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Properties

Label 20.0.43283557601...000.18
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $240.89$
Ramified primes $2, 3, 5, 31$
Class number $208916480$ (GRH)
Class group $[4, 4, 4, 4, 404, 2020]$ (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![2993279551691929, 239656306415480, 665515140558396, 34326601914734, 68428126663536, 1938921689882, 4332722462127, 39816797608, 188810021443, -1101238746, 5970317223, -97484958, 139893917, -3057836, 2420331, -53446, 29939, -512, 243, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 243*x^18 - 512*x^17 + 29939*x^16 - 53446*x^15 + 2420331*x^14 - 3057836*x^13 + 139893917*x^12 - 97484958*x^11 + 5970317223*x^10 - 1101238746*x^9 + 188810021443*x^8 + 39816797608*x^7 + 4332722462127*x^6 + 1938921689882*x^5 + 68428126663536*x^4 + 34326601914734*x^3 + 665515140558396*x^2 + 239656306415480*x + 2993279551691929)
 
gp: K = bnfinit(x^20 - 2*x^19 + 243*x^18 - 512*x^17 + 29939*x^16 - 53446*x^15 + 2420331*x^14 - 3057836*x^13 + 139893917*x^12 - 97484958*x^11 + 5970317223*x^10 - 1101238746*x^9 + 188810021443*x^8 + 39816797608*x^7 + 4332722462127*x^6 + 1938921689882*x^5 + 68428126663536*x^4 + 34326601914734*x^3 + 665515140558396*x^2 + 239656306415480*x + 2993279551691929, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 243 x^{18} - 512 x^{17} + 29939 x^{16} - 53446 x^{15} + 2420331 x^{14} - 3057836 x^{13} + 139893917 x^{12} - 97484958 x^{11} + 5970317223 x^{10} - 1101238746 x^{9} + 188810021443 x^{8} + 39816797608 x^{7} + 4332722462127 x^{6} + 1938921689882 x^{5} + 68428126663536 x^{4} + 34326601914734 x^{3} + 665515140558396 x^{2} + 239656306415480 x + 2993279551691929 \)

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:  \(432835576015865621834099207027388579840000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.89$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(1349,·)$, $\chi_{3720}(2441,·)$, $\chi_{3720}(3629,·)$, $\chi_{3720}(2509,·)$, $\chi_{3720}(1549,·)$, $\chi_{3720}(401,·)$, $\chi_{3720}(2321,·)$, $\chi_{3720}(3161,·)$, $\chi_{3720}(709,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(1069,·)$, $\chi_{3720}(869,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(1709,·)$, $\chi_{3720}(1589,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(2681,·)$, $\chi_{3720}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{201} a^{16} + \frac{25}{201} a^{15} - \frac{22}{201} a^{14} + \frac{8}{67} a^{13} + \frac{28}{201} a^{12} - \frac{2}{67} a^{11} + \frac{2}{67} a^{10} - \frac{70}{201} a^{9} + \frac{79}{201} a^{8} + \frac{13}{67} a^{7} + \frac{10}{67} a^{6} - \frac{50}{201} a^{5} + \frac{43}{201} a^{4} + \frac{11}{201} a^{3} + \frac{100}{201} a^{2} + \frac{1}{3} a + \frac{12}{67}$, $\frac{1}{201} a^{17} + \frac{23}{201} a^{15} - \frac{29}{201} a^{14} + \frac{31}{201} a^{13} + \frac{31}{201} a^{12} + \frac{22}{201} a^{11} - \frac{19}{201} a^{10} + \frac{29}{67} a^{9} + \frac{74}{201} a^{8} - \frac{74}{201} a^{7} + \frac{4}{201} a^{6} + \frac{20}{201} a^{5} + \frac{25}{67} a^{4} + \frac{31}{67} a^{3} + \frac{46}{201} a^{2} - \frac{31}{201} a - \frac{32}{67}$, $\frac{1}{69747} a^{18} + \frac{76}{69747} a^{17} + \frac{109}{69747} a^{16} - \frac{2507}{23249} a^{15} - \frac{3998}{69747} a^{14} - \frac{2983}{23249} a^{13} - \frac{1579}{69747} a^{12} - \frac{3620}{69747} a^{11} - \frac{1330}{23249} a^{10} + \frac{4175}{23249} a^{9} + \frac{4606}{23249} a^{8} - \frac{27458}{69747} a^{7} + \frac{17443}{69747} a^{6} - \frac{32453}{69747} a^{5} + \frac{5806}{69747} a^{4} - \frac{507}{23249} a^{3} + \frac{17224}{69747} a^{2} + \frac{7799}{69747} a + \frac{1146}{23249}$, $\frac{1}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{19} - \frac{27477073921985897932217987858962300888023172057511041201949580788426935051915077573409553}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{18} + \frac{12626515452234634294605903879895090019719965220149718793541395089878835432843663277253529409}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{17} - \frac{25808016106588613072478473058361570229498187431436562610785637400992081822965755521743591966}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{16} + \frac{363032496604555242889161139293208191297667251971559464473579551294557737557218117748391608350}{6167871828166793683901738223942525646235607559255016074164479611874953453697954368532423917631} a^{15} - \frac{2859369505315125957143431077249537557701589259220349256689792227739636417330259953949020718880}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{14} - \frac{2998126294239333266287449425776569946678786816347987861293424927852997377639699536828904089339}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{13} - \frac{311032447138606219584682922485272893846095575590244810843884682942639656267990696571746179850}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{12} + \frac{5616535651452438015913977296277343668851857657342481906641439489649379724006282454354973789}{53324540301153835883876699342442584837771823278861810439462359756843977985861276961375422919} a^{11} + \frac{2928021641576611838870656367298229734055502752425169881831921089115749240135190019116478909627}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{10} - \frac{5719928220214477203834542988929309034399270970799455182018977731780736129108933323505178605309}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{9} + \frac{6331754468251133827873234776320581622764229493108048521692580366849836567751960771799329253214}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{8} + \frac{1978118202293638174519288775114741974175204942572919471490034389296556702238069586019570697984}{6167871828166793683901738223942525646235607559255016074164479611874953453697954368532423917631} a^{7} - \frac{605478865152180692604605730932007895286615544272541082640366221522826510006518267600200438123}{6167871828166793683901738223942525646235607559255016074164479611874953453697954368532423917631} a^{6} + \frac{814560184818093016348624780189303233179719897376499158052957334160583805086393646618948664700}{6167871828166793683901738223942525646235607559255016074164479611874953453697954368532423917631} a^{5} - \frac{7761520070033018828069561654503051673914205604011327893535710754401483209522843639292096971046}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{4} + \frac{4591912108459701189892349898705339053257289721647195754936729649136367080639886909675541208881}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{3} + \frac{7657988955240622486429845400930605291330936633571404468444499384271073535739960229982796338939}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893} a^{2} + \frac{123133326243357430676079506491960186377627376905363379949478928814098398407700334519893653257}{6167871828166793683901738223942525646235607559255016074164479611874953453697954368532423917631} a + \frac{3651616273058614318715482570684413352360634909447699194593005847185632418142565430641436714977}{18503615484500381051705214671827576938706822677765048222493438835624860361093863105597271752893}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{404}\times C_{2020}$, which has order $208916480$ (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:  \( 12967416.87984626 \) (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

\(\Q(\sqrt{-310}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{93}) \), \(\Q(\sqrt{-30}, \sqrt{93})\), 5.5.923521.1, 10.0.2707417309252710400000.1, 10.0.21222658262851891200000.1, 10.10.6424828185043053.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$