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Properties

Label 20.0.16210890375...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 11^{10}$
Root discriminant $144.71$
Ramified primes $2, 5, 11$
Class number $1128300$ (GRH)
Class group $[10, 112830]$ (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![512312849, -374283550, 530072515, -349010210, 274410255, -138059956, 77738210, -30140740, 11425605, -2859690, 770797, -40670, -24075, -11740, 18170, -4732, -505, 330, -5, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 5*x^18 + 330*x^17 - 505*x^16 - 4732*x^15 + 18170*x^14 - 11740*x^13 - 24075*x^12 - 40670*x^11 + 770797*x^10 - 2859690*x^9 + 11425605*x^8 - 30140740*x^7 + 77738210*x^6 - 138059956*x^5 + 274410255*x^4 - 349010210*x^3 + 530072515*x^2 - 374283550*x + 512312849)
 
gp: K = bnfinit(x^20 - 10*x^19 - 5*x^18 + 330*x^17 - 505*x^16 - 4732*x^15 + 18170*x^14 - 11740*x^13 - 24075*x^12 - 40670*x^11 + 770797*x^10 - 2859690*x^9 + 11425605*x^8 - 30140740*x^7 + 77738210*x^6 - 138059956*x^5 + 274410255*x^4 - 349010210*x^3 + 530072515*x^2 - 374283550*x + 512312849, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 5 x^{18} + 330 x^{17} - 505 x^{16} - 4732 x^{15} + 18170 x^{14} - 11740 x^{13} - 24075 x^{12} - 40670 x^{11} + 770797 x^{10} - 2859690 x^{9} + 11425605 x^{8} - 30140740 x^{7} + 77738210 x^{6} - 138059956 x^{5} + 274410255 x^{4} - 349010210 x^{3} + 530072515 x^{2} - 374283550 x + 512312849 \)

