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Properties

Label 20.0.13642860264...5449.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{10}\cdot 47^{10}$
Root discriminant $22.74$
Ramified primes $11, 47$
Class number $1$ (GRH)
Class group Trivial (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![27889, 0, 41965, 0, 51635, 0, 94643, 0, 76742, 0, 38962, 0, 12704, 0, 2732, 0, 373, 0, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 29*x^18 + 373*x^16 + 2732*x^14 + 12704*x^12 + 38962*x^10 + 76742*x^8 + 94643*x^6 + 51635*x^4 + 41965*x^2 + 27889)
 
gp: K = bnfinit(x^20 + 29*x^18 + 373*x^16 + 2732*x^14 + 12704*x^12 + 38962*x^10 + 76742*x^8 + 94643*x^6 + 51635*x^4 + 41965*x^2 + 27889, 1)
 

Normalized defining polynomial

\( x^{20} + 29 x^{18} + 373 x^{16} + 2732 x^{14} + 12704 x^{12} + 38962 x^{10} + 76742 x^{8} + 94643 x^{6} + 51635 x^{4} + 41965 x^{2} + 27889 \)

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:  \(1364286026444870446587635449=11^{10}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.74$
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:  $11, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{12} - \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{3}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{12} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{2} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{3701284276302319037810} a^{18} + \frac{22297066740107349169}{740256855260463807562} a^{16} - \frac{19045272667591586797}{3701284276302319037810} a^{14} - \frac{184881224128502147415}{740256855260463807562} a^{12} + \frac{8814749411945877297}{1850642138151159518905} a^{10} - \frac{185223620214451929157}{1850642138151159518905} a^{8} + \frac{723024793921183060293}{3701284276302319037810} a^{6} - \frac{1}{2} a^{5} - \frac{1590253515635591325319}{3701284276302319037810} a^{4} - \frac{276665966414720303207}{1850642138151159518905} a^{2} - \frac{1}{2} a + \frac{904581539658107673891}{3701284276302319037810}$, $\frac{1}{618114474142487279314270} a^{19} + \frac{14675815558244428572272}{309057237071243639657135} a^{17} - \frac{853199910816292537376}{61811447414248727931427} a^{15} + \frac{13417246996439213414299}{61811447414248727931427} a^{13} + \frac{33690501663763049121368}{309057237071243639657135} a^{11} + \frac{156193877647159191441049}{618114474142487279314270} a^{9} - \frac{15214001581736459320126}{61811447414248727931427} a^{7} - \frac{1}{2} a^{6} + \frac{22381795891117563807731}{61811447414248727931427} a^{5} - \frac{157857913675677999713339}{618114474142487279314270} a^{3} + \frac{115747295976646291864727}{309057237071243639657135} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 186660.272497 \) (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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{517}) \), \(\Q(\sqrt{-11}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.0.785877504731.2 x5, 10.2.36936242722357.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$