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Properties

Label 20.0.94964489254...000.17
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $23917120$ (GRH)
Class group $[2, 4, 2989640]$ (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![3241961064409, -1463656482026, 2015103544347, -900155631148, 398300256029, -81919808394, 35515197703, -6382780148, 3347838129, -307737088, 163217351, -15229510, 7815333, -401156, 206059, -10962, 5549, -136, 69, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 69*x^18 - 136*x^17 + 5549*x^16 - 10962*x^15 + 206059*x^14 - 401156*x^13 + 7815333*x^12 - 15229510*x^11 + 163217351*x^10 - 307737088*x^9 + 3347838129*x^8 - 6382780148*x^7 + 35515197703*x^6 - 81919808394*x^5 + 398300256029*x^4 - 900155631148*x^3 + 2015103544347*x^2 - 1463656482026*x + 3241961064409)
 
gp: K = bnfinit(x^20 - 2*x^19 + 69*x^18 - 136*x^17 + 5549*x^16 - 10962*x^15 + 206059*x^14 - 401156*x^13 + 7815333*x^12 - 15229510*x^11 + 163217351*x^10 - 307737088*x^9 + 3347838129*x^8 - 6382780148*x^7 + 35515197703*x^6 - 81919808394*x^5 + 398300256029*x^4 - 900155631148*x^3 + 2015103544347*x^2 - 1463656482026*x + 3241961064409, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 69 x^{18} - 136 x^{17} + 5549 x^{16} - 10962 x^{15} + 206059 x^{14} - 401156 x^{13} + 7815333 x^{12} - 15229510 x^{11} + 163217351 x^{10} - 307737088 x^{9} + 3347838129 x^{8} - 6382780148 x^{7} + 35515197703 x^{6} - 81919808394 x^{5} + 398300256029 x^{4} - 900155631148 x^{3} + 2015103544347 x^{2} - 1463656482026 x + 3241961064409 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(811,·)$, $\chi_{4620}(391,·)$, $\chi_{4620}(1289,·)$, $\chi_{4620}(4171,·)$, $\chi_{4620}(1231,·)$, $\chi_{4620}(2129,·)$, $\chi_{4620}(2969,·)$, $\chi_{4620}(29,·)$, $\chi_{4620}(2911,·)$, $\chi_{4620}(419,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(839,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(2099,·)$, $\chi_{4620}(2549,·)$, $\chi_{4620}(1259,·)$, $\chi_{4620}(3359,·)$, $\chi_{4620}(2941,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{3}$, $\frac{1}{473} a^{15} + \frac{20}{473} a^{14} - \frac{6}{473} a^{13} + \frac{15}{473} a^{12} + \frac{20}{473} a^{11} + \frac{1}{473} a^{10} + \frac{197}{473} a^{9} + \frac{166}{473} a^{8} + \frac{186}{473} a^{7} - \frac{87}{473} a^{6} + \frac{186}{473} a^{5} + \frac{156}{473} a^{4} - \frac{212}{473} a^{3} - \frac{159}{473} a^{2} + \frac{3}{473} a + \frac{208}{473}$, $\frac{1}{31846727336237} a^{16} + \frac{2308057653}{2895157030567} a^{15} + \frac{815233040965}{31846727336237} a^{14} - \frac{165021390354}{31846727336237} a^{13} + \frac{998564313136}{31846727336237} a^{12} + \frac{1239702815057}{31846727336237} a^{11} + \frac{156945935262}{31846727336237} a^{10} - \frac{6799154577848}{31846727336237} a^{9} - \frac{14501651641647}{31846727336237} a^{8} + \frac{11747059631202}{31846727336237} a^{7} - \frac{9783115760357}{31846727336237} a^{6} - \frac{87148424159}{31846727336237} a^{5} + \frac{13661376136799}{31846727336237} a^{4} + \frac{205351353203}{31846727336237} a^{3} - \frac{844471564102}{31846727336237} a^{2} - \frac{15801525413268}{31846727336237} a - \frac{11247551321426}{31846727336237}$, $\frac{1}{31846727336237} a^{17} - \frac{2338051682}{31846727336237} a^{15} + \frac{593502587094}{31846727336237} a^{14} - \frac{719687166790}{31846727336237} a^{13} - \frac{926651998967}{31846727336237} a^{12} - \frac{962766823532}{31846727336237} a^{11} - \frac{1197115654947}{31846727336237} a^{10} + \frac{5961299478753}{31846727336237} a^{9} + \frac{9411078254157}{31846727336237} a^{8} + \frac{5944098311649}{31846727336237} a^{7} - \frac{9782745461733}{31846727336237} a^{6} - \frac{2366389799617}{31846727336237} a^{5} + \frac{6829686887175}{31846727336237} a^{4} - \frac{13055768602281}{31846727336237} a^{3} + \frac{3852538268765}{31846727336237} a^{2} + \frac{13268157529034}{31846727336237} a - \frac{64880727819}{2895157030567}$, $\frac{1}{31846727336237} a^{18} - \frac{12395857036}{31846727336237} a^{15} + \frac{1262216268824}{31846727336237} a^{14} - \frac{79800949189}{31846727336237} a^{13} - \frac{88564682241}{31846727336237} a^{12} - \frac{1019845229316}{31846727336237} a^{11} + \frac{28988354310}{2895157030567} a^{10} + \frac{1042599382793}{2895157030567} a^{9} + \frac{872880449831}{2895157030567} a^{8} - \frac{12824633528838}{31846727336237} a^{7} - \frac{1013443929649}{2895157030567} a^{6} + \frac{129385124613}{2895157030567} a^{5} - \frac{1716017417600}{31846727336237} a^{4} - \frac{12294680611208}{31846727336237} a^{3} - \frac{2724614686854}{31846727336237} a^{2} + \frac{11558719110134}{31846727336237} a - \frac{109132935933}{740621565959}$, $\frac{1}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{19} - \frac{103405301370656546338175037128779876377680831221195062674}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{18} - \frac{165983491657711086800408094788212599984044598078747597499}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{17} + \frac{140640836477855181851938348533914689548854303886471134936}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{16} - \frac{3964700171254383715972411488229771812019378258021678311626492032669}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{15} - \frac{77765376346751063804865759006514242534132184121905798335475734295059}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{14} - \frac{193753013555012886890802440517075519795411167551841766952029266248364}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{13} + \frac{382013395254586233566526944677963362301292710713715676881302362261438}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{12} - \frac{90656671598303073598905673486478652363020241086203069616743018190938}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{11} - \frac{23813858777912092666111073345111904948617360512361598737085620143849}{1058626348489644266587551984847341104640037541708579521006424082229807} a^{10} + \frac{283413847957034623842410503568609668933309406954962582653825275699409}{1058626348489644266587551984847341104640037541708579521006424082229807} a^{9} - \frac{107725913244250033484865571096480036301420237445739712812196183250110}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{8} - \frac{1950801312262349241270850980242800112772386063048687301022038248125047}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{7} + \frac{2711404228740373120856927090439457978588743562641381153691434273487226}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{6} + \frac{4674322363113414752834301976578676119907376012256037874499682832223313}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{5} - \frac{3433460162818675336310901912207556408182160642067677043070469678935276}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{4} - \frac{3112810690530149515700835624685505220184866837246844884338087804680100}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{3} + \frac{4480689679534978624798975314996055713733798101908719534687695369326844}{11644889833386086932463071833320752151040412958794374731070664904527877} a^{2} - \frac{5372633982617230613172176930659332658957177404677628472742665590383609}{11644889833386086932463071833320752151040412958794374731070664904527877} a - \frac{5039113945477087589135254325416219964825449777747725558126298085957834}{11644889833386086932463071833320752151040412958794374731070664904527877}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{2989640}$, which has order $23917120$ (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:  \( 5362955.973727969 \) (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-77}, \sqrt{-105})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, 10.0.2801482624803139200000.1, 10.10.1790566527853125.1

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 R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$