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:  \(16210890375625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.71$
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$$\chi_{2200}(1,·)$, $\chi_{2200}(1761,·)$, $\chi_{2200}(1869,·)$, $\chi_{2200}(109,·)$, $\chi_{2200}(2001,·)$, $\chi_{2200}(1429,·)$, $\chi_{2200}(1561,·)$, $\chi_{2200}(989,·)$, $\chi_{2200}(1189,·)$, $\chi_{2200}(1121,·)$, $\chi_{2200}(549,·)$, $\chi_{2200}(881,·)$, $\chi_{2200}(681,·)$, $\chi_{2200}(749,·)$, $\chi_{2200}(1629,·)$, $\chi_{2200}(241,·)$, $\chi_{2200}(309,·)$, $\chi_{2200}(1321,·)$, $\chi_{2200}(441,·)$, $\chi_{2200}(2069,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{14} a^{7} + \frac{1}{14} a + \frac{3}{7}$, $\frac{1}{28} a^{8} - \frac{1}{4} a^{4} + \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{4}$, $\frac{1}{28} a^{9} - \frac{1}{4} a^{5} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{4} a$, $\frac{1}{28} a^{10} - \frac{1}{4} a^{6} - \frac{3}{14} a^{4} - \frac{2}{7} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{7} - \frac{3}{14} a^{5} + \frac{3}{14} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{2}{7} a - \frac{3}{14}$, $\frac{1}{392} a^{12} - \frac{3}{196} a^{11} + \frac{5}{392} a^{10} + \frac{1}{196} a^{9} - \frac{3}{196} a^{8} + \frac{1}{196} a^{7} + \frac{29}{392} a^{6} - \frac{1}{4} a^{5} - \frac{33}{196} a^{4} + \frac{23}{196} a^{3} - \frac{71}{392} a^{2} + \frac{95}{196} a + \frac{5}{392}$, $\frac{1}{392} a^{13} - \frac{3}{392} a^{11} + \frac{1}{98} a^{10} + \frac{3}{196} a^{9} - \frac{3}{196} a^{8} + \frac{13}{392} a^{7} + \frac{19}{98} a^{6} - \frac{19}{196} a^{5} - \frac{1}{28} a^{4} - \frac{159}{392} a^{3} - \frac{3}{98} a^{2} - \frac{87}{392} a + \frac{29}{196}$, $\frac{1}{392} a^{14} - \frac{1}{56} a^{10} - \frac{5}{392} a^{8} + \frac{3}{98} a^{7} + \frac{1}{8} a^{6} + \frac{13}{56} a^{4} + \frac{1}{7} a^{3} - \frac{13}{49} a^{2} + \frac{17}{98} a - \frac{13}{392}$, $\frac{1}{2744} a^{15} + \frac{3}{2744} a^{14} + \frac{3}{2744} a^{13} - \frac{1}{1372} a^{12} + \frac{5}{1372} a^{11} - \frac{19}{2744} a^{10} - \frac{5}{2744} a^{9} + \frac{33}{2744} a^{8} - \frac{45}{1372} a^{7} + \frac{513}{2744} a^{6} - \frac{9}{2744} a^{5} + \frac{209}{2744} a^{4} + \frac{853}{2744} a^{3} + \frac{449}{1372} a^{2} - \frac{51}{343} a - \frac{57}{2744}$, $\frac{1}{5488} a^{16} - \frac{3}{2744} a^{14} - \frac{1}{1372} a^{13} + \frac{1}{2744} a^{12} - \frac{3}{196} a^{11} + \frac{5}{2744} a^{10} - \frac{9}{1372} a^{9} + \frac{1}{112} a^{8} - \frac{9}{1372} a^{7} + \frac{661}{2744} a^{6} + \frac{311}{1372} a^{5} + \frac{29}{343} a^{4} - \frac{193}{1372} a^{3} + \frac{10}{343} a^{2} - \frac{113}{343} a + \frac{703}{5488}$, $\frac{1}{5488} a^{17} + \frac{3}{2744} a^{13} + \frac{1}{2744} a^{12} - \frac{3}{196} a^{11} - \frac{5}{2744} a^{10} - \frac{65}{5488} a^{9} - \frac{19}{1372} a^{8} - \frac{39}{1372} a^{7} - \frac{37}{2744} a^{6} + \frac{667}{2744} a^{5} - \frac{247}{1372} a^{4} - \frac{33}{196} a^{3} + \frac{691}{2744} a^{2} + \frac{2441}{5488} a - \frac{535}{2744}$, $\frac{1}{67347976677739617757094512} a^{18} - \frac{9}{67347976677739617757094512} a^{17} + \frac{1142909499851429552371}{16836994169434904439273628} a^{16} - \frac{6014690358093059106585}{33673988338869808878547256} a^{15} + \frac{9020138374999247120399}{16836994169434904439273628} a^{14} - \frac{277150885746181961085}{343612125906834784474972} a^{13} + \frac{4968657078511415863137}{4810569762695686982649608} a^{12} - \frac{72148091249114504796043}{4209248542358726109818407} a^{11} - \frac{696354747165337544076397}{67347976677739617757094512} a^{10} - \frac{409365775373502039358449}{67347976677739617757094512} a^{9} - \frac{165515806407163493228729}{33673988338869808878547256} a^{8} - \frac{49004984600172916385023}{8418497084717452219636814} a^{7} - \frac{7168522990620630972168931}{33673988338869808878547256} a^{6} + \frac{940643930463576690275151}{8418497084717452219636814} a^{5} - \frac{4287650400121884107787579}{33673988338869808878547256} a^{4} + \frac{3613936609456714321015881}{33673988338869808878547256} a^{3} + \frac{630424302825901332927785}{9621139525391373965299216} a^{2} - \frac{26595637731458918942441717}{67347976677739617757094512} a + \frac{366343818060568631996809}{4209248542358726109818407}$, $\frac{1}{29114462969810159016774200443088} a^{19} + \frac{535}{72065502400520195586074753572} a^{18} - \frac{123699312607733088940365208}{1819653935613134938548387527693} a^{17} - \frac{756633519770599927724245127}{29114462969810159016774200443088} a^{16} - \frac{1533167549639214975973806371}{14557231484905079508387100221544} a^{15} - \frac{1189081752009415763403083525}{3639307871226269877096775055386} a^{14} + \frac{1746421103590995221791845}{5225136929255233132945836404} a^{13} + \frac{16084888462119731911123904391}{14557231484905079508387100221544} a^{12} - \frac{22543707343251298484179396293}{29114462969810159016774200443088} a^{11} - \frac{51249138468525517128627467137}{3639307871226269877096775055386} a^{10} + \frac{12521213303274971153471655687}{3639307871226269877096775055386} a^{9} + \frac{163569808536779875580515377355}{29114462969810159016774200443088} a^{8} + \frac{45786423757442223592322327163}{2079604497843582786912442888792} a^{7} - \frac{23940187737338419763534420359}{1039802248921791393456221444396} a^{6} - \frac{198470055411992580406125951412}{1819653935613134938548387527693} a^{5} - \frac{221594029886746180766087316291}{14557231484905079508387100221544} a^{4} - \frac{7115975884377078704172043712151}{29114462969810159016774200443088} a^{3} + \frac{1040918104722124540027639484181}{3639307871226269877096775055386} a^{2} - \frac{2942401392501874368536029587193}{7278615742452539754193550110772} a - \frac{373249493654962068707016591163}{29114462969810159016774200443088}$

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

Class group and class number

$C_{10}\times C_{112830}$, which has order $1128300$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{10}, \sqrt{-11})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.4026275000000000000000.3, 10.0.24574432373046875.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
11Data not computed