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Properties

Label 20.0.13745154533...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 31^{16}$
Root discriminant $180.68$
Ramified primes $2, 3, 5, 31$
Class number $18780224$ (GRH)
Class group $[2, 2, 2, 2, 2, 586882]$ (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![24743548045, -21846268820, 23426527875, -13729323890, 8213541501, -3391191716, 1357191693, -375641044, 106843288, -19460856, 5238630, -895320, 383096, -74980, 25680, -2924, 803, -46, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 33*x^18 - 46*x^17 + 803*x^16 - 2924*x^15 + 25680*x^14 - 74980*x^13 + 383096*x^12 - 895320*x^11 + 5238630*x^10 - 19460856*x^9 + 106843288*x^8 - 375641044*x^7 + 1357191693*x^6 - 3391191716*x^5 + 8213541501*x^4 - 13729323890*x^3 + 23426527875*x^2 - 21846268820*x + 24743548045)
 
gp: K = bnfinit(x^20 - 4*x^19 + 33*x^18 - 46*x^17 + 803*x^16 - 2924*x^15 + 25680*x^14 - 74980*x^13 + 383096*x^12 - 895320*x^11 + 5238630*x^10 - 19460856*x^9 + 106843288*x^8 - 375641044*x^7 + 1357191693*x^6 - 3391191716*x^5 + 8213541501*x^4 - 13729323890*x^3 + 23426527875*x^2 - 21846268820*x + 24743548045, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 33 x^{18} - 46 x^{17} + 803 x^{16} - 2924 x^{15} + 25680 x^{14} - 74980 x^{13} + 383096 x^{12} - 895320 x^{11} + 5238630 x^{10} - 19460856 x^{9} + 106843288 x^{8} - 375641044 x^{7} + 1357191693 x^{6} - 3391191716 x^{5} + 8213541501 x^{4} - 13729323890 x^{3} + 23426527875 x^{2} - 21846268820 x + 24743548045 \)

