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Properties

Label 20.0.15056807481...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $64.40$
Ramified primes $2, 5, 11$
Class number $22082$ (GRH)
Class group $[22082]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90594151, -43101732, 92992732, -38741504, 47151361, -18262392, 15991074, -5429272, 3826719, -1188452, 710860, -198776, 101774, -24620, 10834, -2164, 815, -124, 40, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151)
 
gp: K = bnfinit(x^20 - 4*x^19 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 40 x^{18} - 124 x^{17} + 815 x^{16} - 2164 x^{15} + 10834 x^{14} - 24620 x^{13} + 101774 x^{12} - 198776 x^{11} + 710860 x^{10} - 1188452 x^{9} + 3826719 x^{8} - 5429272 x^{7} + 15991074 x^{6} - 18262392 x^{5} + 47151361 x^{4} - 38741504 x^{3} + 92992732 x^{2} - 43101732 x + 90594151 \)

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:  \(1505680748169532571648000000000000000=2^{30}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.40$
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:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(133,·)$, $\chi_{440}(201,·)$, $\chi_{440}(397,·)$, $\chi_{440}(333,·)$, $\chi_{440}(81,·)$, $\chi_{440}(37,·)$, $\chi_{440}(213,·)$, $\chi_{440}(89,·)$, $\chi_{440}(93,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(357,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(157,·)$, $\chi_{440}(49,·)$, $\chi_{440}(53,·)$, $\chi_{440}(9,·)$, $\chi_{440}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{129465048205969732082} a^{18} - \frac{4386745894764693593}{129465048205969732082} a^{17} - \frac{5795495127824378987}{129465048205969732082} a^{16} + \frac{5398581827088625317}{64732524102984866041} a^{15} - \frac{21681594856348756439}{129465048205969732082} a^{14} - \frac{32028476571840651089}{129465048205969732082} a^{13} - \frac{6280515428730923949}{64732524102984866041} a^{12} - \frac{12687981328305388033}{129465048205969732082} a^{11} + \frac{1773135110523987587}{64732524102984866041} a^{10} - \frac{7023735395622664239}{64732524102984866041} a^{9} - \frac{51687944860507675625}{129465048205969732082} a^{8} - \frac{18260704970265208431}{64732524102984866041} a^{7} - \frac{55289415541152622783}{129465048205969732082} a^{6} - \frac{21079642912051569583}{129465048205969732082} a^{5} + \frac{5280954985400113009}{64732524102984866041} a^{4} - \frac{3080758734537280111}{64732524102984866041} a^{3} - \frac{54623750072342076023}{129465048205969732082} a^{2} + \frac{17748939790426915670}{64732524102984866041} a - \frac{495418086098270553}{1187752735834584698}$, $\frac{1}{118216937413906326588751968930290650256977427042} a^{19} + \frac{242871368242008535795300241}{118216937413906326588751968930290650256977427042} a^{18} + \frac{1837540030207052918567970715856992886530091086}{59108468706953163294375984465145325128488713521} a^{17} + \frac{15713336633476176543057408629450503108448619595}{118216937413906326588751968930290650256977427042} a^{16} + \frac{10349421815976805838494631863951486648687496702}{59108468706953163294375984465145325128488713521} a^{15} + \frac{12212983205610533853809473769392323671916251943}{118216937413906326588751968930290650256977427042} a^{14} + \frac{8810687717641064865010810271809837618801171283}{59108468706953163294375984465145325128488713521} a^{13} + \frac{24524706890264878875897618227086030440520906639}{118216937413906326588751968930290650256977427042} a^{12} - \frac{9721766949743151714862250607794620943669023271}{118216937413906326588751968930290650256977427042} a^{11} - \frac{3648414194016155056274698987350592885583083664}{59108468706953163294375984465145325128488713521} a^{10} + \frac{16872727046540713825079493045297365977864495096}{59108468706953163294375984465145325128488713521} a^{9} + \frac{10181665844792234318827804085070446932962560357}{59108468706953163294375984465145325128488713521} a^{8} - \frac{2851284670857578054430499291581143338196148906}{59108468706953163294375984465145325128488713521} a^{7} - \frac{28296158741921968782542484904219599335743315139}{59108468706953163294375984465145325128488713521} a^{6} + \frac{16578445401604141872333719181260202576469609776}{59108468706953163294375984465145325128488713521} a^{5} + \frac{19970654894339195018018102061956072077582406055}{59108468706953163294375984465145325128488713521} a^{4} + \frac{47954903666644538858106256986359388494750806319}{118216937413906326588751968930290650256977427042} a^{3} - \frac{40298407922832348658119151761570923342055104899}{118216937413906326588751968930290650256977427042} a^{2} + \frac{22312438400537867903317311213178258542784684101}{118216937413906326588751968930290650256977427042} a + \frac{248500831794186747839388456732310079102169891}{542279529421588654076843894175645184665034069}$

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

Class group and class number

$C_{22082}$, which has order $22082$ (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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$