↑

Properties

Label 20.0.81976336236...8176.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 163^{10}$
Root discriminant $44.23$
Ramified primes $2, 3, 163$
Class number $242$ (GRH)
Class group $[11, 22]$ (GRH)
Galois group $D_{10}$ (as 20T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![147456, 0, 61440, 0, -155888, 0, 144184, 0, -72343, 0, 11490, 0, 2015, 0, -780, 0, 167, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 + 167*x^16 - 780*x^14 + 2015*x^12 + 11490*x^10 - 72343*x^8 + 144184*x^6 - 155888*x^4 + 61440*x^2 + 147456)
 
gp: K = bnfinit(x^20 - 14*x^18 + 167*x^16 - 780*x^14 + 2015*x^12 + 11490*x^10 - 72343*x^8 + 144184*x^6 - 155888*x^4 + 61440*x^2 + 147456, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} + 167 x^{16} - 780 x^{14} + 2015 x^{12} + 11490 x^{10} - 72343 x^{8} + 144184 x^{6} - 155888 x^{4} + 61440 x^{2} + 147456 \)

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:  \(819763362363033867895669628338176=2^{20}\cdot 3^{10}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $2, 3, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{64} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{3}{128} a^{5} - \frac{7}{32} a^{4} + \frac{7}{32} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{768} a^{14} - \frac{1}{256} a^{13} + \frac{1}{384} a^{12} - \frac{1}{64} a^{11} + \frac{1}{128} a^{10} - \frac{3}{128} a^{9} - \frac{5}{192} a^{8} - \frac{5}{256} a^{6} + \frac{7}{256} a^{5} - \frac{23}{384} a^{4} - \frac{15}{64} a^{3} - \frac{7}{96} a^{2} - \frac{1}{4} a$, $\frac{1}{768} a^{15} - \frac{1}{768} a^{13} - \frac{1}{128} a^{12} - \frac{1}{128} a^{11} + \frac{5}{384} a^{9} - \frac{1}{64} a^{8} - \frac{5}{256} a^{7} - \frac{1}{32} a^{6} - \frac{73}{768} a^{5} + \frac{3}{128} a^{4} + \frac{37}{192} a^{3} + \frac{1}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{3072} a^{16} + \frac{1}{3072} a^{14} + \frac{11}{1536} a^{12} - \frac{1}{1536} a^{10} - \frac{1}{32} a^{9} + \frac{65}{3072} a^{8} - \frac{1}{32} a^{7} + \frac{113}{3072} a^{6} - \frac{3}{32} a^{5} - \frac{59}{384} a^{4} + \frac{1}{32} a^{3} + \frac{65}{192} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{9216} a^{17} - \frac{1}{3072} a^{15} + \frac{13}{4608} a^{13} - \frac{1}{128} a^{12} - \frac{37}{4608} a^{11} - \frac{263}{9216} a^{9} - \frac{1}{64} a^{8} + \frac{461}{9216} a^{7} - \frac{1}{32} a^{6} + \frac{3}{256} a^{5} - \frac{29}{128} a^{4} - \frac{2}{9} a^{3} - \frac{7}{32} a^{2} - \frac{1}{6} a$, $\frac{1}{1819243806741504} a^{18} - \frac{47979772175}{303207301123584} a^{16} + \frac{332115453503}{1819243806741504} a^{14} - \frac{1}{256} a^{13} + \frac{249356707787}{41346450153216} a^{12} - \frac{1}{64} a^{11} - \frac{53552497474001}{1819243806741504} a^{10} + \frac{1}{128} a^{9} - \frac{10833307920701}{909621903370752} a^{8} + \frac{1}{32} a^{7} + \frac{12104164845849}{202138200749056} a^{6} + \frac{31}{256} a^{5} - \frac{25390609208849}{113702737921344} a^{4} - \frac{1}{64} a^{3} + \frac{1240827918937}{37900912640448} a^{2} - \frac{1}{8} a + \frac{130535142685}{394801173338}$, $\frac{1}{14553950453932032} a^{19} + \frac{250861856813}{7276975226966016} a^{17} - \frac{9143112706609}{14553950453932032} a^{15} - \frac{299718598823}{110257200408576} a^{13} - \frac{1}{128} a^{12} + \frac{15932509033487}{14553950453932032} a^{11} - \frac{14319873194307}{808552802996224} a^{9} - \frac{1}{64} a^{8} - \frac{500635528021231}{14553950453932032} a^{7} - \frac{1}{32} a^{6} - \frac{27685349777425}{1819243806741504} a^{5} - \frac{29}{128} a^{4} - \frac{157751196138431}{909621903370752} a^{3} + \frac{9}{32} a^{2} - \frac{3951207478663}{9475228160112} a - \frac{1}{2}$

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

Class group and class number

$C_{11}\times C_{22}$, which has order $242$ (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:  \( \frac{36484847}{18706877190144} a^{19} - \frac{336321437}{9353438595072} a^{17} + \frac{7814939137}{18706877190144} a^{15} - \frac{372283897}{141718766592} a^{13} + \frac{124580474497}{18706877190144} a^{11} + \frac{18384683155}{1039270955008} a^{9} - \frac{4922072346977}{18706877190144} a^{7} + \frac{1169346437449}{2338359648768} a^{5} - \frac{522067905649}{1169179824384} a^{3} + \frac{8765163601}{12178956504} a \) (order $4$)
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:  \( 1141791581.42 \) (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

\(\Q(\sqrt{-489}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{489}) \), \(\Q(i, \sqrt{489})\), 5.1.3825936.1 x5, 10.0.28631509956043776.1, 10.0.58551145104384.1 x5, 10.2.7157877489010944.1 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 R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$163$163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$