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Properties

Label 20.0.33618693882...0000.7
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 $44888$ (GRH)
Class group $[2, 22444]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![261399601, -87181604, 223626538, -59924902, 92642076, -20925574, 25267471, -4931470, 5053831, -862794, 776698, -115100, 92914, -11788, 8702, -936, 618, -50, 31, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 31*x^18 - 50*x^17 + 618*x^16 - 936*x^15 + 8702*x^14 - 11788*x^13 + 92914*x^12 - 115100*x^11 + 776698*x^10 - 862794*x^9 + 5053831*x^8 - 4931470*x^7 + 25267471*x^6 - 20925574*x^5 + 92642076*x^4 - 59924902*x^3 + 223626538*x^2 - 87181604*x + 261399601)
 
gp: K = bnfinit(x^20 - 2*x^19 + 31*x^18 - 50*x^17 + 618*x^16 - 936*x^15 + 8702*x^14 - 11788*x^13 + 92914*x^12 - 115100*x^11 + 776698*x^10 - 862794*x^9 + 5053831*x^8 - 4931470*x^7 + 25267471*x^6 - 20925574*x^5 + 92642076*x^4 - 59924902*x^3 + 223626538*x^2 - 87181604*x + 261399601, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 31 x^{18} - 50 x^{17} + 618 x^{16} - 936 x^{15} + 8702 x^{14} - 11788 x^{13} + 92914 x^{12} - 115100 x^{11} + 776698 x^{10} - 862794 x^{9} + 5053831 x^{8} - 4931470 x^{7} + 25267471 x^{6} - 20925574 x^{5} + 92642076 x^{4} - 59924902 x^{3} + 223626538 x^{2} - 87181604 x + 261399601 \)

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$$\chi_{660}(1,·)$, $\chi_{660}(619,·)$, $\chi_{660}(199,·)$, $\chi_{660}(461,·)$, $\chi_{660}(659,·)$, $\chi_{660}(281,·)$, $\chi_{660}(239,·)$, $\chi_{660}(101,·)$, $\chi_{660}(161,·)$, $\chi_{660}(421,·)$, $\chi_{660}(359,·)$, $\chi_{660}(41,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(499,·)$, $\chi_{660}(181,·)$, $\chi_{660}(479,·)$, $\chi_{660}(361,·)$, $\chi_{660}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{727} a^{18} - \frac{154}{727} a^{17} + \frac{210}{727} a^{16} - \frac{283}{727} a^{15} + \frac{94}{727} a^{14} + \frac{316}{727} a^{13} + \frac{310}{727} a^{12} + \frac{134}{727} a^{11} - \frac{209}{727} a^{10} - \frac{126}{727} a^{9} - \frac{260}{727} a^{8} - \frac{358}{727} a^{7} - \frac{31}{727} a^{6} - \frac{50}{727} a^{5} - \frac{169}{727} a^{4} + \frac{341}{727} a^{3} + \frac{152}{727} a^{2} - \frac{10}{727} a + \frac{181}{727}$, $\frac{1}{273266414566686629236028794334485415241535560565388644925339} a^{19} + \frac{139561566323376720671229228154775648454917488534686341183}{273266414566686629236028794334485415241535560565388644925339} a^{18} - \frac{72472875804831424424583785674097402330757747145398618501232}{273266414566686629236028794334485415241535560565388644925339} a^{17} - \frac{1126337814817362855057911543182813668375414007623343799919}{273266414566686629236028794334485415241535560565388644925339} a^{16} - \frac{96388363713519854055185879545132899202811062647810451286054}{273266414566686629236028794334485415241535560565388644925339} a^{15} - \frac{111138271186388176830846095341389188357605343401444122441819}{273266414566686629236028794334485415241535560565388644925339} a^{14} - \frac{129416297924739215767907101660790408141059441779804501842937}{273266414566686629236028794334485415241535560565388644925339} a^{13} + \frac{70445712567632238380098975613764191645182093232623052647275}{273266414566686629236028794334485415241535560565388644925339} a^{12} + \frac{93539071571368664041892949938373640294488316569750613186100}{273266414566686629236028794334485415241535560565388644925339} a^{11} + \frac{1424132953115713998242883918074178016936481882773033001046}{4078603202487860137851176034843065899127396426349084252617} a^{10} - \frac{60026333848168218956676293042971374615659843381068438636642}{273266414566686629236028794334485415241535560565388644925339} a^{9} - \frac{44468299459021259095381238204460124550438571489796365073175}{273266414566686629236028794334485415241535560565388644925339} a^{8} - \frac{108063968707072526152418411093112824026997161960626190240783}{273266414566686629236028794334485415241535560565388644925339} a^{7} - \frac{90017576089768326760494883451099552992535043620139278626739}{273266414566686629236028794334485415241535560565388644925339} a^{6} - \frac{50488207949321935614716154223735350453243647002241687735656}{273266414566686629236028794334485415241535560565388644925339} a^{5} - \frac{73356454836751464449327837182725394273336659842387631559297}{273266414566686629236028794334485415241535560565388644925339} a^{4} + \frac{33982264002972253276623924144670983532076173399795801764656}{273266414566686629236028794334485415241535560565388644925339} a^{3} - \frac{83425264631456339404606000395360926129081261456377664619056}{273266414566686629236028794334485415241535560565388644925339} a^{2} + \frac{131630362781487381624791572849649108720864634595930173616924}{273266414566686629236028794334485415241535560565388644925339} a - \frac{50189001092850480857747560004323832587025702790590447356100}{273266414566686629236028794334485415241535560565388644925339}$

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

Class group and class number

$C_{2}\times C_{22444}$, which has order $44888$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-5}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.0.685948419200000.1, \(\Q(\zeta_{33})^+\), 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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
11Data not computed