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Properties

Label 20.0.22041958999...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 71^{18}$
Root discriminant $293.20$
Ramified primes $2, 5, 71$
Class number $255637184$ (GRH)
Class group $[11, 22, 1056352]$ (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![137299627081, -30561478764, 84765787672, -99362801904, 8493008276, -1669082756, 2139997020, 1166178954, 276729750, 53231750, 5436326, -5082384, -133550, -591120, 29109, -18678, 5225, -296, 131, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 131*x^18 - 296*x^17 + 5225*x^16 - 18678*x^15 + 29109*x^14 - 591120*x^13 - 133550*x^12 - 5082384*x^11 + 5436326*x^10 + 53231750*x^9 + 276729750*x^8 + 1166178954*x^7 + 2139997020*x^6 - 1669082756*x^5 + 8493008276*x^4 - 99362801904*x^3 + 84765787672*x^2 - 30561478764*x + 137299627081)
 
gp: K = bnfinit(x^20 - 2*x^19 + 131*x^18 - 296*x^17 + 5225*x^16 - 18678*x^15 + 29109*x^14 - 591120*x^13 - 133550*x^12 - 5082384*x^11 + 5436326*x^10 + 53231750*x^9 + 276729750*x^8 + 1166178954*x^7 + 2139997020*x^6 - 1669082756*x^5 + 8493008276*x^4 - 99362801904*x^3 + 84765787672*x^2 - 30561478764*x + 137299627081, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 131 x^{18} - 296 x^{17} + 5225 x^{16} - 18678 x^{15} + 29109 x^{14} - 591120 x^{13} - 133550 x^{12} - 5082384 x^{11} + 5436326 x^{10} + 53231750 x^{9} + 276729750 x^{8} + 1166178954 x^{7} + 2139997020 x^{6} - 1669082756 x^{5} + 8493008276 x^{4} - 99362801904 x^{3} + 84765787672 x^{2} - 30561478764 x + 137299627081 \)

