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Properties

Label 20.4.45653946346...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 61^{10}$
Root discriminant $30.41$
Ramified primes $2, 5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 90, -1200, 5786, -13793, 16842, -8502, -846, 896, 1338, -138, -926, 648, -202, 246, -218, 71, -20, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 20*x^17 + 71*x^16 - 218*x^15 + 246*x^14 - 202*x^13 + 648*x^12 - 926*x^11 - 138*x^10 + 1338*x^9 + 896*x^8 - 846*x^7 - 8502*x^6 + 16842*x^5 - 13793*x^4 + 5786*x^3 - 1200*x^2 + 90*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 20*x^17 + 71*x^16 - 218*x^15 + 246*x^14 - 202*x^13 + 648*x^12 - 926*x^11 - 138*x^10 + 1338*x^9 + 896*x^8 - 846*x^7 - 8502*x^6 + 16842*x^5 - 13793*x^4 + 5786*x^3 - 1200*x^2 + 90*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 20 x^{17} + 71 x^{16} - 218 x^{15} + 246 x^{14} - 202 x^{13} + 648 x^{12} - 926 x^{11} - 138 x^{10} + 1338 x^{9} + 896 x^{8} - 846 x^{7} - 8502 x^{6} + 16842 x^{5} - 13793 x^{4} + 5786 x^{3} - 1200 x^{2} + 90 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(456539463464244864640000000000=2^{16}\cdot 5^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.41$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{5}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{12}$, $\frac{1}{120} a^{15} + \frac{1}{40} a^{14} - \frac{1}{8} a^{13} + \frac{1}{24} a^{12} - \frac{3}{40} a^{11} - \frac{3}{40} a^{10} - \frac{7}{40} a^{9} - \frac{7}{120} a^{8} - \frac{1}{120} a^{7} - \frac{19}{120} a^{6} + \frac{7}{120} a^{5} + \frac{9}{40} a^{4} + \frac{17}{120} a^{3} + \frac{3}{40} a^{2} - \frac{59}{120} a + \frac{31}{120}$, $\frac{1}{360} a^{16} - \frac{1}{360} a^{15} - \frac{7}{360} a^{14} - \frac{1}{24} a^{13} - \frac{1}{40} a^{12} - \frac{43}{360} a^{11} - \frac{7}{72} a^{10} + \frac{67}{360} a^{9} - \frac{23}{360} a^{8} - \frac{5}{24} a^{7} + \frac{1}{120} a^{6} - \frac{1}{360} a^{5} + \frac{169}{360} a^{4} - \frac{169}{360} a^{3} + \frac{35}{72} a^{2} - \frac{41}{120} a - \frac{37}{180}$, $\frac{1}{3600} a^{17} - \frac{1}{1200} a^{16} + \frac{1}{360} a^{15} - \frac{19}{900} a^{14} + \frac{31}{600} a^{13} - \frac{17}{180} a^{12} - \frac{49}{600} a^{11} + \frac{2}{225} a^{10} - \frac{29}{450} a^{9} + \frac{353}{1800} a^{8} + \frac{113}{600} a^{7} - \frac{223}{900} a^{6} + \frac{61}{600} a^{5} + \frac{139}{300} a^{4} + \frac{193}{600} a^{3} + \frac{7}{225} a^{2} + \frac{1087}{3600} a + \frac{1753}{3600}$, $\frac{1}{54000} a^{18} + \frac{1}{10800} a^{17} - \frac{1}{1000} a^{16} + \frac{37}{27000} a^{15} - \frac{313}{13500} a^{14} - \frac{1901}{27000} a^{13} - \frac{344}{3375} a^{12} - \frac{73}{600} a^{11} - \frac{1523}{27000} a^{10} + \frac{527}{2700} a^{9} - \frac{191}{13500} a^{8} + \frac{5551}{27000} a^{7} - \frac{583}{2700} a^{6} + \frac{173}{27000} a^{5} - \frac{2681}{6750} a^{4} - \frac{6827}{27000} a^{3} + \frac{9653}{54000} a^{2} + \frac{2911}{6000} a + \frac{2957}{27000}$, $\frac{1}{14501765084958000} a^{19} + \frac{1429885679}{181272063561975} a^{18} - \frac{797716160197}{7250882542479000} a^{17} + \frac{18668935940459}{14501765084958000} a^{16} - \frac{5233809052507}{2416960847493000} a^{15} + \frac{50232473164909}{1208480423746500} a^{14} - \frac{115607587409537}{7250882542479000} a^{13} + \frac{81932285601421}{725088254247900} a^{12} + \frac{34899418148354}{906360317809875} a^{11} + \frac{6689968972084}{181272063561975} a^{10} - \frac{119568640890589}{2416960847493000} a^{9} + \frac{296638362085301}{7250882542479000} a^{8} - \frac{15281122901807}{290035301699160} a^{7} + \frac{36315013284091}{402826807915500} a^{6} + \frac{1674161346606701}{7250882542479000} a^{5} + \frac{269461171192063}{1208480423746500} a^{4} + \frac{988466624850541}{4833921694986000} a^{3} - \frac{482444088451567}{1812720635619750} a^{2} + \frac{321662292131843}{1812720635619750} a + \frac{1117627972876903}{2900353016991600}$

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:  $11$
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:  \( 46607660.7666 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 10.2.5405416326400.1

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
Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61Data not computed