000442637181130484107} a^{7} - \frac{69064089594317629851437}{3939001327911543391452321} a^{6} + \frac{462495656121060943652494}{3939001327911543391452321} a^{5} - \frac{398241602423407159861171}{3939001327911543391452321} a^{4} - \frac{264657363151349765221391}{1313000442637181130484107} a^{3} + \frac{64441131033114593240135}{3939001327911543391452321} a^{2} - \frac{1280978231480711650317487}{3939001327911543391452321} a + \frac{460515611627316058324264}{1313000442637181130484107}$, $\frac{1}{49803066592551848058668562392217} a^{19} + \frac{6321779}{49803066592551848058668562392217} a^{18} - \frac{60853334766636801221428058939}{49803066592551848058668562392217} a^{17} + \frac{241248778615678412121076903961}{49803066592551848058668562392217} a^{16} - \frac{267413765917171976672810500874}{16601022197517282686222854130739} a^{15} + \frac{796957056716341189977218250314}{49803066592551848058668562392217} a^{14} + \frac{557874541513783695947290120553}{49803066592551848058668562392217} a^{13} + \frac{563201843071845879511631192362}{49803066592551848058668562392217} a^{12} + \frac{179746408165041209487976910081}{5533674065839094228740951376913} a^{11} - \frac{362642333552256884998985125103}{16601022197517282686222854130739} a^{10} + \frac{1385643895795317083498767362374}{49803066592551848058668562392217} a^{9} + \frac{2559431937628513683618768115063}{49803066592551848058668562392217} a^{8} - \frac{4450465778046698682571477362143}{49803066592551848058668562392217} a^{7} + \frac{2707361082817855227171440039864}{49803066592551848058668562392217} a^{6} - \frac{1562464098387432576894698719474}{16601022197517282686222854130739} a^{5} - \frac{5833676444888766342459723951715}{49803066592551848058668562392217} a^{4} + \frac{5521871407408618985365997443832}{49803066592551848058668562392217} a^{3} + \frac{21805120229214727287061771046086}{49803066592551848058668562392217} a^{2} - \frac{22274543871904300159430268491581}{49803066592551848058668562392217} a - \frac{14832497940871236632365769141053}{49803066592551848058668562392217}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{7964}$, which has order $31856$ (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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.586732839846912.1, 10.0.1833540124521600000.1

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$