↑

Properties

Label 20.0.30249544679...0864.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 19^{10}$
Root discriminant $83.95$
Ramified primes $2, 11, 19$
Class number $209715$ (GRH)
Class group $[209715]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![761311783, -373781678, 694330259, -269621164, 268615445, -86582502, 61342885, -17457004, 9747738, -2525960, 1160478, -280194, 111312, -26146, 8948, -1820, 507, -86, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 27*x^18 - 86*x^17 + 507*x^16 - 1820*x^15 + 8948*x^14 - 26146*x^13 + 111312*x^12 - 280194*x^11 + 1160478*x^10 - 2525960*x^9 + 9747738*x^8 - 17457004*x^7 + 61342885*x^6 - 86582502*x^5 + 268615445*x^4 - 269621164*x^3 + 694330259*x^2 - 373781678*x + 761311783)
 
gp: K = bnfinit(x^20 - 6*x^19 + 27*x^18 - 86*x^17 + 507*x^16 - 1820*x^15 + 8948*x^14 - 26146*x^13 + 111312*x^12 - 280194*x^11 + 1160478*x^10 - 2525960*x^9 + 9747738*x^8 - 17457004*x^7 + 61342885*x^6 - 86582502*x^5 + 268615445*x^4 - 269621164*x^3 + 694330259*x^2 - 373781678*x + 761311783, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 27 x^{18} - 86 x^{17} + 507 x^{16} - 1820 x^{15} + 8948 x^{14} - 26146 x^{13} + 111312 x^{12} - 280194 x^{11} + 1160478 x^{10} - 2525960 x^{9} + 9747738 x^{8} - 17457004 x^{7} + 61342885 x^{6} - 86582502 x^{5} + 268615445 x^{4} - 269621164 x^{3} + 694330259 x^{2} - 373781678 x + 761311783 \)

