Properties

Label 20.0.15913686444...8669.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1609^{4}\cdot 4021$
Root discriminant $11.48$
Ramified primes $3, 1609, 4021$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -2, 15, -1, 2, 6, -59, 17, 31, 42, -38, -42, 8, 50, -19, -13, 0, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} - 13 x^{16} - 19 x^{15} + 50 x^{14} + 8 x^{13} - 42 x^{12} - 38 x^{11} + 42 x^{10} + 31 x^{9} + 17 x^{8} - 59 x^{7} + 6 x^{6} + 2 x^{5} - x^{4} + 15 x^{3} - 2 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1591368644495819328669=3^{10}\cdot 1609^{4}\cdot 4021\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.48$
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:  $3, 1609, 4021$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{73} a^{18} - \frac{22}{73} a^{17} - \frac{23}{73} a^{16} + \frac{27}{73} a^{15} - \frac{35}{73} a^{14} - \frac{26}{73} a^{13} + \frac{33}{73} a^{12} + \frac{26}{73} a^{11} + \frac{17}{73} a^{10} + \frac{20}{73} a^{9} + \frac{22}{73} a^{8} - \frac{11}{73} a^{7} + \frac{23}{73} a^{6} + \frac{23}{73} a^{5} + \frac{31}{73} a^{4} - \frac{28}{73} a^{3} + \frac{15}{73} a^{2} - \frac{2}{73} a + \frac{7}{73}$, $\frac{1}{14026147} a^{19} + \frac{46492}{14026147} a^{18} + \frac{2911223}{14026147} a^{17} + \frac{2392303}{14026147} a^{16} - \frac{2529864}{14026147} a^{15} + \frac{6752822}{14026147} a^{14} + \frac{5119915}{14026147} a^{13} - \frac{1906743}{14026147} a^{12} - \frac{4495788}{14026147} a^{11} + \frac{2465816}{14026147} a^{10} - \frac{2667284}{14026147} a^{9} + \frac{4850906}{14026147} a^{8} + \frac{267352}{14026147} a^{7} + \frac{5511968}{14026147} a^{6} - \frac{5833173}{14026147} a^{5} + \frac{5215422}{14026147} a^{4} + \frac{2296085}{14026147} a^{3} + \frac{6128470}{14026147} a^{2} - \frac{4364178}{14026147} a + \frac{2440481}{14026147}$

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:  \( \frac{82}{73} a^{19} - \frac{435}{73} a^{18} + \frac{627}{73} a^{17} + 4 a^{16} - \frac{655}{73} a^{15} - \frac{1940}{73} a^{14} + \frac{2590}{73} a^{13} + \frac{1830}{73} a^{12} - \frac{1410}{73} a^{11} - \frac{3630}{73} a^{10} + \frac{714}{73} a^{9} + \frac{2060}{73} a^{8} + \frac{2887}{73} a^{7} - \frac{2616}{73} a^{6} - \frac{354}{73} a^{5} - \frac{591}{73} a^{4} - \frac{237}{73} a^{3} + \frac{661}{73} a^{2} + \frac{99}{73} a - \frac{53}{73} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 427.42684139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.1609.1, 10.0.629098083.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: 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 $20$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
3Data not computed
1609Data not computed
4021Data not computed