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Properties

Label 20.0.35744090881...7856.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 19^{10}$
Root discriminant $75.45$
Ramified primes $2, 11, 19$
Class number $112750$ (GRH)
Class group $[5, 22550]$ (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![260403529, -189992882, 266043355, -172598574, 141674005, -78655554, 48560433, -23401988, 11762064, -4931916, 2094610, -764038, 276168, -86794, 26676, -7028, 1771, -372, 73, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 73*x^18 - 372*x^17 + 1771*x^16 - 7028*x^15 + 26676*x^14 - 86794*x^13 + 276168*x^12 - 764038*x^11 + 2094610*x^10 - 4931916*x^9 + 11762064*x^8 - 23401988*x^7 + 48560433*x^6 - 78655554*x^5 + 141674005*x^4 - 172598574*x^3 + 266043355*x^2 - 189992882*x + 260403529)
 
gp: K = bnfinit(x^20 - 10*x^19 + 73*x^18 - 372*x^17 + 1771*x^16 - 7028*x^15 + 26676*x^14 - 86794*x^13 + 276168*x^12 - 764038*x^11 + 2094610*x^10 - 4931916*x^9 + 11762064*x^8 - 23401988*x^7 + 48560433*x^6 - 78655554*x^5 + 141674005*x^4 - 172598574*x^3 + 266043355*x^2 - 189992882*x + 260403529, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 73 x^{18} - 372 x^{17} + 1771 x^{16} - 7028 x^{15} + 26676 x^{14} - 86794 x^{13} + 276168 x^{12} - 764038 x^{11} + 2094610 x^{10} - 4931916 x^{9} + 11762064 x^{8} - 23401988 x^{7} + 48560433 x^{6} - 78655554 x^{5} + 141674005 x^{4} - 172598574 x^{3} + 266043355 x^{2} - 189992882 x + 260403529 \)

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:  \(35744090881435427178177640826961657856=2^{20}\cdot 11^{18}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.45$
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:  \(836=2^{2}\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{836}(799,·)$, $\chi_{836}(1,·)$, $\chi_{836}(835,·)$, $\chi_{836}(571,·)$, $\chi_{836}(265,·)$, $\chi_{836}(151,·)$, $\chi_{836}(685,·)$, $\chi_{836}(723,·)$, $\chi_{836}(533,·)$, $\chi_{836}(343,·)$, $\chi_{836}(797,·)$, $\chi_{836}(229,·)$, $\chi_{836}(609,·)$, $\chi_{836}(227,·)$, $\chi_{836}(37,·)$, $\chi_{836}(39,·)$, $\chi_{836}(493,·)$, $\chi_{836}(303,·)$, $\chi_{836}(113,·)$, $\chi_{836}(607,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{67} a^{16} - \frac{8}{67} a^{15} - \frac{20}{67} a^{14} + \frac{12}{67} a^{13} - \frac{33}{67} a^{12} + \frac{13}{67} a^{11} - \frac{3}{67} a^{10} - \frac{15}{67} a^{9} + \frac{4}{67} a^{8} - \frac{8}{67} a^{7} - \frac{29}{67} a^{6} + \frac{6}{67} a^{5} - \frac{10}{67} a^{4} + \frac{4}{67} a^{3} - \frac{25}{67} a^{2} - \frac{23}{67} a - \frac{6}{67}$, $\frac{1}{67} a^{17} - \frac{17}{67} a^{15} - \frac{14}{67} a^{14} - \frac{4}{67} a^{13} + \frac{17}{67} a^{12} - \frac{33}{67} a^{11} + \frac{28}{67} a^{10} + \frac{18}{67} a^{9} + \frac{24}{67} a^{8} - \frac{26}{67} a^{7} - \frac{25}{67} a^{6} - \frac{29}{67} a^{5} - \frac{9}{67} a^{4} + \frac{7}{67} a^{3} - \frac{22}{67} a^{2} + \frac{11}{67} a + \frac{19}{67}$, $\frac{1}{25595305666416690405982243} a^{18} - \frac{9}{25595305666416690405982243} a^{17} - \frac{85011622883943320487541}{25595305666416690405982243} a^{16} + \frac{680092983071546563900532}{25595305666416690405982243} a^{15} + \frac{12255663122613488166280888}{25595305666416690405982243} a^{14} + \frac{4689953603620279591702732}{25595305666416690405982243} a^{13} - \frac{10801934589694354473024939}{25595305666416690405982243} a^{12} + \frac{9191135754441170397508733}{25595305666416690405982243} a^{11} + \frac{4999631025153294149850470}{25595305666416690405982243} a^{10} - \frac{5997386416372545172038291}{25595305666416690405982243} a^{9} + \frac{11740042168549859530332422}{25595305666416690405982243} a^{8} + \frac{4167288996594936998346920}{25595305666416690405982243} a^{7} - \frac{4626307181090892921425601}{25595305666416690405982243} a^{6} + \frac{11622850153206071833103849}{25595305666416690405982243} a^{5} + \frac{5430546647520482430705}{234819318040520095467727} a^{4} - \frac{6300228918489714881274153}{25595305666416690405982243} a^{3} - \frac{5069907689516984859905010}{25595305666416690405982243} a^{2} - \frac{21828437423361996743815}{382019487558458065760929} a + \frac{7664236997431083131838797}{25595305666416690405982243}$, $\frac{1}{386405956414781837303202832992493} a^{19} + \frac{7548366}{386405956414781837303202832992493} a^{18} + \frac{1033189956025629760894200582176}{386405956414781837303202832992493} a^{17} + \frac{31502863940672069464338154399}{5767253080817639362734370641679} a^{16} + \frac{53563293803340551288364733578736}{386405956414781837303202832992493} a^{15} + \frac{141387291647359114830713624937629}{386405956414781837303202832992493} a^{14} - \frac{4781518380179471781677130278651}{386405956414781837303202832992493} a^{13} - \frac{16914064969273916049757676687904}{386405956414781837303202832992493} a^{12} - \frac{128879228450847160211898018649962}{386405956414781837303202832992493} a^{11} + \frac{159706113659959298711265025030658}{386405956414781837303202832992493} a^{10} - \frac{206499971486276035320279368925}{3545008774447539791772503054977} a^{9} + \frac{23997348129449283669276841596732}{386405956414781837303202832992493} a^{8} + \frac{159388581456601651996500226017596}{386405956414781837303202832992493} a^{7} - \frac{19250063683635976000591821477582}{386405956414781837303202832992493} a^{6} + \frac{17467700925067171204333090329215}{386405956414781837303202832992493} a^{5} + \frac{96277173051478254268385753880205}{386405956414781837303202832992493} a^{4} + \frac{160811628507517936746961723460387}{386405956414781837303202832992493} a^{3} - \frac{138123025993934556796302556728137}{386405956414781837303202832992493} a^{2} + \frac{181601889230969693068149456238716}{386405956414781837303202832992493} a + \frac{41607206281613293046816788261574}{386405956414781837303202832992493}$

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

Class group and class number

$C_{5}\times C_{22550}$, which has order $112750$ (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:  \( 281202.490766 \) (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

\(\Q(\sqrt{-209}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{11}, \sqrt{-19})\), \(\Q(\zeta_{11})^+\), 10.0.5978636205811106816.3, \(\Q(\zeta_{44})^+\), 10.0.530773810885219.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$19$19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$