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Properties

Label 20.0.35583470806...000.21
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $534.03$
Ramified primes $2, 5, 11$
Class number $1762444860$ (GRH)
Class group $[22, 80111130]$ (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![3859291108515249, 0, -506924202145680, 0, 16018850939850, 0, 1227599011980, 0, 13313608825, 0, -523936776, 0, 12022560, 0, 12100, 0, -7260, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7260*x^16 + 12100*x^14 + 12022560*x^12 - 523936776*x^10 + 13313608825*x^8 + 1227599011980*x^6 + 16018850939850*x^4 - 506924202145680*x^2 + 3859291108515249)
 
gp: K = bnfinit(x^20 - 7260*x^16 + 12100*x^14 + 12022560*x^12 - 523936776*x^10 + 13313608825*x^8 + 1227599011980*x^6 + 16018850939850*x^4 - 506924202145680*x^2 + 3859291108515249, 1)
 

Normalized defining polynomial

\( x^{20} - 7260 x^{16} + 12100 x^{14} + 12022560 x^{12} - 523936776 x^{10} + 13313608825 x^{8} + 1227599011980 x^{6} + 16018850939850 x^{4} - 506924202145680 x^{2} + 3859291108515249 \)

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:  \(3558347080635028147840000000000000000000000000000000000=2^{40}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $534.03$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(811,·)$, $\chi_{2200}(1281,·)$, $\chi_{2200}(2121,·)$, $\chi_{2200}(1931,·)$, $\chi_{2200}(909,·)$, $\chi_{2200}(789,·)$, $\chi_{2200}(1879,·)$, $\chi_{2200}(199,·)$, $\chi_{2200}(491,·)$, $\chi_{2200}(549,·)$, $\chi_{2200}(1961,·)$, $\chi_{2200}(1451,·)$, $\chi_{2200}(1841,·)$, $\chi_{2200}(839,·)$, $\chi_{2200}(1971,·)$, $\chi_{2200}(629,·)$, $\chi_{2200}(1159,·)$, $\chi_{2200}(1469,·)$, $\chi_{2200}(1919,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{891} a^{10} + \frac{1}{27} a^{8} - \frac{1}{27} a^{6} + \frac{13}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{891} a^{11} - \frac{1}{27} a^{7} - \frac{5}{81} a^{5} + \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{2673} a^{12} - \frac{1}{2673} a^{10} + \frac{4}{81} a^{8} + \frac{7}{243} a^{6} - \frac{1}{243} a^{4} - \frac{2}{27} a^{2}$, $\frac{1}{2673} a^{13} - \frac{1}{2673} a^{11} - \frac{11}{243} a^{7} - \frac{10}{243} a^{5} - \frac{11}{81} a^{3} + \frac{2}{9} a$, $\frac{1}{24057} a^{14} - \frac{2}{24057} a^{12} - \frac{2}{24057} a^{10} - \frac{86}{2187} a^{8} - \frac{8}{2187} a^{6} + \frac{253}{2187} a^{4} + \frac{56}{243} a^{2} + \frac{1}{3}$, $\frac{1}{216513} a^{15} + \frac{34}{216513} a^{13} + \frac{97}{216513} a^{11} + \frac{22}{19683} a^{9} + \frac{811}{19683} a^{7} - \frac{701}{19683} a^{5} - \frac{340}{2187} a^{3} + \frac{13}{27} a$, $\frac{1}{9310059} a^{16} - \frac{1}{846369} a^{14} - \frac{542}{9310059} a^{12} + \frac{4706}{9310059} a^{10} - \frac{8441}{846369} a^{8} - \frac{5444}{846369} a^{6} - \frac{1507}{31347} a^{4} - \frac{811}{10449} a^{2} + \frac{4}{129}$, $\frac{1}{567913599} a^{17} + \frac{1064}{567913599} a^{15} - \frac{47584}{567913599} a^{13} + \frac{31846}{63101511} a^{11} + \frac{77903}{51628509} a^{9} - \frac{659173}{51628509} a^{7} + \frac{1170148}{51628509} a^{5} - \frac{668854}{5736501} a^{3} + \frac{31426}{70821} a$, $\frac{1}{278589569602000724015243228993667962126881520973} a^{18} - \frac{4122264846804374101318619743634078606}{1146459134164612032984540037010979268011858111} a^{16} - \frac{902948634396131190542329192561459503571610}{92863189867333574671747742997889320708960506991} a^{14} - \frac{12327354190956153722926443861272123875628048}{278589569602000724015243228993667962126881520973} a^{12} + \frac{13550181963658571967759782102708514525651692}{30954396622444524890582580999296440236320168997} a^{10} - \frac{369002209699873321703597553163441441571008046}{8442108169757597697431612999808120064450955181} a^{8} - \frac{1357584805297313562512897869491075954698039851}{25326324509272793092294838999424360193352865543} a^{6} + \frac{301734676526004144754684271045891762750703643}{2814036056585865899143870999936040021483651727} a^{4} - \frac{6594729586232835218020967275047323346992023}{34741185883776122211652728394272099030662367} a^{2} + \frac{1679587316558258962480965250864606874817}{7031205400480899051133926005721938682587}$, $\frac{1}{106142626018362275849807670246587493570341859490713} a^{19} + \frac{67362742490083825559140663025252441845}{92863189867333574671747742997889320708960506991} a^{17} - \frac{37575426773639227790403029681102903133769676}{35380875339454091949935890082195831190113953163571} a^{15} - \frac{12578894336728129790493154603717518179796136682}{106142626018362275849807670246587493570341859490713} a^{13} + \frac{143210036945723529967667023746848264059937120}{1072147737559214907573814850975631248185271307987} a^{11} + \frac{18897041499884413120590828331054831669023464926}{3216443212677644722721444552926893744555813923961} a^{9} + \frac{431258326726525953081730824566956616108477225156}{9649329638032934168164333658780681233667441771883} a^{7} + \frac{133775595053839232599047831542387618558227775917}{1072147737559214907573814850975631248185271307987} a^{5} + \frac{5905835782681821277451481068789824324689287323}{39709175465156107687919068554653009192047085481} a^{3} - \frac{46873495405938632361380678144324392419465940}{490236734137729724542210722896950730766013401} a$

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

Class group and class number

$C_{22}\times C_{80111130}$, which has order $1762444860$ (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:  \( 13680393897245.906 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-5}, \sqrt{22})\), 5.5.5719140625.2, 10.0.167467875781250000000000.4, 10.0.58948692275000000000000000.1, 10.10.11789738455000000000000000.2

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed