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Properties

Label 20.0.20038303066...000.38
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $2885714480$ (GRH)
Class group $[2, 22, 65584420]$ (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![165264883180898149, -4212066055510280, 14175271751235155, -40333066477110, 628697885710730, 1638272498456, 20069393424575, 39024267810, 474391226650, 1806845810, 8913543045, 1637130, 140041165, 297660, 1650440, -902, 18755, 0, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 18755*x^16 - 902*x^15 + 1650440*x^14 + 297660*x^13 + 140041165*x^12 + 1637130*x^11 + 8913543045*x^10 + 1806845810*x^9 + 474391226650*x^8 + 39024267810*x^7 + 20069393424575*x^6 + 1638272498456*x^5 + 628697885710730*x^4 - 40333066477110*x^3 + 14175271751235155*x^2 - 4212066055510280*x + 165264883180898149)
 
gp: K = bnfinit(x^20 + 110*x^18 + 18755*x^16 - 902*x^15 + 1650440*x^14 + 297660*x^13 + 140041165*x^12 + 1637130*x^11 + 8913543045*x^10 + 1806845810*x^9 + 474391226650*x^8 + 39024267810*x^7 + 20069393424575*x^6 + 1638272498456*x^5 + 628697885710730*x^4 - 40333066477110*x^3 + 14175271751235155*x^2 - 4212066055510280*x + 165264883180898149, 1)
 

