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Properties

Label 20.0.43438845422...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 23^{10}$
Root discriminant $30.33$
Ramified primes $2, 5, 23$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -8192, 30304, -66080, 96184, -101248, 82996, -61168, 50334, -39472, 22037, -6746, 2485, -2752, 1682, -364, 20, -48, 37, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 37*x^18 - 48*x^17 + 20*x^16 - 364*x^15 + 1682*x^14 - 2752*x^13 + 2485*x^12 - 6746*x^11 + 22037*x^10 - 39472*x^9 + 50334*x^8 - 61168*x^7 + 82996*x^6 - 101248*x^5 + 96184*x^4 - 66080*x^3 + 30304*x^2 - 8192*x + 1024)
 
gp: K = bnfinit(x^20 - 10*x^19 + 37*x^18 - 48*x^17 + 20*x^16 - 364*x^15 + 1682*x^14 - 2752*x^13 + 2485*x^12 - 6746*x^11 + 22037*x^10 - 39472*x^9 + 50334*x^8 - 61168*x^7 + 82996*x^6 - 101248*x^5 + 96184*x^4 - 66080*x^3 + 30304*x^2 - 8192*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 37 x^{18} - 48 x^{17} + 20 x^{16} - 364 x^{15} + 1682 x^{14} - 2752 x^{13} + 2485 x^{12} - 6746 x^{11} + 22037 x^{10} - 39472 x^{9} + 50334 x^{8} - 61168 x^{7} + 82996 x^{6} - 101248 x^{5} + 96184 x^{4} - 66080 x^{3} + 30304 x^{2} - 8192 x + 1024 \)

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:  \(434388454223632138240000000000=2^{30}\cdot 5^{10}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.33$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{10} - \frac{3}{40} a^{6} - \frac{1}{20} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{10} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{40} a^{11} - \frac{3}{40} a^{7} - \frac{1}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} - \frac{3}{80} a^{8} + \frac{1}{10} a^{7} - \frac{7}{80} a^{6} - \frac{1}{10} a^{5} - \frac{7}{40} a^{4} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{11} - \frac{3}{80} a^{9} - \frac{1}{40} a^{8} - \frac{7}{80} a^{7} - \frac{1}{10} a^{6} + \frac{3}{40} a^{5} - \frac{1}{40} a^{4} - \frac{1}{4} a^{3} + \frac{3}{20} a^{2} + \frac{3}{10} a$, $\frac{1}{320} a^{14} + \frac{1}{320} a^{13} + \frac{1}{320} a^{12} - \frac{3}{320} a^{11} + \frac{3}{320} a^{10} + \frac{3}{64} a^{9} - \frac{1}{64} a^{8} + \frac{7}{320} a^{7} + \frac{1}{80} a^{6} - \frac{1}{40} a^{5} - \frac{1}{16} a^{4} + \frac{9}{80} a^{3} - \frac{7}{20} a^{2} + \frac{1}{4} a - \frac{1}{5}$, $\frac{1}{1600} a^{15} - \frac{1}{400} a^{13} - \frac{1}{400} a^{12} + \frac{9}{800} a^{11} - \frac{1}{400} a^{10} - \frac{11}{200} a^{9} + \frac{3}{80} a^{8} + \frac{81}{1600} a^{7} - \frac{27}{400} a^{6} + \frac{99}{400} a^{5} + \frac{43}{200} a^{4} + \frac{27}{80} a^{3} - \frac{19}{50} a^{2} - \frac{7}{20} a - \frac{3}{25}$, $\frac{1}{6400} a^{16} - \frac{1}{1600} a^{14} - \frac{3}{800} a^{13} + \frac{9}{3200} a^{12} + \frac{1}{400} a^{11} - \frac{1}{800} a^{10} + \frac{1}{20} a^{9} + \frac{321}{6400} a^{8} - \frac{3}{25} a^{7} + \frac{179}{1600} a^{6} - \frac{17}{800} a^{5} + \frac{59}{320} a^{4} - \frac{63}{400} a^{3} + \frac{21}{80} a^{2} + \frac{3}{25} a + \frac{1}{5}$, $\frac{1}{6400} a^{17} - \frac{1}{1600} a^{14} + \frac{11}{3200} a^{13} + \frac{1}{320} a^{12} + \frac{1}{1600} a^{11} + \frac{11}{1600} a^{10} + \frac{269}{6400} a^{9} + \frac{43}{1600} a^{8} - \frac{21}{320} a^{7} + \frac{59}{800} a^{6} - \frac{389}{1600} a^{5} + \frac{3}{25} a^{4} + \frac{17}{80} a^{3} - \frac{4}{25} a^{2} - \frac{3}{10} a - \frac{3}{25}$, $\frac{1}{119104000} a^{18} - \frac{9}{119104000} a^{17} + \frac{2603}{59552000} a^{16} + \frac{8249}{29776000} a^{15} + \frac{11897}{59552000} a^{14} + \frac{130119}{59552000} a^{13} - \frac{118837}{29776000} a^{12} - \frac{97047}{14888000} a^{11} + \frac{32653}{23820800} a^{10} - \frac{5964729}{119104000} a^{9} + \frac{624539}{59552000} a^{8} - \frac{959939}{29776000} a^{7} - \frac{857337}{29776000} a^{6} - \frac{6177111}{29776000} a^{5} - \frac{2206389}{14888000} a^{4} - \frac{3649617}{7444000} a^{3} - \frac{554807}{3722000} a^{2} - \frac{2157}{37220} a + \frac{60301}{232625}$, $\frac{1}{169961408000} a^{19} + \frac{11}{2655647000} a^{18} - \frac{2197191}{169961408000} a^{17} - \frac{2110103}{84980704000} a^{16} - \frac{23062949}{84980704000} a^{15} + \frac{70249}{84980704} a^{14} + \frac{31980233}{84980704000} a^{13} + \frac{45611621}{8498070400} a^{12} - \frac{1301365943}{169961408000} a^{11} - \frac{82648619}{10622588000} a^{10} - \frac{213790999}{169961408000} a^{9} + \frac{3489051809}{84980704000} a^{8} - \frac{991652261}{10622588000} a^{7} - \frac{350954009}{5311294000} a^{6} - \frac{5881153341}{42490352000} a^{5} + \frac{2701520309}{21245176000} a^{4} - \frac{1000929299}{2124517600} a^{3} + \frac{1097882729}{5311294000} a^{2} + \frac{127272799}{1327823500} a - \frac{98283202}{331955875}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 161938702.069 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-230}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{10}, \sqrt{-23})\), 5.1.846400.1 x5, 10.0.659081523200000.1, 10.0.16477038080000.1 x5, 10.2.28655718400000.1 x5

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

Sibling fields

Degree 10 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$