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Properties

Label 20.0.15097568161...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{34}\cdot 11^{10}$
Root discriminant $51.16$
Ramified primes $5, 11$
Class number $3550$ (GRH)
Class group $[5, 710]$ (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![4451401, -4116040, 6692935, -5360940, 4994295, -3425825, 2414710, -1427265, 837095, -432640, 218518, -99585, 43615, -17485, 6655, -2250, 715, -210, 55, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 55*x^18 - 210*x^17 + 715*x^16 - 2250*x^15 + 6655*x^14 - 17485*x^13 + 43615*x^12 - 99585*x^11 + 218518*x^10 - 432640*x^9 + 837095*x^8 - 1427265*x^7 + 2414710*x^6 - 3425825*x^5 + 4994295*x^4 - 5360940*x^3 + 6692935*x^2 - 4116040*x + 4451401)
 
gp: K = bnfinit(x^20 - 10*x^19 + 55*x^18 - 210*x^17 + 715*x^16 - 2250*x^15 + 6655*x^14 - 17485*x^13 + 43615*x^12 - 99585*x^11 + 218518*x^10 - 432640*x^9 + 837095*x^8 - 1427265*x^7 + 2414710*x^6 - 3425825*x^5 + 4994295*x^4 - 5360940*x^3 + 6692935*x^2 - 4116040*x + 4451401, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 55 x^{18} - 210 x^{17} + 715 x^{16} - 2250 x^{15} + 6655 x^{14} - 17485 x^{13} + 43615 x^{12} - 99585 x^{11} + 218518 x^{10} - 432640 x^{9} + 837095 x^{8} - 1427265 x^{7} + 2414710 x^{6} - 3425825 x^{5} + 4994295 x^{4} - 5360940 x^{3} + 6692935 x^{2} - 4116040 x + 4451401 \)

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:  \(15097568161436356604099273681640625=5^{34}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.16$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(275=5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{275}(1,·)$, $\chi_{275}(131,·)$, $\chi_{275}(199,·)$, $\chi_{275}(76,·)$, $\chi_{275}(144,·)$, $\chi_{275}(274,·)$, $\chi_{275}(21,·)$, $\chi_{275}(89,·)$, $\chi_{275}(219,·)$, $\chi_{275}(221,·)$, $\chi_{275}(34,·)$, $\chi_{275}(164,·)$, $\chi_{275}(166,·)$, $\chi_{275}(109,·)$, $\chi_{275}(111,·)$, $\chi_{275}(241,·)$, $\chi_{275}(54,·)$, $\chi_{275}(56,·)$, $\chi_{275}(186,·)$, $\chi_{275}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{1757} a^{18} - \frac{22}{1757} a^{17} + \frac{708}{1757} a^{16} + \frac{55}{1757} a^{15} + \frac{622}{1757} a^{14} - \frac{367}{1757} a^{13} - \frac{493}{1757} a^{12} + \frac{284}{1757} a^{11} + \frac{285}{1757} a^{10} + \frac{694}{1757} a^{9} - \frac{68}{251} a^{8} - \frac{87}{1757} a^{7} - \frac{629}{1757} a^{6} - \frac{521}{1757} a^{5} + \frac{111}{1757} a^{4} - \frac{617}{1757} a^{3} + \frac{29}{1757} a^{2} + \frac{15}{1757} a - \frac{425}{1757}$, $\frac{1}{4558006837289471498546712843645966763595674792799} a^{19} - \frac{117954069124256541366600347854149238486603832}{651143833898495928363816120520852394799382113257} a^{18} + \frac{312594371329615736065031998588471968462041887466}{4558006837289471498546712843645966763595674792799} a^{17} - \frac{1001279671494232459389947322190391030525315984782}{4558006837289471498546712843645966763595674792799} a^{16} + \frac{1150718449152378818009385884601660867226470601961}{4558006837289471498546712843645966763595674792799} a^{15} + \frac{1284034946271907972800113582338184772162707088186}{4558006837289471498546712843645966763595674792799} a^{14} + \frac{1632111378948569788888285986826555997965498655027}{4558006837289471498546712843645966763595674792799} a^{13} - \frac{261786404986226920765780642370833383581974209552}{4558006837289471498546712843645966763595674792799} a^{12} - \frac{274535780717859654711463910986777814151195732897}{4558006837289471498546712843645966763595674792799} a^{11} + \frac{514509422997995869011249359904476301487114925357}{4558006837289471498546712843645966763595674792799} a^{10} - \frac{520410715971362398116570357204118404762609571106}{4558006837289471498546712843645966763595674792799} a^{9} - \frac{1264994617579358520743477022669372964035954861616}{4558006837289471498546712843645966763595674792799} a^{8} - \frac{457994810211810616373836281577607102245826340309}{4558006837289471498546712843645966763595674792799} a^{7} + \frac{1632378738672959928798811299429092264342961003099}{4558006837289471498546712843645966763595674792799} a^{6} + \frac{70468719393148369173439535111409900079739240810}{651143833898495928363816120520852394799382113257} a^{5} - \frac{2021247308909061350289988656769977894328002582472}{4558006837289471498546712843645966763595674792799} a^{4} - \frac{197117098887339104508827396450632554518754482432}{651143833898495928363816120520852394799382113257} a^{3} + \frac{758352704124536154044345015028884499301192387027}{4558006837289471498546712843645966763595674792799} a^{2} - \frac{829456136367718619159278253738665386034781779649}{4558006837289471498546712843645966763595674792799} a + \frac{767804915426236012836113819182133915828329404686}{4558006837289471498546712843645966763595674792799}$

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

Class group and class number

$C_{5}\times C_{710}$, which has order $3550$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.122872161865234375.1, 10.0.24574432373046875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$