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Properties

Label 20.0.96363567882...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{34}\cdot 11^{16}$
Root discriminant $706.64$
Ramified primes $2, 5, 11$
Class number $2387272210$ (GRH)
Class group $[2387272210]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![224628092046401, 360221472820440, 409475710606620, 308747918975760, 178487748476450, 78601752107604, 26769996852420, 7113309760260, 1441734783805, 207299277900, 16318627506, -794674980, -428470840, -50497920, -181050, 573804, 44290, -1980, -340, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 340*x^18 - 1980*x^17 + 44290*x^16 + 573804*x^15 - 181050*x^14 - 50497920*x^13 - 428470840*x^12 - 794674980*x^11 + 16318627506*x^10 + 207299277900*x^9 + 1441734783805*x^8 + 7113309760260*x^7 + 26769996852420*x^6 + 78601752107604*x^5 + 178487748476450*x^4 + 308747918975760*x^3 + 409475710606620*x^2 + 360221472820440*x + 224628092046401)
 
gp: K = bnfinit(x^20 - 340*x^18 - 1980*x^17 + 44290*x^16 + 573804*x^15 - 181050*x^14 - 50497920*x^13 - 428470840*x^12 - 794674980*x^11 + 16318627506*x^10 + 207299277900*x^9 + 1441734783805*x^8 + 7113309760260*x^7 + 26769996852420*x^6 + 78601752107604*x^5 + 178487748476450*x^4 + 308747918975760*x^3 + 409475710606620*x^2 + 360221472820440*x + 224628092046401, 1)
 

Normalized defining polynomial

\( x^{20} - 340 x^{18} - 1980 x^{17} + 44290 x^{16} + 573804 x^{15} - 181050 x^{14} - 50497920 x^{13} - 428470840 x^{12} - 794674980 x^{11} + 16318627506 x^{10} + 207299277900 x^{9} + 1441734783805 x^{8} + 7113309760260 x^{7} + 26769996852420 x^{6} + 78601752107604 x^{5} + 178487748476450 x^{4} + 308747918975760 x^{3} + 409475710606620 x^{2} + 360221472820440 x + 224628092046401 \)

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:  \(963635678828500845854720000000000000000000000000000000000=2^{55}\cdot 5^{34}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $706.64$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4400=2^{4}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(3019,·)$, $\chi_{4400}(2259,·)$, $\chi_{4400}(4161,·)$, $\chi_{4400}(4041,·)$, $\chi_{4400}(779,·)$, $\chi_{4400}(1281,·)$, $\chi_{4400}(1939,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(3481,·)$, $\chi_{4400}(4321,·)$, $\chi_{4400}(2979,·)$, $\chi_{4400}(1961,·)$, $\chi_{4400}(4139,·)$, $\chi_{4400}(1841,·)$, $\chi_{4400}(1299,·)$, $\chi_{4400}(2121,·)$, $\chi_{4400}(59,·)$, $\chi_{4400}(3499,·)$, $\chi_{4400}(819,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{25} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{25}$, $\frac{1}{25} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{25} a$, $\frac{1}{25} a^{7} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{25} a^{2} + \frac{2}{5}$, $\frac{1}{25} a^{8} + \frac{2}{5} a^{4} + \frac{1}{25} a^{3} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{625} a^{10} + \frac{1}{125} a^{8} + \frac{2}{125} a^{7} - \frac{1}{125} a^{6} + \frac{2}{625} a^{5} + \frac{3}{25} a^{4} - \frac{54}{125} a^{3} - \frac{3}{125} a^{2} - \frac{21}{125} a - \frac{24}{625}$, $\frac{1}{625} a^{11} + \frac{1}{125} a^{9} + \frac{2}{125} a^{8} - \frac{1}{125} a^{7} + \frac{2}{625} a^{6} - \frac{54}{125} a^{4} + \frac{22}{125} a^{3} + \frac{29}{125} a^{2} - \frac{149}{625} a - \frac{3}{25}$, $\frac{1}{1875} a^{12} - \frac{1}{1875} a^{10} - \frac{1}{125} a^{9} + \frac{1}{125} a^{8} - \frac{11}{625} a^{7} + \frac{2}{125} a^{6} + \frac{6}{625} a^{5} + \frac{52}{375} a^{4} + \frac{21}{125} a^{3} - \frac{659}{1875} a^{2} - \frac{13}{125} a - \frac{556}{1875}$, $\frac{1}{1875} a^{13} - \frac{1}{1875} a^{11} + \frac{1}{125} a^{9} - \frac{11}{625} a^{8} + \frac{2}{125} a^{7} + \frac{6}{625} a^{6} - \frac{2}{375} a^{5} - \frac{54}{125} a^{4} + \frac{841}{1875} a^{3} + \frac{37}{125} a^{2} - \frac{181}{1875} a + \frac{6}{125}$, $\frac{1}{9375} a^{14} - \frac{1}{9375} a^{13} + \frac{1}{9375} a^{12} + \frac{4}{9375} a^{11} + \frac{1}{9375} a^{10} + \frac{54}{3125} a^{9} + \frac{21}{3125} a^{8} + \frac{29}{3125} a^{7} - \frac{52}{9375} a^{6} + \frac{187}{9375} a^{5} + \frac{3686}{9375} a^{4} + \frac{164}{9375} a^{3} + \frac{736}{9375} a^{2} + \frac{3544}{9375} a + \frac{586}{9375}$, $\frac{1}{46875} a^{15} - \frac{1}{9375} a^{13} - \frac{2}{9375} a^{12} - \frac{4}{9375} a^{11} + \frac{28}{46875} a^{10} - \frac{7}{625} a^{9} - \frac{39}{3125} a^{8} - \frac{119}{9375} a^{7} + \frac{44}{3125} a^{6} - \frac{772}{46875} a^{5} + \frac{829}{1875} a^{4} + \frac{284}{9375} a^{3} + \frac{4558}{9375} a^{2} + \frac{4586}{9375} a - \frac{17924}{46875}$, $\frac{1}{2015625} a^{16} + \frac{6}{671875} a^{15} - \frac{2}{80625} a^{14} + \frac{34}{403125} a^{13} + \frac{2}{134375} a^{12} + \frac{238}{2015625} a^{11} - \frac{1091}{2015625} a^{10} - \frac{151}{26875} a^{9} + \frac{6748}{403125} a^{8} + \frac{1089}{134375} a^{7} + \frac{12698}{2015625} a^{6} - \frac{21011}{2015625} a^{5} - \frac{187}{1875} a^{4} + \frac{162439}{403125} a^{3} + \frac{8587}{134375} a^{2} - \frac{413039}{2015625} a - \frac{913652}{2015625}$, $\frac{1}{2015625} a^{17} + \frac{13}{2015625} a^{15} - \frac{1}{403125} a^{14} + \frac{82}{403125} a^{13} + \frac{128}{2015625} a^{12} + \frac{4}{9375} a^{11} - \frac{67}{671875} a^{10} + \frac{7528}{403125} a^{9} + \frac{1408}{134375} a^{8} - \frac{8497}{2015625} a^{7} + \frac{1384}{403125} a^{6} + \frac{9278}{671875} a^{5} - \frac{35146}{403125} a^{4} - \frac{84608}{403125} a^{3} + \frac{835751}{2015625} a^{2} + \frac{18037}{403125} a - \frac{276734}{671875}$, $\frac{1}{11624468501559369901471212762234375} a^{18} - \frac{656867680410582169028490539}{3874822833853123300490404254078125} a^{17} - \frac{1329064461121699802273764733}{11624468501559369901471212762234375} a^{16} + \frac{93170466648460939517542279978}{11624468501559369901471212762234375} a^{15} - \frac{32522136898081008922324614623}{774964566770624660098080850815625} a^{14} + \frac{334612339774853838948419941906}{3874822833853123300490404254078125} a^{13} - \frac{1073211791616167154816004836481}{11624468501559369901471212762234375} a^{12} + \frac{4125472521530694463812999871856}{11624468501559369901471212762234375} a^{11} - \frac{8654440514675739736174488469601}{11624468501559369901471212762234375} a^{10} - \frac{14879945762665352670405001136261}{774964566770624660098080850815625} a^{9} + \frac{60147197029863228976360836056933}{11624468501559369901471212762234375} a^{8} - \frac{655382981627203573297009061888}{247329117054454678754706654515625} a^{7} + \frac{124630959498571046187030415812236}{11624468501559369901471212762234375} a^{6} - \frac{4458172478653638208007185396537}{3874822833853123300490404254078125} a^{5} - \frac{877134379941257272459877167487939}{2324893700311873980294242552446875} a^{4} - \frac{302517462965476186393887175847053}{3874822833853123300490404254078125} a^{3} + \frac{2226454874385037475604215990101628}{11624468501559369901471212762234375} a^{2} - \frac{2203596358564458861293987468117353}{11624468501559369901471212762234375} a + \frac{639609076751395335218300353305218}{11624468501559369901471212762234375}$, $\frac{1}{832317283377451580351551746148103260165927146788719028671875} a^{19} + \frac{968747577348310664558268}{277439094459150526783850582049367753388642382262906342890625} a^{18} - \frac{35170282653579562822424054305878099987097454832673279}{832317283377451580351551746148103260165927146788719028671875} a^{17} - \frac{92298300967570801377125727275883138702904744491008531}{832317283377451580351551746148103260165927146788719028671875} a^{16} - \frac{7938439648979448666541355153168899677689369120721835199}{832317283377451580351551746148103260165927146788719028671875} a^{15} - \frac{2667033537404881873177164581880908060132473457462040107}{832317283377451580351551746148103260165927146788719028671875} a^{14} - \frac{2658940320811317538460675461940547694675452752495139476}{277439094459150526783850582049367753388642382262906342890625} a^{13} - \frac{41606873976605892490082818633646707859042874674346163224}{277439094459150526783850582049367753388642382262906342890625} a^{12} + \frac{514273324399206179840343353565436886656007235096695530117}{832317283377451580351551746148103260165927146788719028671875} a^{11} - \frac{33876245434313044669725636427830295821090177157980469832}{832317283377451580351551746148103260165927146788719028671875} a^{10} + \frac{16078673973725668691302353009631820385963125906041853227858}{832317283377451580351551746148103260165927146788719028671875} a^{9} - \frac{4381373600726697572539785548818824133506486244617174841468}{832317283377451580351551746148103260165927146788719028671875} a^{8} + \frac{6464450674586327369489079377123452437263556539402991386443}{832317283377451580351551746148103260165927146788719028671875} a^{7} + \frac{13507579313660506724630031485485095182043662323225785318752}{832317283377451580351551746148103260165927146788719028671875} a^{6} + \frac{4474777903058078774499759614098401539877897417676776300336}{277439094459150526783850582049367753388642382262906342890625} a^{5} + \frac{321324860019800930889246815915794721564270051731034968874966}{832317283377451580351551746148103260165927146788719028671875} a^{4} - \frac{3154628084586506962349198282104433146638613315692317813412}{277439094459150526783850582049367753388642382262906342890625} a^{3} - \frac{95011167268930203457181736540005487988102528854390132187763}{277439094459150526783850582049367753388642382262906342890625} a^{2} - \frac{89846998125148561470017981600876566743614272624448164420021}{832317283377451580351551746148103260165927146788719028671875} a + \frac{37032668109016450018490923614889226121940280764891157995141}{832317283377451580351551746148103260165927146788719028671875}$

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

Class group and class number

$C_{2387272210}$, which has order $2387272210$ (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:  \( 59692546406.70596 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.51200.2, 5.5.5719140625.2, 10.10.1071794405000000000000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ $20$ $20$

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
11Data not computed