Normalized defining polynomial

\( x^{20} + 110 x^{18} + 18755 x^{16} - 902 x^{15} + 1650440 x^{14} + 297660 x^{13} + 140041165 x^{12} + 1637130 x^{11} + 8913543045 x^{10} + 1806845810 x^{9} + 474391226650 x^{8} + 39024267810 x^{7} + 20069393424575 x^{6} + 1638272498456 x^{5} + 628697885710730 x^{4} - 40333066477110 x^{3} + 14175271751235155 x^{2} - 4212066055510280 x + 165264883180898149 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(709,·)$, $\chi_{3300}(1031,·)$, $\chi_{3300}(841,·)$, $\chi_{3300}(1679,·)$, $\chi_{3300}(2219,·)$, $\chi_{3300}(1849,·)$, $\chi_{3300}(1081,·)$, $\chi_{3300}(2459,·)$, $\chi_{3300}(2269,·)$, $\chi_{3300}(2591,·)$, $\chi_{3300}(3299,·)$, $\chi_{3300}(2471,·)$, $\chi_{3300}(361,·)$, $\chi_{3300}(2411,·)$, $\chi_{3300}(889,·)$, $\chi_{3300}(2939,·)$, $\chi_{3300}(829,·)$, $\chi_{3300}(1621,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{121} a^{9} + \frac{3}{11} a^{4}$, $\frac{1}{3751} a^{10} + \frac{5}{341} a^{8} + \frac{2}{341} a^{6} - \frac{10}{341} a^{5} - \frac{10}{31} a^{4} + \frac{9}{31} a^{3} + \frac{11}{31} a^{2} - \frac{3}{31} a + \frac{1}{31}$, $\frac{1}{3751} a^{11} - \frac{7}{3751} a^{9} + \frac{2}{341} a^{7} - \frac{10}{341} a^{6} + \frac{14}{341} a^{5} - \frac{87}{341} a^{4} + \frac{11}{31} a^{3} - \frac{3}{31} a^{2} + \frac{1}{31} a$, $\frac{1}{3751} a^{12} + \frac{6}{341} a^{8} - \frac{10}{341} a^{7} - \frac{3}{341} a^{6} - \frac{2}{341} a^{5} + \frac{3}{31} a^{4} - \frac{2}{31} a^{3} - \frac{15}{31} a^{2} + \frac{10}{31} a + \frac{7}{31}$, $\frac{1}{41261} a^{13} + \frac{6}{3751} a^{9} + \frac{83}{3751} a^{8} + \frac{1}{31} a^{7} - \frac{3}{341} a^{6} + \frac{3}{341} a^{5} + \frac{29}{341} a^{4} - \frac{46}{341} a^{3} + \frac{15}{31} a^{2} - \frac{5}{31} a$, $\frac{1}{41261} a^{14} - \frac{10}{3751} a^{9} + \frac{12}{341} a^{8} - \frac{3}{341} a^{7} - \frac{9}{341} a^{6} - \frac{4}{341} a^{5} - \frac{6}{341} a^{4} - \frac{8}{31} a^{3} - \frac{9}{31} a^{2} - \frac{13}{31} a - \frac{6}{31}$, $\frac{1}{63088069} a^{15} - \frac{46}{5735279} a^{14} - \frac{32}{5735279} a^{13} - \frac{28}{521389} a^{12} - \frac{47}{521389} a^{11} + \frac{61}{5735279} a^{10} + \frac{206}{521389} a^{9} - \frac{14688}{521389} a^{8} - \frac{398}{47399} a^{7} + \frac{20}{1529} a^{6} + \frac{11207}{521389} a^{5} + \frac{20271}{47399} a^{4} + \frac{19214}{47399} a^{3} + \frac{190}{4309} a^{2} + \frac{1575}{4309} a + \frac{1039}{4309}$, $\frac{1}{2586610829} a^{16} - \frac{135}{21376949} a^{14} + \frac{1}{5735279} a^{13} + \frac{1492}{21376949} a^{12} - \frac{2}{5735279} a^{11} + \frac{232}{21376949} a^{10} + \frac{689}{521389} a^{9} + \frac{876528}{21376949} a^{8} + \frac{948}{47399} a^{7} + \frac{642651}{21376949} a^{6} + \frac{1365}{47399} a^{5} - \frac{89863}{1943359} a^{4} + \frac{9447}{47399} a^{3} - \frac{80344}{176669} a^{2} + \frac{1583}{4309} a + \frac{36593}{176669}$, $\frac{1}{2586610829} a^{17} - \frac{17}{2586610829} a^{15} + \frac{41}{5735279} a^{14} + \frac{1447}{235146439} a^{13} - \frac{266}{5735279} a^{12} + \frac{241}{1943359} a^{11} - \frac{252}{5735279} a^{10} + \frac{26475}{21376949} a^{9} - \frac{12130}{521389} a^{8} + \frac{542529}{21376949} a^{7} + \frac{10}{47399} a^{6} - \frac{474968}{21376949} a^{5} + \frac{21699}{47399} a^{4} + \frac{886473}{1943359} a^{3} + \frac{1865}{4309} a^{2} + \frac{29254}{176669} a - \frac{27}{139}$, $\frac{1}{28452719119} a^{18} - \frac{2033}{235146439} a^{14} - \frac{134}{63088069} a^{13} - \frac{640}{21376949} a^{12} - \frac{96}{5735279} a^{11} + \frac{2251}{21376949} a^{10} - \frac{384}{521389} a^{9} - \frac{5980471}{235146439} a^{8} + \frac{270}{47399} a^{7} - \frac{79542}{21376949} a^{6} - \frac{1284}{47399} a^{5} - \frac{642373}{1943359} a^{4} - \frac{3233}{47399} a^{3} - \frac{84452}{176669} a^{2} - \frac{615}{4309} a - \frac{76153}{176669}$, $\frac{1}{74646723796241905916817094751613053681139629884798902199820547311272545701125337} a^{19} + \frac{358685047382805207673882519825775479436017659722504479500854391797}{48820617263729173261489270602755430792112249761150361150961770641774065206753} a^{18} - \frac{476669182939199994957966768238685226815354514142302555262692851478555}{6786065799658355083347008613783004880103602716799900199983686119206595063738667} a^{17} - \frac{1014026718532365694099890612552985743025464909314290386251444576829795}{6786065799658355083347008613783004880103602716799900199983686119206595063738667} a^{16} - \frac{37994058651306051151284515099136400107222715928007643885292248534766630}{6786065799658355083347008613783004880103602716799900199983686119206595063738667} a^{15} + \frac{56839967769405089006039885621741171563206087567683725977480123244644182539}{6786065799658355083347008613783004880103602716799900199983686119206595063738667} a^{14} + \frac{284672131172848274078669850077571756181868922918444406563850785683648387}{56083188426928554407826517469281032066971923279338018181683356357079298047427} a^{13} + \frac{18118040869579696446946817883242676446660966767760142789831750602575646951}{616915072696214098486091692162091352736691156072718199998516919927872278521697} a^{12} + \frac{28868048040340658490144540336517244351767500724149794583942773803416198889}{616915072696214098486091692162091352736691156072718199998516919927872278521697} a^{11} + \frac{61024224893987355458700266878561416751227565586097006878634911324099978172}{616915072696214098486091692162091352736691156072718199998516919927872278521697} a^{10} - \frac{563532095072148776761706110401650431929932090387948647043332646115125154478}{616915072696214098486091692162091352736691156072718199998516919927872278521697} a^{9} - \frac{779819616613462472722403177071220439107401687474270582069703168836824353945}{56083188426928554407826517469281032066971923279338018181683356357079298047427} a^{8} + \frac{1876506895189700502403164054069126261686470854123233824165678406434126435956}{56083188426928554407826517469281032066971923279338018181683356357079298047427} a^{7} + \frac{27175771395149026014721593558280448260859260990786781152641337046012306150}{56083188426928554407826517469281032066971923279338018181683356357079298047427} a^{6} + \frac{2260613559747176505194305724002583593483073584310902258058557110374244130711}{56083188426928554407826517469281032066971923279338018181683356357079298047427} a^{5} + \frac{2190742393511849651804749112121063832543971450862581325568969533485031290276}{5098471675175323127984228860843730187906538479939819834698486941552663458857} a^{4} - \frac{2474999337272439978437315025506743599447649415722126545502195641087663537578}{5098471675175323127984228860843730187906538479939819834698486941552663458857} a^{3} - \frac{186778474632164245514941799883010993911003289343993487431393866462409041585}{463497425015938466180384441894884562536958043630892712245316994686605768987} a^{2} + \frac{229494053573083122348578892773032013960282122451246702396889634667667198383}{463497425015938466180384441894884562536958043630892712245316994686605768987} a + \frac{75998722749035915039544419406899384363407273789141194515170459400588518059}{463497425015938466180384441894884562536958043630892712245316994686605768987}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{65584420}$, which has order $2885714480$ (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:  \( 860169583.1577827 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), 5.5.5719140625.1, 10.10.163542847442626953125.2, 10.0.447641631963281250000000000.3, 10.0.89528326392656250000000000.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
5Data not computed
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$