↑

Properties

Label 20.0.18374514268...6464.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 3^{14}\cdot 389^{4}$
Root discriminant $16.34$
Ramified primes $2, 3, 389$
Class number $1$
Class group Trivial
Galois group 20T872

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 26, -38, -1, 98, -165, 128, -23, -106, 202, -198, 153, -104, 47, -28, 15, -4, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 - 4*x^17 + 15*x^16 - 28*x^15 + 47*x^14 - 104*x^13 + 153*x^12 - 198*x^11 + 202*x^10 - 106*x^9 - 23*x^8 + 128*x^7 - 165*x^6 + 98*x^5 - x^4 - 38*x^3 + 26*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 + 6*x^18 - 4*x^17 + 15*x^16 - 28*x^15 + 47*x^14 - 104*x^13 + 153*x^12 - 198*x^11 + 202*x^10 - 106*x^9 - 23*x^8 + 128*x^7 - 165*x^6 + 98*x^5 - x^4 - 38*x^3 + 26*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 6 x^{18} - 4 x^{17} + 15 x^{16} - 28 x^{15} + 47 x^{14} - 104 x^{13} + 153 x^{12} - 198 x^{11} + 202 x^{10} - 106 x^{9} - 23 x^{8} + 128 x^{7} - 165 x^{6} + 98 x^{5} - x^{4} - 38 x^{3} + 26 x^{2} - 8 x + 1 \)

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:  \(1837451426888215601086464=2^{24}\cdot 3^{14}\cdot 389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.34$
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:  $2, 3, 389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{247911} a^{19} - \frac{51382}{247911} a^{18} + \frac{105691}{247911} a^{17} + \frac{123400}{247911} a^{16} + \frac{32951}{247911} a^{15} - \frac{2041}{4861} a^{14} - \frac{28105}{247911} a^{13} + \frac{9431}{247911} a^{12} + \frac{82516}{247911} a^{11} - \frac{63388}{247911} a^{10} - \frac{52300}{247911} a^{9} - \frac{25582}{82637} a^{8} + \frac{90583}{247911} a^{7} - \frac{54464}{247911} a^{6} + \frac{49715}{247911} a^{5} + \frac{6304}{82637} a^{4} + \frac{74735}{247911} a^{3} + \frac{107582}{247911} a^{2} - \frac{35577}{82637} a + \frac{13003}{247911}$

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:  \( \frac{1638443}{14583} a^{19} + \frac{1341187}{14583} a^{18} + \frac{10880846}{14583} a^{17} + \frac{2295932}{14583} a^{16} + \frac{26116660}{14583} a^{15} - \frac{8231714}{4861} a^{14} + \frac{55947313}{14583} a^{13} - \frac{124208861}{14583} a^{12} + \frac{147501605}{14583} a^{11} - \frac{200610908}{14583} a^{10} + \frac{163577008}{14583} a^{9} - \frac{11676769}{4861} a^{8} - \frac{69432055}{14583} a^{7} + \frac{152963888}{14583} a^{6} - \frac{143326520}{14583} a^{5} + \frac{13221433}{4861} a^{4} + \frac{33502405}{14583} a^{3} - \frac{34991156}{14583} a^{2} + \frac{4344785}{4861} a - \frac{1846987}{14583} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28290.2541963 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.112960521216.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
389Data not computed