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:  \(22041958999702857966168416136346583695360000000000=2^{30}\cdot 5^{10}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $293.20$
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:  $2, 5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2840=2^{3}\cdot 5\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{2840}(1,·)$, $\chi_{2840}(709,·)$, $\chi_{2840}(341,·)$, $\chi_{2840}(2561,·)$, $\chi_{2840}(1161,·)$, $\chi_{2840}(1421,·)$, $\chi_{2840}(2129,·)$, $\chi_{2840}(2389,·)$, $\chi_{2840}(989,·)$, $\chi_{2840}(1761,·)$, $\chi_{2840}(869,·)$, $\chi_{2840}(369,·)$, $\chi_{2840}(2409,·)$, $\chi_{2840}(1261,·)$, $\chi_{2840}(2289,·)$, $\chi_{2840}(1141,·)$, $\chi_{2840}(969,·)$, $\chi_{2840}(2681,·)$, $\chi_{2840}(1789,·)$, $\chi_{2840}(2581,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} + \frac{1}{23} a^{10} + \frac{6}{23} a^{9} + \frac{4}{23} a^{8} + \frac{2}{23} a^{7} + \frac{11}{23} a^{6} - \frac{7}{23} a^{4} - \frac{6}{23} a^{3} - \frac{9}{23} a^{2} - \frac{8}{23} a + \frac{5}{23}$, $\frac{1}{23} a^{13} + \frac{1}{23} a^{11} + \frac{6}{23} a^{10} + \frac{4}{23} a^{9} + \frac{2}{23} a^{8} + \frac{11}{23} a^{7} - \frac{7}{23} a^{5} - \frac{6}{23} a^{4} - \frac{9}{23} a^{3} - \frac{8}{23} a^{2} + \frac{5}{23} a$, $\frac{1}{23} a^{14} + \frac{6}{23} a^{11} + \frac{3}{23} a^{10} - \frac{4}{23} a^{9} + \frac{7}{23} a^{8} - \frac{2}{23} a^{7} + \frac{5}{23} a^{6} - \frac{6}{23} a^{5} - \frac{2}{23} a^{4} - \frac{2}{23} a^{3} - \frac{9}{23} a^{2} + \frac{8}{23} a - \frac{5}{23}$, $\frac{1}{23} a^{15} + \frac{3}{23} a^{11} - \frac{10}{23} a^{10} - \frac{6}{23} a^{9} - \frac{3}{23} a^{8} - \frac{7}{23} a^{7} - \frac{3}{23} a^{6} - \frac{2}{23} a^{5} - \frac{6}{23} a^{4} + \frac{4}{23} a^{3} - \frac{7}{23} a^{2} - \frac{3}{23} a - \frac{7}{23}$, $\frac{1}{2323} a^{16} - \frac{22}{2323} a^{15} + \frac{6}{2323} a^{14} - \frac{9}{2323} a^{13} + \frac{27}{2323} a^{12} + \frac{710}{2323} a^{11} + \frac{869}{2323} a^{10} + \frac{443}{2323} a^{9} - \frac{235}{2323} a^{8} + \frac{571}{2323} a^{7} + \frac{220}{2323} a^{6} - \frac{1016}{2323} a^{5} + \frac{171}{2323} a^{4} - \frac{538}{2323} a^{3} - \frac{1059}{2323} a^{2} - \frac{705}{2323} a + \frac{612}{2323}$, $\frac{1}{2323} a^{17} + \frac{27}{2323} a^{15} + \frac{22}{2323} a^{14} + \frac{31}{2323} a^{13} - \frac{9}{2323} a^{12} - \frac{984}{2323} a^{11} + \frac{169}{2323} a^{10} - \frac{185}{2323} a^{9} - \frac{54}{2323} a^{8} - \frac{247}{2323} a^{7} - \frac{1024}{2323} a^{6} - \frac{769}{2323} a^{5} - \frac{917}{2323} a^{4} + \frac{33}{2323} a^{3} - \frac{167}{2323} a^{2} - \frac{1061}{2323} a - \frac{777}{2323}$, $\frac{1}{1750926591488834807} a^{18} + \frac{109903210762961}{1750926591488834807} a^{17} + \frac{299587849308571}{1750926591488834807} a^{16} - \frac{34967290412887742}{1750926591488834807} a^{15} + \frac{7526072154213921}{1750926591488834807} a^{14} - \frac{27245927883673634}{1750926591488834807} a^{13} + \frac{19578466488049699}{1750926591488834807} a^{12} + \frac{711586930070296529}{1750926591488834807} a^{11} + \frac{30345258399622059}{76127243108210209} a^{10} - \frac{464061638024757283}{1750926591488834807} a^{9} - \frac{679426448000317381}{1750926591488834807} a^{8} + \frac{113884008884758017}{1750926591488834807} a^{7} - \frac{98353130794946805}{1750926591488834807} a^{6} + \frac{222180077842101402}{1750926591488834807} a^{5} + \frac{587539122520481657}{1750926591488834807} a^{4} - \frac{116696626223470399}{1750926591488834807} a^{3} + \frac{116241105785735578}{1750926591488834807} a^{2} - \frac{567052601798816668}{1750926591488834807} a + \frac{562145555291503006}{1750926591488834807}$, $\frac{1}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{19} + \frac{140753321472844368418678216627295395777848481803058396794007371}{662864083093964502070394871620833711690402875245765910047339329436971573040061187} a^{18} - \frac{1348928368619725594118850477814696926492191367259675058548727359490249232240102}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{17} - \frac{2491256627108247147746714540307989503536123471070867621707421800611522063607452}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{16} - \frac{93476418751827909885268072390185175715099642036957426399290524985206005309448672}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{15} - \frac{238708937000020326879632448299358750472400808842412809203496797063808878410871114}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{14} + \frac{212717706609792835777064570212854615197774783807952351467089485118020810268203075}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{13} - \frac{138841539938198967247724905236733350060198451268580326350867223566648403483283398}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{12} - \frac{5661771039815493159143524397717939584205369799860968691412694250661730369068890814}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{11} + \frac{4191659368514982521694579201397196999455006058842651286044796325280911732196321693}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{10} - \frac{107063604556348627592052521806513580532245532978285586634165084140739254087911188}{662864083093964502070394871620833711690402875245765910047339329436971573040061187} a^{9} - \frac{118133702223583489069536468468845860776867073870551861159207346930553291051435299}{662864083093964502070394871620833711690402875245765910047339329436971573040061187} a^{8} + \frac{3946888373172081451342842194763868514057664549925610032343199819358365780464310528}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{7} - \frac{2410307460970478608580233601455033411332801741651028089995432351148020161743533295}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{6} + \frac{4775821370071054057579318639728000030556643084674960064261646604114675487269142434}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{5} - \frac{5800414813384614102434101978249740075278186162980774423932514904534507561949275362}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{4} - \frac{4162717630339371607626254187499038119425155719280199346212949075374487519509380365}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{3} - \frac{1211293369964624343146635512460159908268055598539816960940747210390505578785963936}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a^{2} - \frac{7483231359689176986326005488907308583271814876042546947988936614001214811316763397}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301} a + \frac{1130717045260337570388834824016303577514588922063715832821991640510164792280411921}{15245873911161183547619082047279175368879266130652615931088804577050346179921407301}$

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

Class group and class number

$C_{11}\times C_{22}\times C_{1056352}$, which has order $255637184$ (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:  \( 116573225.49574807 \) (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

\(\Q(\sqrt{-710}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-355}) \), \(\Q(\sqrt{2}, \sqrt{-355})\), 5.5.25411681.1, 10.0.4694886473569180774400000.1, 10.10.21160051711861096448.1, 10.0.143276564745153221875.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$71$71.10.9.8$x^{10} + 2272$$10$$1$$9$$C_{10}$$[\ ]_{10}$
71.10.9.8$x^{10} + 2272$$10$$1$$9$$C_{10}$$[\ ]_{10}$