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Properties

Label 20.0.87989866733...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{26}$
Root discriminant $14.04$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
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![9, -45, 150, -345, 595, -794, 915, -975, 905, -560, -30, 575, -735, 475, -75, -172, 200, -120, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 200*x^16 - 172*x^15 - 75*x^14 + 475*x^13 - 735*x^12 + 575*x^11 - 30*x^10 - 560*x^9 + 905*x^8 - 975*x^7 + 915*x^6 - 794*x^5 + 595*x^4 - 345*x^3 + 150*x^2 - 45*x + 9)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 200*x^16 - 172*x^15 - 75*x^14 + 475*x^13 - 735*x^12 + 575*x^11 - 30*x^10 - 560*x^9 + 905*x^8 - 975*x^7 + 915*x^6 - 794*x^5 + 595*x^4 - 345*x^3 + 150*x^2 - 45*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 120 x^{17} + 200 x^{16} - 172 x^{15} - 75 x^{14} + 475 x^{13} - 735 x^{12} + 575 x^{11} - 30 x^{10} - 560 x^{9} + 905 x^{8} - 975 x^{7} + 915 x^{6} - 794 x^{5} + 595 x^{4} - 345 x^{3} + 150 x^{2} - 45 x + 9 \)

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:  \(87989866733551025390625=3^{10}\cdot 5^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.04$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{21} a^{15} + \frac{1}{7} a^{14} - \frac{8}{21} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{21} a^{10} + \frac{1}{21} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{4}{21} a^{6} - \frac{2}{21} a^{5} + \frac{3}{7} a^{4} + \frac{2}{21} a^{3} - \frac{5}{21} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{21} a^{16} - \frac{1}{7} a^{14} - \frac{8}{21} a^{13} + \frac{1}{3} a^{12} + \frac{2}{7} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{3}{7} a^{7} - \frac{1}{3} a^{6} + \frac{1}{21} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{4}{21} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{21} a^{17} + \frac{1}{21} a^{14} + \frac{4}{21} a^{13} - \frac{2}{7} a^{12} - \frac{1}{21} a^{11} + \frac{1}{21} a^{10} + \frac{10}{21} a^{9} - \frac{1}{21} a^{7} - \frac{8}{21} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{21} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{777483} a^{18} - \frac{1}{86387} a^{17} + \frac{4363}{259161} a^{16} + \frac{729}{86387} a^{15} - \frac{114736}{777483} a^{14} + \frac{26830}{111069} a^{13} - \frac{14723}{111069} a^{12} - \frac{355762}{777483} a^{11} + \frac{359435}{777483} a^{10} - \frac{346946}{777483} a^{9} - \frac{233504}{777483} a^{8} - \frac{81727}{777483} a^{7} - \frac{238151}{777483} a^{6} + \frac{55249}{777483} a^{5} - \frac{299897}{777483} a^{4} + \frac{26696}{777483} a^{3} - \frac{42516}{86387} a^{2} + \frac{21233}{259161} a + \frac{12289}{86387}$, $\frac{1}{59866191} a^{19} + \frac{29}{59866191} a^{18} - \frac{9232}{407253} a^{17} - \frac{263954}{19955397} a^{16} - \frac{91843}{8552313} a^{15} - \frac{5282848}{59866191} a^{14} + \frac{666383}{2850771} a^{13} - \frac{21709913}{59866191} a^{12} + \frac{8077903}{19955397} a^{11} + \frac{543730}{5442381} a^{10} + \frac{2105269}{19955397} a^{9} + \frac{496906}{1392237} a^{8} + \frac{2172650}{59866191} a^{7} + \frac{328018}{2850771} a^{6} + \frac{4561316}{19955397} a^{5} - \frac{10739999}{59866191} a^{4} - \frac{16620914}{59866191} a^{3} + \frac{5725964}{19955397} a^{2} - \frac{6857063}{19955397} a - \frac{2371448}{6651799}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{16560}{154693} a^{19} - \frac{157320}{154693} a^{18} + \frac{284890}{66297} a^{17} - \frac{4915975}{464079} a^{16} + \frac{149032}{9471} a^{15} - \frac{4589680}{464079} a^{14} - \frac{292610}{22099} a^{13} + \frac{19921915}{464079} a^{12} - \frac{8377410}{154693} a^{11} + \frac{1350508}{42189} a^{10} + \frac{1772335}{154693} a^{9} - \frac{22875590}{464079} a^{8} + \frac{10490570}{154693} a^{7} - \frac{4629385}{66297} a^{6} + \frac{30140738}{464079} a^{5} - \frac{24958210}{464079} a^{4} + \frac{17204465}{464079} a^{3} - \frac{2910020}{154693} a^{2} + \frac{1233810}{154693} a - \frac{211840}{154693} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4953.47319009 \)
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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.1.140625.1 x5, 10.0.296630859375.1, 10.2.98876953125.1 x5, 10.0.59326171875.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
5Data not computed