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Properties

Label 20.8.17533216858...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8469, -36765, 173745, -34890, -148436, -131853, 56747, 137657, -138550, 51084, 3142, -20534, 13715, -2712, -919, 918, -314, 22, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 + 22*x^17 - 314*x^16 + 918*x^15 - 919*x^14 - 2712*x^13 + 13715*x^12 - 20534*x^11 + 3142*x^10 + 51084*x^9 - 138550*x^8 + 137657*x^7 + 56747*x^6 - 131853*x^5 - 148436*x^4 - 34890*x^3 + 173745*x^2 - 36765*x - 8469)
 
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 + 22*x^17 - 314*x^16 + 918*x^15 - 919*x^14 - 2712*x^13 + 13715*x^12 - 20534*x^11 + 3142*x^10 + 51084*x^9 - 138550*x^8 + 137657*x^7 + 56747*x^6 - 131853*x^5 - 148436*x^4 - 34890*x^3 + 173745*x^2 - 36765*x - 8469, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 11 x^{18} + 22 x^{17} - 314 x^{16} + 918 x^{15} - 919 x^{14} - 2712 x^{13} + 13715 x^{12} - 20534 x^{11} + 3142 x^{10} + 51084 x^{9} - 138550 x^{8} + 137657 x^{7} + 56747 x^{6} - 131853 x^{5} - 148436 x^{4} - 34890 x^{3} + 173745 x^{2} - 36765 x - 8469 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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:  $3, 5, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{18} + \frac{1}{15} a^{17} + \frac{1}{15} a^{16} + \frac{2}{15} a^{15} + \frac{7}{15} a^{14} - \frac{2}{15} a^{13} + \frac{2}{15} a^{12} + \frac{2}{15} a^{11} + \frac{1}{3} a^{10} - \frac{1}{15} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{3} a^{6} - \frac{4}{15} a^{5} + \frac{1}{5} a^{4} - \frac{1}{15} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{10217199550268764055143797379019540597760213927704202765} a^{19} + \frac{76836438977875746608478450656046135507627153509650277}{3405733183422921351714599126339846865920071309234734255} a^{18} - \frac{694696430225252455016483208661167521978303622072001639}{10217199550268764055143797379019540597760213927704202765} a^{17} - \frac{139068945342217436617536891445810068507890015318125348}{10217199550268764055143797379019540597760213927704202765} a^{16} + \frac{4131710927050846330022962078458802104655474653111729017}{10217199550268764055143797379019540597760213927704202765} a^{15} + \frac{4732404687466238879808627746089016504637346706679162523}{10217199550268764055143797379019540597760213927704202765} a^{14} + \frac{1750009575243915479173424911971462089477743591214719207}{10217199550268764055143797379019540597760213927704202765} a^{13} - \frac{1506321529274201496933174949466430322229654379491565163}{10217199550268764055143797379019540597760213927704202765} a^{12} - \frac{72242025022890213811947638454748061283907455767965097}{227048878894861423447639941755989791061338087282315617} a^{11} - \frac{2810259642018024632831352843078338024712597678703275536}{10217199550268764055143797379019540597760213927704202765} a^{10} - \frac{236308191871268202053460974173739744998089467329564916}{1135244394474307117238199708779948955306690436411578085} a^{9} + \frac{1656014698012004893524142043245443866201072751400821712}{3405733183422921351714599126339846865920071309234734255} a^{8} + \frac{379663628542598006258656070150439380212870867465937416}{2043439910053752811028759475803908119552042785540840553} a^{7} - \frac{1018864181047897635044065223325205699474071065215799008}{3405733183422921351714599126339846865920071309234734255} a^{6} + \frac{4770975203407411037560986174726200352636616590868668968}{10217199550268764055143797379019540597760213927704202765} a^{5} - \frac{3396371745571908530390866478145983968783380840236972696}{10217199550268764055143797379019540597760213927704202765} a^{4} + \frac{80591251274892287945804500962518424934609184452383814}{681146636684584270342919825267969373184014261846946851} a^{3} + \frac{502210314334899009199459353286890403431058292097757124}{1135244394474307117238199708779948955306690436411578085} a^{2} + \frac{256485109248590359869194693228097053383928045466238329}{1135244394474307117238199708779948955306690436411578085} a + \frac{64390261156707732164965799045130962101555818042353366}{227048878894861423447639941755989791061338087282315617}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 30328116268.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed