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Properties

Label 20.0.26166731193...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{32}\cdot 41^{18}$
Root discriminant $742.83$
Ramified primes $2, 5, 41$
Class number $2420202500$ (GRH)
Class group $[5, 5, 5, 19361620]$ (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![2188557863100625, 0, 3128076535624375, 0, 955070487099375, 0, 119492300657500, 0, 7204928006375, 0, 232074088051, 0, 4245354840, 0, 44893975, 0, 267320, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 267320*x^16 + 44893975*x^14 + 4245354840*x^12 + 232074088051*x^10 + 7204928006375*x^8 + 119492300657500*x^6 + 955070487099375*x^4 + 3128076535624375*x^2 + 2188557863100625)
 
gp: K = bnfinit(x^20 + 820*x^18 + 267320*x^16 + 44893975*x^14 + 4245354840*x^12 + 232074088051*x^10 + 7204928006375*x^8 + 119492300657500*x^6 + 955070487099375*x^4 + 3128076535624375*x^2 + 2188557863100625, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 267320 x^{16} + 44893975 x^{14} + 4245354840 x^{12} + 232074088051 x^{10} + 7204928006375 x^{8} + 119492300657500 x^{6} + 955070487099375 x^{4} + 3128076535624375 x^{2} + 2188557863100625 \)

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:  \(2616673119324506454012530994165039062500000000000000000000=2^{20}\cdot 5^{32}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $742.83$
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:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(901,·)$, $\chi_{4100}(3911,·)$, $\chi_{4100}(3721,·)$, $\chi_{4100}(1911,·)$, $\chi_{4100}(1671,·)$, $\chi_{4100}(3981,·)$, $\chi_{4100}(141,·)$, $\chi_{4100}(1991,·)$, $\chi_{4100}(1431,·)$, $\chi_{4100}(3481,·)$, $\chi_{4100}(2191,·)$, $\chi_{4100}(1861,·)$, $\chi_{4100}(871,·)$, $\chi_{4100}(2921,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(4041,·)$, $\chi_{4100}(3961,·)$, $\chi_{4100}(1931,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10}$, $\frac{1}{41} a^{11}$, $\frac{1}{41} a^{12}$, $\frac{1}{205} a^{13} + \frac{1}{5} a^{3}$, $\frac{1}{205} a^{14} + \frac{1}{5} a^{4}$, $\frac{1}{1025} a^{15} - \frac{1}{205} a^{11} - \frac{2}{5} a^{7} + \frac{11}{25} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{1025} a^{16} - \frac{1}{205} a^{12} - \frac{2}{5} a^{8} + \frac{11}{25} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{1025} a^{17} - \frac{2}{5} a^{9} + \frac{11}{25} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{13154386981282414149497033705479816838507852944148830640125} a^{18} - \frac{400417294537306394410923352537245512048226133221688032}{2630877396256482829899406741095963367701570588829766128025} a^{16} - \frac{4207847108953091865978651583990204047216510559712413481}{2630877396256482829899406741095963367701570588829766128025} a^{14} - \frac{700542065361561628604330688514799347632476101869937683}{105235095850259313195976269643838534708062823553190645121} a^{12} - \frac{11076016979903894870643824929867137629949086636367787032}{2630877396256482829899406741095963367701570588829766128025} a^{10} + \frac{124076122245338974281017701921865348456818116621966385786}{320838706860546686573098383060483337524581779125581235125} a^{8} + \frac{4983819024627365018842095475184891836259694522640695619}{64167741372109337314619676612096667504916355825116247025} a^{6} - \frac{868593627842969831214280916882966621044480572795744047}{2566709654884373492584787064483866700196654233004649881} a^{4} + \frac{610282267020465855804542154415406927120907541969211798}{2566709654884373492584787064483866700196654233004649881} a^{2} - \frac{748300103777378183157393101134161103832262272966951861}{2566709654884373492584787064483866700196654233004649881}$, $\frac{1}{24615554424721137232084958729423977133379813565179767949083750125} a^{19} + \frac{2671279209129738127236854043211753126706558661921340681576}{24015175048508426567887764614072172813053476648955871169837805} a^{17} + \frac{53882269730499776816337331387754585710293021181792059969823}{120075875242542132839438823070360864065267383244779355849189025} a^{15} - \frac{29915023856962054115457411208910151515914882298505202778629}{24015175048508426567887764614072172813053476648955871169837805} a^{13} + \frac{1129328815012800260588839795810979667651974137859781360121878}{120075875242542132839438823070360864065267383244779355849189025} a^{11} - \frac{127396520444081097576014871717979826802219823630038596273391439}{600379376212710664197194115351804320326336916223896779245945125} a^{9} - \frac{164979398492998167456191503775186135034256465772552448552404}{2928679883964442264376556660252704001591887396214130630468025} a^{7} + \frac{450441614544017361848021441654948237460762220600715101982039}{2928679883964442264376556660252704001591887396214130630468025} a^{5} - \frac{52566953142133881189749233703094619611063258008211307962623}{585735976792888452875311332050540800318377479242826126093605} a^{3} + \frac{17602476561974990061459075605471793582116853943061182372877}{117147195358577690575062266410108160063675495848565225218721} a$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}\times C_{19361620}$, which has order $2420202500$ (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:  \( \frac{1316349024241720099564}{3315354594870947968852575074650774225} a^{19} + \frac{26458601614259757060698}{80862307191974340703721343284165225} a^{17} + \frac{8668606873816674881977051}{80862307191974340703721343284165225} a^{15} + \frac{292017559664114460861694436}{16172461438394868140744268656833045} a^{13} + \frac{27499776648535088231996009186}{16172461438394868140744268656833045} a^{11} + \frac{7361876346100312205641447023074}{80862307191974340703721343284165225} a^{9} + \frac{5290949023857616572845517384163}{1972251394926203431798081543516225} a^{7} + \frac{79917320468761716416211381135246}{1972251394926203431798081543516225} a^{5} + \frac{24433714793237670921582282165965}{78890055797048137271923261740649} a^{3} + \frac{101708058820860379596316990091225}{78890055797048137271923261740649} a \) (order $4$)
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:  \( 326379971244.5939 \) (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

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{41}) \), \(\Q(i, \sqrt{41})\), 5.5.1103812890625.3, 10.0.51153427249056406250000000000.2, 10.0.1247644567050156250000000000.3, 10.10.49954518797906646728515625.4

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.5.8.5$x^{5} - 5 x^{4} + 80$$5$$1$$8$$C_5$$[2]$
5.5.8.5$x^{5} - 5 x^{4} + 80$$5$$1$$8$$C_5$$[2]$
5.5.8.5$x^{5} - 5 x^{4} + 80$$5$$1$$8$$C_5$$[2]$
5.5.8.5$x^{5} - 5 x^{4} + 80$$5$$1$$8$$C_5$$[2]$
$41$41.10.9.2$x^{10} - 1476$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.2$x^{10} - 1476$$10$$1$$9$$C_{10}$$[\ ]_{10}$