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Properties

Label 20.0.38393724531...2176.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $33.82$
Ramified primes $2, 11, 13$
Class number $12$ (GRH)
Class group $[12]$ (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![21025, -82360, 161216, -242232, 262425, -108680, -108124, 126032, 8422, -60492, 22752, 6116, -8045, 2396, 752, -672, 39, 68, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 10*x^18 + 68*x^17 + 39*x^16 - 672*x^15 + 752*x^14 + 2396*x^13 - 8045*x^12 + 6116*x^11 + 22752*x^10 - 60492*x^9 + 8422*x^8 + 126032*x^7 - 108124*x^6 - 108680*x^5 + 262425*x^4 - 242232*x^3 + 161216*x^2 - 82360*x + 21025)
 
gp: K = bnfinit(x^20 - 4*x^19 - 10*x^18 + 68*x^17 + 39*x^16 - 672*x^15 + 752*x^14 + 2396*x^13 - 8045*x^12 + 6116*x^11 + 22752*x^10 - 60492*x^9 + 8422*x^8 + 126032*x^7 - 108124*x^6 - 108680*x^5 + 262425*x^4 - 242232*x^3 + 161216*x^2 - 82360*x + 21025, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 10 x^{18} + 68 x^{17} + 39 x^{16} - 672 x^{15} + 752 x^{14} + 2396 x^{13} - 8045 x^{12} + 6116 x^{11} + 22752 x^{10} - 60492 x^{9} + 8422 x^{8} + 126032 x^{7} - 108124 x^{6} - 108680 x^{5} + 262425 x^{4} - 242232 x^{3} + 161216 x^{2} - 82360 x + 21025 \)

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:  \(3839372453113070701618634162176=2^{30}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.82$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{3}{10} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{1}{5} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{15} - \frac{1}{5} a^{12} + \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{3}{10} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{3}{10} a$, $\frac{1}{50} a^{17} - \frac{1}{25} a^{16} - \frac{1}{50} a^{13} + \frac{3}{25} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{19}{50} a^{8} - \frac{21}{50} a^{7} + \frac{1}{10} a^{6} + \frac{4}{25} a^{5} + \frac{4}{25} a^{4} + \frac{2}{25} a^{3} - \frac{1}{10} a^{2} + \frac{11}{50} a - \frac{1}{10}$, $\frac{1}{50} a^{18} + \frac{1}{50} a^{16} - \frac{1}{50} a^{14} - \frac{1}{50} a^{13} - \frac{4}{25} a^{12} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{12}{25} a^{9} + \frac{11}{50} a^{8} + \frac{13}{50} a^{7} + \frac{9}{25} a^{6} - \frac{21}{50} a^{5} + \frac{2}{5} a^{4} + \frac{4}{25} a^{3} + \frac{3}{25} a^{2} - \frac{3}{50} a - \frac{1}{5}$, $\frac{1}{148575140954216128587763339516100606048227250} a^{19} + \frac{305073472714198426117131449806103616142421}{74287570477108064293881669758050303024113625} a^{18} - \frac{717392643008053448659216498398925754010504}{74287570477108064293881669758050303024113625} a^{17} + \frac{1072683688626146712550026113586821061763803}{29715028190843225717552667903220121209645450} a^{16} + \frac{493790627327158502146575330349305814557119}{148575140954216128587763339516100606048227250} a^{15} - \frac{3471943840582005846875908782926347936475024}{74287570477108064293881669758050303024113625} a^{14} + \frac{1624496212286109016262828064615146758040449}{148575140954216128587763339516100606048227250} a^{13} + \frac{1469082199023771935676649646386283595908954}{14857514095421612858776333951610060604822725} a^{12} - \frac{418296337810041140106459947816786146262487}{2971502819084322571755266790322012120964545} a^{11} + \frac{15962440176010424530807580642340925592214491}{148575140954216128587763339516100606048227250} a^{10} + \frac{14434413147629375895286615832034802345077494}{74287570477108064293881669758050303024113625} a^{9} + \frac{31893076708191989687170002363948985123675901}{148575140954216128587763339516100606048227250} a^{8} + \frac{4068540423540460287970315876560499035487353}{148575140954216128587763339516100606048227250} a^{7} + \frac{1341743231033102366227452233079967330592409}{14857514095421612858776333951610060604822725} a^{6} + \frac{5238657311140589181469219677812778377713338}{74287570477108064293881669758050303024113625} a^{5} + \frac{74136458884937659834688893187175227889634841}{148575140954216128587763339516100606048227250} a^{4} + \frac{19172838038483817015852670061812400456675578}{74287570477108064293881669758050303024113625} a^{3} + \frac{12908100897933348961859285018252862548542282}{74287570477108064293881669758050303024113625} a^{2} - \frac{1691345763645835035477187942631725602310199}{29715028190843225717552667903220121209645450} a + \frac{9908848980093731795159924925757046392231}{20493122890236707391415633036703531868721}$

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

Class group and class number

$C_{12}$, 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:  \( 698842.003029 \) (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

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-286}) \), \(\Q(\sqrt{2}, \sqrt{-143})\), 5.1.20449.1 x5, 10.0.59797108943.2, 10.2.13702319341568.1 x5, 10.0.1959431665844224.1 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$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$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$