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Properties

Label 20.0.34425542535...000.24
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $94.81$
Ramified primes $2, 3, 5, 11$
Class number $1836440$ (GRH)
Class group $[2, 2, 459110]$ (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![62528029201, -11369175884, 37904480108, -5857664802, 10981722576, -1464272744, 2012905741, -232860430, 258929781, -25909114, 24460118, -2091920, 1723054, -123108, 89802, -5136, 3338, -140, 81, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 81*x^18 - 140*x^17 + 3338*x^16 - 5136*x^15 + 89802*x^14 - 123108*x^13 + 1723054*x^12 - 2091920*x^11 + 24460118*x^10 - 25909114*x^9 + 258929781*x^8 - 232860430*x^7 + 2012905741*x^6 - 1464272744*x^5 + 10981722576*x^4 - 5857664802*x^3 + 37904480108*x^2 - 11369175884*x + 62528029201)
 
gp: K = bnfinit(x^20 - 2*x^19 + 81*x^18 - 140*x^17 + 3338*x^16 - 5136*x^15 + 89802*x^14 - 123108*x^13 + 1723054*x^12 - 2091920*x^11 + 24460118*x^10 - 25909114*x^9 + 258929781*x^8 - 232860430*x^7 + 2012905741*x^6 - 1464272744*x^5 + 10981722576*x^4 - 5857664802*x^3 + 37904480108*x^2 - 11369175884*x + 62528029201, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 81 x^{18} - 140 x^{17} + 3338 x^{16} - 5136 x^{15} + 89802 x^{14} - 123108 x^{13} + 1723054 x^{12} - 2091920 x^{11} + 24460118 x^{10} - 25909114 x^{9} + 258929781 x^{8} - 232860430 x^{7} + 2012905741 x^{6} - 1464272744 x^{5} + 10981722576 x^{4} - 5857664802 x^{3} + 37904480108 x^{2} - 11369175884 x + 62528029201 \)

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:  \(3442554253548220860035959357440000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.81$
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:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(899,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(1219,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(761,·)$, $\chi_{1320}(281,·)$, $\chi_{1320}(859,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(1121,·)$, $\chi_{1320}(1019,·)$, $\chi_{1320}(161,·)$, $\chi_{1320}(299,·)$, $\chi_{1320}(619,·)$, $\chi_{1320}(1139,·)$, $\chi_{1320}(499,·)$, $\chi_{1320}(41,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{263} a^{18} - \frac{107}{263} a^{17} + \frac{68}{263} a^{16} - \frac{131}{263} a^{15} - \frac{62}{263} a^{14} - \frac{42}{263} a^{13} - \frac{42}{263} a^{12} - \frac{17}{263} a^{11} - \frac{110}{263} a^{10} - \frac{25}{263} a^{9} - \frac{74}{263} a^{8} + \frac{1}{263} a^{7} + \frac{84}{263} a^{6} - \frac{47}{263} a^{5} + \frac{64}{263} a^{4} - \frac{112}{263} a^{3} - \frac{77}{263} a^{2} + \frac{6}{263} a + \frac{67}{263}$, $\frac{1}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{19} + \frac{176767269186873235449705942682477153769471734051990207450772872499207}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{18} - \frac{235699703732917081710816201895141217314989581463365998127026181976839693}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{17} - \frac{236256811864128626566090578961011658418718731182186728045136694128834113}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{16} + \frac{44944207201425713528682959522587807073639033000432989862491493574341790}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{15} - \frac{116141043784796509103773239251644687725179818709748724328175838160484602}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{14} - \frac{85282284158753861440563056553474400591727773603816427398780597103106181}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{13} - \frac{211887055984929126939933237430011242761979541842923026693303443292169386}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{12} - \frac{230357722529584659150593016798918026610206341866784243937727237435318469}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{11} - \frac{44798669930115842313645370302490422104991251710963307708370009726845672}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{10} + \frac{27869606047391885365081537456318228937569270953742349923922284339778737}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{9} + \frac{248592681871711608444295792413129550110537645607852205766110395699173881}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{8} + \frac{33872229346077932695036362783253343004834359014797115553111260883875232}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{7} - \frac{221830416541523940755175237332789169624303571786333219826476504014333927}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{6} - \frac{34554734000550014900343500924811629813176299530883133639972854207222158}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{5} - \frac{189217803060538878358042511466911468357122172158672282829034575424104100}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{4} + \frac{187038009823607816425367752837082608631894561840282308802649872501627248}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{3} - \frac{129565823472505751435165143862568355631321053978159835340350120974513901}{520429170565427120431105019742080474075969942098346131948294671883940111} a^{2} - \frac{29941134312005119906681227606731691789438785524044818885245288607828906}{520429170565427120431105019742080474075969942098346131948294671883940111} a - \frac{178866062326700216437060938744019663415103021092869018431144983913234551}{520429170565427120431105019742080474075969942098346131948294671883940111}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{459110}$, which has order $1836440$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-330}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-10}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.0.58673283984691200000.1, \(\Q(\zeta_{33})^+\), 10.0.21950349414400000.4

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
3Data not computed
5Data not computed
$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}$