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:  \(302495446798263449838462018238088740864=2^{30}\cdot 11^{16}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.95$
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:  $2, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1672=2^{3}\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1672}(1,·)$, $\chi_{1672}(837,·)$, $\chi_{1672}(1633,·)$, $\chi_{1672}(1445,·)$, $\chi_{1672}(1101,·)$, $\chi_{1672}(493,·)$, $\chi_{1672}(533,·)$, $\chi_{1672}(1369,·)$, $\chi_{1672}(265,·)$, $\chi_{1672}(797,·)$, $\chi_{1672}(37,·)$, $\chi_{1672}(609,·)$, $\chi_{1672}(229,·)$, $\chi_{1672}(1329,·)$, $\chi_{1672}(1065,·)$, $\chi_{1672}(685,·)$, $\chi_{1672}(113,·)$, $\chi_{1672}(1521,·)$, $\chi_{1672}(949,·)$, $\chi_{1672}(873,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{27} a^{15} + \frac{1}{3} a^{14} - \frac{10}{27} a^{13} - \frac{4}{9} a^{12} - \frac{5}{27} a^{11} + \frac{1}{3} a^{10} - \frac{8}{27} a^{9} + \frac{5}{27} a^{7} + \frac{1}{3} a^{6} + \frac{13}{27} a^{5} - \frac{4}{27} a^{3} + \frac{1}{9} a^{2} - \frac{13}{27} a - \frac{10}{27}$, $\frac{1}{621} a^{16} + \frac{1}{621} a^{15} - \frac{28}{621} a^{14} - \frac{148}{621} a^{13} - \frac{125}{621} a^{12} - \frac{167}{621} a^{11} - \frac{296}{621} a^{10} + \frac{10}{621} a^{9} - \frac{211}{621} a^{8} - \frac{31}{621} a^{7} + \frac{157}{621} a^{6} + \frac{220}{621} a^{5} + \frac{1}{27} a^{4} - \frac{19}{621} a^{3} - \frac{145}{621} a^{2} - \frac{257}{621} a + \frac{26}{621}$, $\frac{1}{621} a^{17} - \frac{2}{207} a^{15} + \frac{29}{207} a^{14} - \frac{1}{3} a^{13} + \frac{101}{207} a^{12} - \frac{244}{621} a^{11} - \frac{4}{23} a^{10} + \frac{8}{23} a^{9} + \frac{20}{69} a^{8} + \frac{101}{207} a^{7} + \frac{10}{23} a^{6} + \frac{34}{207} a^{5} - \frac{14}{207} a^{4} - \frac{218}{621} a^{3} - \frac{43}{621} a^{2} - \frac{16}{621} a - \frac{256}{621}$, $\frac{1}{20577029120096216751} a^{18} + \frac{4403024537039042}{20577029120096216751} a^{17} + \frac{427538678637040}{762112189633193213} a^{16} - \frac{260321578602677606}{20577029120096216751} a^{15} - \frac{3124845376138467355}{6859009706698738917} a^{14} - \frac{3024064071171763207}{20577029120096216751} a^{13} - \frac{1774305818572271278}{20577029120096216751} a^{12} + \frac{10101015979480334954}{20577029120096216751} a^{11} - \frac{135434716579697030}{298217813334727779} a^{10} + \frac{1793190787158300538}{20577029120096216751} a^{9} + \frac{846210467191172945}{2286336568899579639} a^{8} + \frac{8061263317819372640}{20577029120096216751} a^{7} - \frac{369016843187157100}{762112189633193213} a^{6} - \frac{9439129267264662194}{20577029120096216751} a^{5} + \frac{256838153667324496}{20577029120096216751} a^{4} + \frac{1369681511078302270}{6859009706698738917} a^{3} - \frac{389223496288059611}{2286336568899579639} a^{2} + \frac{4711214019300170789}{20577029120096216751} a + \frac{603646573866540052}{6859009706698738917}$, $\frac{1}{4103330804794573466338334238967915534785189473847} a^{19} - \frac{87804176388316195991868089270}{4103330804794573466338334238967915534785189473847} a^{18} + \frac{139803628126789721910515613414264306737118406}{1367776934931524488779444746322638511595063157949} a^{17} + \frac{1869751353705814068446027232318248848436763344}{4103330804794573466338334238967915534785189473847} a^{16} - \frac{37340051294219820832949943754653517398047961227}{4103330804794573466338334238967915534785189473847} a^{15} - \frac{35203631919389304194805989865699870977156438417}{4103330804794573466338334238967915534785189473847} a^{14} - \frac{1060722202483128936461446362603852322698433821173}{4103330804794573466338334238967915534785189473847} a^{13} - \frac{858483678329828638893731412624360726575247293741}{4103330804794573466338334238967915534785189473847} a^{12} + \frac{2311102659410464189723610735114667102077460892}{20619752787912429479087106728481987611985876753} a^{11} - \frac{968338268505978803170764428479085287088769830776}{4103330804794573466338334238967915534785189473847} a^{10} + \frac{2030139731076178484172942233693857634868424099012}{4103330804794573466338334238967915534785189473847} a^{9} + \frac{1725868330070710613580850078821250277067689898727}{4103330804794573466338334238967915534785189473847} a^{8} - \frac{1364826660531210862892155821734553362437441744072}{4103330804794573466338334238967915534785189473847} a^{7} - \frac{72005380881734183532751771224626563007733430940}{178405687164981455058188445172518066729790846689} a^{6} - \frac{1711423292851565219814145259508329637632453926621}{4103330804794573466338334238967915534785189473847} a^{5} + \frac{1200889866730356216805104872584587956636593648846}{4103330804794573466338334238967915534785189473847} a^{4} + \frac{63429797945692076242529465084466950369010599465}{151975214992391609864382749591404279066118128661} a^{3} + \frac{1847094354755496976480922860273240882782507400058}{4103330804794573466338334238967915534785189473847} a^{2} + \frac{987422688461284106223803258252230806159913467743}{4103330804794573466338334238967915534785189473847} a + \frac{1158761326467978069358564331923523319017082176789}{4103330804794573466338334238967915534785189473847}$

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

Class group and class number

$C_{209715}$, which has order $209715$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.0.530773810885219.1, 10.0.17392396235086856192.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
19Data not computed