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:  \(1374515453313585364601855186208000000000000000=2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $180.68$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1860=2^{2}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1860}(1,·)$, $\chi_{1860}(901,·)$, $\chi_{1860}(841,·)$, $\chi_{1860}(1163,·)$, $\chi_{1860}(1489,·)$, $\chi_{1860}(467,·)$, $\chi_{1860}(469,·)$, $\chi_{1860}(407,·)$, $\chi_{1860}(721,·)$, $\chi_{1860}(47,·)$, $\chi_{1860}(349,·)$, $\chi_{1860}(287,·)$, $\chi_{1860}(481,·)$, $\chi_{1860}(529,·)$, $\chi_{1860}(683,·)$, $\chi_{1860}(1427,·)$, $\chi_{1860}(109,·)$, $\chi_{1860}(1583,·)$, $\chi_{1860}(1523,·)$, $\chi_{1860}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{16} - \frac{2}{15} a^{15} + \frac{1}{15} a^{12} + \frac{1}{15} a^{10} + \frac{1}{3} a^{9} + \frac{1}{15} a^{8} + \frac{1}{3} a^{6} + \frac{2}{15} a^{5} + \frac{2}{5} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{4575} a^{17} - \frac{19}{915} a^{16} + \frac{386}{4575} a^{15} + \frac{13}{915} a^{14} + \frac{391}{4575} a^{13} + \frac{472}{4575} a^{12} - \frac{364}{4575} a^{11} + \frac{422}{4575} a^{10} + \frac{896}{4575} a^{9} - \frac{1988}{4575} a^{8} - \frac{52}{305} a^{7} - \frac{13}{75} a^{6} + \frac{211}{915} a^{5} + \frac{1022}{4575} a^{4} + \frac{277}{915} a^{3} + \frac{439}{915} a^{2} - \frac{65}{183} a - \frac{52}{915}$, $\frac{1}{349735249169587449794292675} a^{18} + \frac{2248301311167842858771}{23315683277972496652952845} a^{17} - \frac{3022106670871390079807374}{349735249169587449794292675} a^{16} + \frac{2808195627767251103013764}{69947049833917489958858535} a^{15} - \frac{26481748628476235562269834}{349735249169587449794292675} a^{14} + \frac{1586223786049536426045219}{116578416389862483264764225} a^{13} + \frac{48663895017655768076074846}{349735249169587449794292675} a^{12} + \frac{23515658025412594984021007}{349735249169587449794292675} a^{11} + \frac{28363964712801447059241931}{349735249169587449794292675} a^{10} + \frac{15875365230259694830186674}{116578416389862483264764225} a^{9} + \frac{21786301477652127570487376}{69947049833917489958858535} a^{8} - \frac{51507070985135292896034143}{349735249169587449794292675} a^{7} + \frac{3920919043014512161002131}{13989409966783497991771707} a^{6} - \frac{603483376246990109950741}{116578416389862483264764225} a^{5} + \frac{22966583524541220514170044}{69947049833917489958858535} a^{4} - \frac{2187579955389112850654882}{23315683277972496652952845} a^{3} - \frac{2695626058286868613678039}{13989409966783497991771707} a^{2} + \frac{27167402148138698993882768}{69947049833917489958858535} a - \frac{1386170161876661651862444}{4663136655594499330590569}$, $\frac{1}{46579703604010716065909390174188996643299388876909014575} a^{19} + \frac{1064134977163484962529939563}{3105313573600714404393959344945933109553292591793934305} a^{18} + \frac{978807913775259108421058246796741878155838059997306}{46579703604010716065909390174188996643299388876909014575} a^{17} - \frac{170782975071866154761699288281500262255909988004453727}{9315940720802143213181878034837799328659877775381802915} a^{16} - \frac{1847279462642021599087604920666337029620960903146067509}{46579703604010716065909390174188996643299388876909014575} a^{15} - \frac{2537056479493226826352810709565857774772209598080238621}{15526567868003572021969796724729665547766462958969671525} a^{14} - \frac{4070584721822840127629953787728466171845465130933222764}{46579703604010716065909390174188996643299388876909014575} a^{13} + \frac{1586528604202695469025227401937581433255668152782889437}{46579703604010716065909390174188996643299388876909014575} a^{12} - \frac{2004543163303197793605688550390805871132641849344582479}{46579703604010716065909390174188996643299388876909014575} a^{11} - \frac{1139814174952694075945381788383989290738708076601925966}{15526567868003572021969796724729665547766462958969671525} a^{10} + \frac{1886319058275841766819601626899673817612447995527790539}{9315940720802143213181878034837799328659877775381802915} a^{9} - \frac{2748551386459880594235164918782116637543539423464437231}{15526567868003572021969796724729665547766462958969671525} a^{8} - \frac{2117600909033913643847109342888591141662978387816089333}{9315940720802143213181878034837799328659877775381802915} a^{7} + \frac{4555124415785292743972295785000896426356502042750008037}{46579703604010716065909390174188996643299388876909014575} a^{6} - \frac{4283508193849603260869502763788016171710284282650955683}{9315940720802143213181878034837799328659877775381802915} a^{5} + \frac{275933921868705239966765377313385056918407713088995863}{3105313573600714404393959344945933109553292591793934305} a^{4} + \frac{396675801977090658935916862381111353414905942726064022}{1863188144160428642636375606967559865731975555076360583} a^{3} - \frac{901778068683258144655248906312058457978846144440365494}{3105313573600714404393959344945933109553292591793934305} a^{2} - \frac{115458408438292086426225844999340182358428151030137648}{1863188144160428642636375606967559865731975555076360583} a + \frac{1166134326252282329279474676092659190022117795431194}{621062714720142880878791868989186621910658518358786861}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{586882}$, which has order $18780224$ (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:  \( 24173706.832424585 \) (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

\(\Q(\sqrt{5}) \), 4.0.18000.1, 5.5.923521.1, 10.10.2665284492003125.1

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.8.1$x^{10} - 20491 x^{5} + 239127552$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
31.10.8.1$x^{10} - 20491 x^{5} + 239127552$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$