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Properties

Label 20.0.11238525471...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $178.87$
Ramified primes $2, 5, 41$
Class number $85656080$ (GRH)
Class group $[2, 42828040]$ (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![1088922609, 828313128, 1685592342, 973121202, 969501555, 373408974, 188740539, 23685348, -11917972, -10923148, -3987980, 388520, 904237, 57086, -78390, -6398, 3905, 190, -97, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 97*x^18 + 190*x^17 + 3905*x^16 - 6398*x^15 - 78390*x^14 + 57086*x^13 + 904237*x^12 + 388520*x^11 - 3987980*x^10 - 10923148*x^9 - 11917972*x^8 + 23685348*x^7 + 188740539*x^6 + 373408974*x^5 + 969501555*x^4 + 973121202*x^3 + 1685592342*x^2 + 828313128*x + 1088922609)
 
gp: K = bnfinit(x^20 - 2*x^19 - 97*x^18 + 190*x^17 + 3905*x^16 - 6398*x^15 - 78390*x^14 + 57086*x^13 + 904237*x^12 + 388520*x^11 - 3987980*x^10 - 10923148*x^9 - 11917972*x^8 + 23685348*x^7 + 188740539*x^6 + 373408974*x^5 + 969501555*x^4 + 973121202*x^3 + 1685592342*x^2 + 828313128*x + 1088922609, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 97 x^{18} + 190 x^{17} + 3905 x^{16} - 6398 x^{15} - 78390 x^{14} + 57086 x^{13} + 904237 x^{12} + 388520 x^{11} - 3987980 x^{10} - 10923148 x^{9} - 11917972 x^{8} + 23685348 x^{7} + 188740539 x^{6} + 373408974 x^{5} + 969501555 x^{4} + 973121202 x^{3} + 1685592342 x^{2} + 828313128 x + 1088922609 \)

