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Properties

Label 20.0.26166731193...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{32}\cdot 41^{18}$
Root discriminant $742.83$
Ramified primes $2, 5, 41$
Class number $3385277500$ (GRH)
Class group $[5, 5, 1045, 129580]$ (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![815454150625, 0, 5199142836875, 0, 9468336511875, 0, 3916459359000, 0, 585061753875, 0, 41080774551, 0, 1471426040, 0, 26751475, 0, 230420, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 230420*x^16 + 26751475*x^14 + 1471426040*x^12 + 41080774551*x^10 + 585061753875*x^8 + 3916459359000*x^6 + 9468336511875*x^4 + 5199142836875*x^2 + 815454150625)
 
gp: K = bnfinit(x^20 + 820*x^18 + 230420*x^16 + 26751475*x^14 + 1471426040*x^12 + 41080774551*x^10 + 585061753875*x^8 + 3916459359000*x^6 + 9468336511875*x^4 + 5199142836875*x^2 + 815454150625, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 230420 x^{16} + 26751475 x^{14} + 1471426040 x^{12} + 41080774551 x^{10} + 585061753875 x^{8} + 3916459359000 x^{6} + 9468336511875 x^{4} + 5199142836875 x^{2} + 815454150625 \)

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:  \(2616673119324506454012530994165039062500000000000000000000=2^{20}\cdot 5^{32}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $742.83$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(1091,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(901,·)$, $\chi_{4100}(961,·)$, $\chi_{4100}(3331,·)$, $\chi_{4100}(1281,·)$, $\chi_{4100}(1041,·)$, $\chi_{4100}(3091,·)$, $\chi_{4100}(3141,·)$, $\chi_{4100}(2081,·)$, $\chi_{4100}(31,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(1521,·)$, $\chi_{4100}(3571,·)$, $\chi_{4100}(761,·)$, $\chi_{4100}(2811,·)$, $\chi_{4100}(1021,·)$, $\chi_{4100}(3011,·)$, $\chi_{4100}(3071,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10}$, $\frac{1}{41} a^{11}$, $\frac{1}{451} a^{12} - \frac{1}{451} a^{10} + \frac{3}{11} a^{6} + \frac{2}{11} a^{4} + \frac{3}{11} a^{2} + \frac{5}{11}$, $\frac{1}{2255} a^{13} + \frac{2}{451} a^{11} + \frac{5}{11} a^{7} - \frac{4}{11} a^{5} - \frac{19}{55} a^{3} + \frac{1}{11} a$, $\frac{1}{2255} a^{14} + \frac{2}{451} a^{10} + \frac{5}{11} a^{8} + \frac{1}{11} a^{6} + \frac{16}{55} a^{4} - \frac{5}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11275} a^{15} + \frac{24}{2255} a^{11} + \frac{1}{11} a^{9} + \frac{23}{55} a^{7} - \frac{39}{275} a^{5} + \frac{6}{55} a^{3} - \frac{2}{11} a$, $\frac{1}{124025} a^{16} + \frac{4}{24805} a^{14} - \frac{6}{24805} a^{12} - \frac{5}{451} a^{10} - \frac{262}{605} a^{8} + \frac{436}{3025} a^{6} - \frac{20}{121} a^{4} + \frac{26}{121} a^{2} + \frac{40}{121}$, $\frac{1}{124025} a^{17} - \frac{2}{124025} a^{15} + \frac{1}{4961} a^{13} - \frac{8}{2255} a^{11} + \frac{233}{605} a^{9} - \frac{719}{3025} a^{7} - \frac{742}{3025} a^{5} - \frac{211}{605} a^{3} - \frac{26}{121} a$, $\frac{1}{41183672544291392011005135712710778057926856848625} a^{18} + \frac{16373366355817757634517609423186580613532721}{8236734508858278402201027142542155611585371369725} a^{16} + \frac{1779524419730251288190443311300541607493343229}{8236734508858278402201027142542155611585371369725} a^{14} + \frac{1060735159885663814570268351809553573324550837}{1647346901771655680440205428508431122317074273945} a^{12} - \frac{10132498873696718167940909074755105546824538492}{8236734508858278402201027142542155611585371369725} a^{10} + \frac{19623005299062481288227743304506807278135980261}{1004479818153448585634271602749043367266508703625} a^{8} + \frac{61496681305546984582677750064240895303773164352}{200895963630689717126854320549808673453301740725} a^{6} + \frac{10683998501940422138920960121575921888201953506}{40179192726137943425370864109961734690660348145} a^{4} - \frac{3840698212503403839611296654071054778333349212}{8035838545227588685074172821992346938132069629} a^{2} + \frac{12423077617912712624881457660837046882916420}{8035838545227588685074172821992346938132069629}$, $\frac{1}{1487595435972349370829516507078826014230375996229183625} a^{19} + \frac{18088730137449211206389249049370841329243108044}{7256563102304143272339104912579639093806712176727725} a^{17} - \frac{88392935819190344493675381963646200412555918364}{7256563102304143272339104912579639093806712176727725} a^{15} - \frac{245843899178744108281766920012023661712697959753}{1451312620460828654467820982515927818761342435345545} a^{13} - \frac{83195497406746000368244902889221327084016289495997}{7256563102304143272339104912579639093806712176727725} a^{11} + \frac{401854717994400793699007597870478010949499682267111}{36282815511520716361695524562898195469033560883638625} a^{9} - \frac{9963465326649066045896859264576974173666473555366}{35397868791727528157751731280876288262471766715745} a^{7} + \frac{51628841072490724576591633435561454416178737658757}{176989343958637640788758656404381441312358833578725} a^{5} + \frac{1238774755603117931742517921796169750861229204371}{7079573758345505631550346256175257652494353343149} a^{3} - \frac{3146747249913136233370198750682278499174784801766}{7079573758345505631550346256175257652494353343149} a$

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

Class group and class number

$C_{5}\times C_{5}\times C_{1045}\times C_{129580}$, which has order $3385277500$ (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:  \( \frac{214804316808189134424}{19112265072612867672849228616225} a^{19} + \frac{390351294616513255898}{42377527877190393953102502475} a^{17} + \frac{1205384576170629986501731}{466152806649094333484127527225} a^{15} + \frac{681207890870252589926646}{2273916129995582114556719645} a^{13} + \frac{1530191599303333398465061766}{93230561329818866696825505445} a^{11} + \frac{212110490985555283197669346084}{466152806649094333484127527225} a^{9} + \frac{72714958775025680481777210843}{11369580649977910572783598225} a^{7} + \frac{472800272570700087555565845476}{11369580649977910572783598225} a^{5} + \frac{41527390072044484032138905205}{454783225999116422911343929} a^{3} + \frac{12143983309696784146727460005}{454783225999116422911343929} a \) (order $4$)
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:  \( 332013200621.1798 \) (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

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{41})\), 5.5.1103812890625.1, 10.0.51153427249056406250000000000.3, 10.10.49954518797906646728515625.1, 10.0.1247644567050156250000000000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
$41$41.10.9.5$x^{10} - 68864256$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.5$x^{10} - 68864256$$10$$1$$9$$C_{10}$$[\ ]_{10}$