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Properties

Label 20.0.14339596913...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 61^{18}$
Root discriminant $255.76$
Ramified primes $2, 5, 61$
Class number $622274532$ (GRH)
Class group $[2, 311137266]$ (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![431056760849, -239603764752, 237984762200, -88916147662, 51507614372, -14260973952, 6430262311, -1401208766, 555153715, -94607688, 34853312, -4345934, 1652989, -141064, 62644, -2746, 1749, -14, 47, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 47*x^18 - 14*x^17 + 1749*x^16 - 2746*x^15 + 62644*x^14 - 141064*x^13 + 1652989*x^12 - 4345934*x^11 + 34853312*x^10 - 94607688*x^9 + 555153715*x^8 - 1401208766*x^7 + 6430262311*x^6 - 14260973952*x^5 + 51507614372*x^4 - 88916147662*x^3 + 237984762200*x^2 - 239603764752*x + 431056760849)
 
gp: K = bnfinit(x^20 - 2*x^19 + 47*x^18 - 14*x^17 + 1749*x^16 - 2746*x^15 + 62644*x^14 - 141064*x^13 + 1652989*x^12 - 4345934*x^11 + 34853312*x^10 - 94607688*x^9 + 555153715*x^8 - 1401208766*x^7 + 6430262311*x^6 - 14260973952*x^5 + 51507614372*x^4 - 88916147662*x^3 + 237984762200*x^2 - 239603764752*x + 431056760849, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 47 x^{18} - 14 x^{17} + 1749 x^{16} - 2746 x^{15} + 62644 x^{14} - 141064 x^{13} + 1652989 x^{12} - 4345934 x^{11} + 34853312 x^{10} - 94607688 x^{9} + 555153715 x^{8} - 1401208766 x^{7} + 6430262311 x^{6} - 14260973952 x^{5} + 51507614372 x^{4} - 88916147662 x^{3} + 237984762200 x^{2} - 239603764752 x + 431056760849 \)

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:  \(1433959691353304658297222439453390274560000000000=2^{30}\cdot 5^{10}\cdot 61^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.76$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2440=2^{3}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{2440}(1,·)$, $\chi_{2440}(1219,·)$, $\chi_{2440}(1099,·)$, $\chi_{2440}(81,·)$, $\chi_{2440}(339,·)$, $\chi_{2440}(601,·)$, $\chi_{2440}(1179,·)$, $\chi_{2440}(979,·)$, $\chi_{2440}(881,·)$, $\chi_{2440}(1699,·)$, $\chi_{2440}(2321,·)$, $\chi_{2440}(41,·)$, $\chi_{2440}(619,·)$, $\chi_{2440}(241,·)$, $\chi_{2440}(1139,·)$, $\chi_{2440}(1961,·)$, $\chi_{2440}(1681,·)$, $\chi_{2440}(121,·)$, $\chi_{2440}(1979,·)$, $\chi_{2440}(1339,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{4}{13} a^{9} + \frac{3}{13} a^{7} - \frac{1}{13} a^{5} - \frac{4}{13} a^{3} - \frac{3}{13} a$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{10} + \frac{3}{13} a^{8} - \frac{1}{13} a^{6} - \frac{4}{13} a^{4} - \frac{3}{13} a^{2}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{611} a^{16} + \frac{2}{611} a^{15} - \frac{6}{611} a^{14} + \frac{2}{611} a^{13} + \frac{20}{611} a^{12} - \frac{1}{47} a^{11} - \frac{258}{611} a^{10} + \frac{8}{47} a^{9} + \frac{99}{611} a^{8} - \frac{8}{47} a^{7} - \frac{111}{611} a^{6} - \frac{15}{47} a^{5} + \frac{140}{611} a^{4} + \frac{297}{611} a^{3} - \frac{93}{611} a^{2} + \frac{232}{611} a + \frac{2}{47}$, $\frac{1}{7943} a^{17} + \frac{2}{7943} a^{16} - \frac{100}{7943} a^{15} + \frac{11}{611} a^{14} + \frac{302}{7943} a^{13} + \frac{175}{7943} a^{12} + \frac{24}{7943} a^{11} + \frac{2689}{7943} a^{10} + \frac{3671}{7943} a^{9} + \frac{3515}{7943} a^{8} + \frac{2568}{7943} a^{7} - \frac{1605}{7943} a^{6} - \frac{3808}{7943} a^{5} + \frac{12}{611} a^{4} + \frac{1928}{7943} a^{3} - \frac{1695}{7943} a^{2} - \frac{1713}{7943} a$, $\frac{1}{87373} a^{18} - \frac{4}{87373} a^{17} + \frac{4}{7943} a^{16} - \frac{1389}{87373} a^{15} + \frac{952}{87373} a^{14} + \frac{1730}{87373} a^{13} + \frac{2094}{87373} a^{12} - \frac{54}{1859} a^{11} + \frac{42605}{87373} a^{10} + \frac{33151}{87373} a^{9} + \frac{20751}{87373} a^{8} + \frac{37028}{87373} a^{7} - \frac{3551}{87373} a^{6} - \frac{4361}{87373} a^{5} - \frac{40712}{87373} a^{4} - \frac{7868}{87373} a^{3} + \frac{1394}{7943} a^{2} - \frac{42736}{87373} a + \frac{15}{47}$, $\frac{1}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{19} - \frac{8932915064037540001700716197198226050482609881816347154236749070}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{18} + \frac{7533665469822130664600976085895962709529132105286178381726129061}{196099087613027609293896445263185846227888626626736302424813170531483} a^{17} + \frac{1717166689262715629415729361871240008268175997194317283908613868922}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{16} + \frac{53381561895169066701637596896164568920351840720739739698946630989474}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{15} - \frac{64558608209658752550913057630836866188263441607842049664471985092042}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{14} + \frac{48707725350820741148276168936229455532798290612926986103252677644159}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{13} + \frac{49720226680373615811878068128432147972728188870784173898655630285715}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{12} - \frac{44323427139234386125188561561903639678862692261938200855524673102077}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{11} - \frac{483099280011812538194756988367495066605376818507065254703036006937173}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{10} + \frac{668974072885862085943003350161499581478896934743863444957195059919031}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{9} - \frac{562182256416199125883406757129634773265131225877371049271202413649135}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{8} - \frac{1070642010248435453560354934241561780500038265701330238388089128604360}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{7} - \frac{223235650507817035631461755646123003059120220967447921969747033110853}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{6} + \frac{457617240501121866774834970214082611693688386665310899708335662593099}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{5} - \frac{60812047362442614874037136308327163655802701066022346618264132232237}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{4} - \frac{66802129918170626452820232477720453506890170009398456052962387712558}{196099087613027609293896445263185846227888626626736302424813170531483} a^{3} + \frac{771897402320283440407340032235059559920486282408095048370131496094852}{2157089963743303702232860897895044308506774892894099326672944875846313} a^{2} + \frac{94667556257678545713852089148455600939874616060909756330957765627503}{196099087613027609293896445263185846227888626626736302424813170531483} a + \frac{167566469329600444002975277527323508320663954847813306402520062}{3061608524660467585110247228976688049021695627339718387297827833}$

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

Class group and class number

$C_{2}\times C_{311137266}$, which has order $622274532$ (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:  \( 36549838.47150319 \) (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

\(\Q(\sqrt{-610}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-10}, \sqrt{61})\), 5.5.13845841.1, 10.0.1197480559906216038400000.1, 10.0.19630828850921574400000.1, 10.10.11694146092834141.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61Data not computed