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:  \(1123852547182106083132474859412520960000000000=2^{30}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $178.87$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1640=2^{3}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(1451,·)$, $\chi_{1640}(1089,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(1419,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(209,·)$, $\chi_{1640}(409,·)$, $\chi_{1640}(819,·)$, $\chi_{1640}(1371,·)$, $\chi_{1640}(1179,·)$, $\chi_{1640}(1499,·)$, $\chi_{1640}(611,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(619,·)$, $\chi_{1640}(411,·)$, $\chi_{1640}(1009,·)$, $\chi_{1640}(51,·)$, $\chi_{1640}(769,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{2}{27} a^{5} + \frac{4}{27} a^{4} - \frac{1}{27} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{2}{27} a^{5} - \frac{7}{81} a^{4} - \frac{1}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{2187} a^{14} - \frac{1}{2187} a^{13} + \frac{2}{2187} a^{12} + \frac{13}{2187} a^{11} + \frac{1}{243} a^{10} + \frac{1}{729} a^{9} - \frac{40}{2187} a^{8} + \frac{76}{2187} a^{7} - \frac{14}{2187} a^{6} + \frac{197}{2187} a^{5} - \frac{49}{729} a^{4} + \frac{13}{243} a^{3} - \frac{26}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{2187} a^{15} + \frac{1}{2187} a^{13} - \frac{1}{729} a^{12} - \frac{5}{2187} a^{11} + \frac{1}{729} a^{10} - \frac{10}{2187} a^{9} + \frac{1}{243} a^{8} + \frac{62}{2187} a^{7} + \frac{4}{729} a^{6} + \frac{320}{2187} a^{5} - \frac{70}{729} a^{4} + \frac{40}{243} a^{3} - \frac{20}{81} a^{2}$, $\frac{1}{2187} a^{16} - \frac{2}{2187} a^{13} + \frac{2}{2187} a^{12} - \frac{10}{2187} a^{11} - \frac{1}{2187} a^{10} + \frac{2}{729} a^{9} - \frac{2}{729} a^{8} - \frac{64}{2187} a^{7} - \frac{107}{2187} a^{6} + \frac{322}{2187} a^{5} - \frac{116}{729} a^{4} + \frac{8}{243} a^{3} - \frac{4}{81} a^{2} + \frac{1}{9} a$, $\frac{1}{19683} a^{17} - \frac{1}{19683} a^{16} + \frac{1}{19683} a^{15} + \frac{2}{19683} a^{14} - \frac{35}{19683} a^{13} + \frac{20}{19683} a^{12} - \frac{70}{19683} a^{11} - \frac{89}{19683} a^{10} - \frac{10}{19683} a^{9} + \frac{196}{19683} a^{8} + \frac{872}{19683} a^{7} + \frac{1087}{19683} a^{6} + \frac{818}{6561} a^{5} - \frac{29}{243} a^{4} + \frac{79}{729} a^{3} - \frac{70}{243} a^{2} + \frac{2}{27} a$, $\frac{1}{62769087} a^{18} + \frac{136}{62769087} a^{17} + \frac{9656}{62769087} a^{16} + \frac{4630}{62769087} a^{15} - \frac{40}{62769087} a^{14} + \frac{78745}{62769087} a^{13} + \frac{3281}{20923029} a^{12} - \frac{117319}{62769087} a^{11} + \frac{247114}{62769087} a^{10} - \frac{127408}{62769087} a^{9} + \frac{64534}{62769087} a^{8} - \frac{252949}{62769087} a^{7} - \frac{589174}{62769087} a^{6} - \frac{3048449}{20923029} a^{5} + \frac{125603}{774927} a^{4} + \frac{328847}{2324781} a^{3} - \frac{116939}{774927} a^{2} - \frac{2429}{86103} a + \frac{319}{1063}$, $\frac{1}{4275103289165909357953014980145249107935024649619} a^{19} + \frac{1140102884543004192461727803808523595203}{4275103289165909357953014980145249107935024649619} a^{18} - \frac{20742642917841036020771274202974047099972710}{4275103289165909357953014980145249107935024649619} a^{17} - \frac{744787809156685014316129254805721605361932775}{4275103289165909357953014980145249107935024649619} a^{16} + \frac{134402311642436985487052547232146336355114997}{4275103289165909357953014980145249107935024649619} a^{15} + \frac{576672978730901409138392791892990897081075035}{4275103289165909357953014980145249107935024649619} a^{14} - \frac{747711760452708083140047289678043298556467188}{1425034429721969785984338326715083035978341549873} a^{13} - \frac{4067587025160425012118679305346298850057905995}{4275103289165909357953014980145249107935024649619} a^{12} + \frac{26149530660270566968494455251337774755698827281}{4275103289165909357953014980145249107935024649619} a^{11} - \frac{24551950544909435250391838385467830029143050807}{4275103289165909357953014980145249107935024649619} a^{10} - \frac{16512581533394706922146409629326802538294939031}{4275103289165909357953014980145249107935024649619} a^{9} - \frac{14959337172464377278412159275715596106616565622}{4275103289165909357953014980145249107935024649619} a^{8} - \frac{219896925861079790076287922398828084633005234633}{4275103289165909357953014980145249107935024649619} a^{7} - \frac{13445690028744854096970019382426024651867806705}{475011476573989928661446108905027678659447183291} a^{6} - \frac{41102996365263102719482499289403241448243807401}{475011476573989928661446108905027678659447183291} a^{5} - \frac{5221480890652710886086617302409880971693639899}{158337158857996642887148702968342559553149061097} a^{4} + \frac{1138509023768029672295142278061178678593861526}{17593017650888515876349855885371395505905451233} a^{3} + \frac{2531095188259935835049893624570814793154049953}{17593017650888515876349855885371395505905451233} a^{2} + \frac{463568862820119384866265132352827894399612190}{1954779738987612875149983987263488389545050137} a + \frac{7095907532487756112302562566546850394004981}{24133083197377936730246715892141831969691977}$

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

Class group and class number

$C_{2}\times C_{42828040}$, which has order $85656080$ (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:  \( 3541438824.6395073 \) (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

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-2}, \sqrt{205})\), 5.5.2825761.1, 10.10.1023068544981128125.1, 10.0.261650029907836928.1, 10.0.33523910081941606400000.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$