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Properties

Label 20.20.9245359332...5456.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 13127^{8}$
Root discriminant $88.78$
Ramified primes $2, 13127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_6$ (as 20T89)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130054, -1836338, 6241202, -535552, -15500919, 6647840, 14029271, -6302156, -6237455, 2757056, 1526976, -689278, -210818, 104446, 15084, -9484, -319, 474, -21, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 21*x^18 + 474*x^17 - 319*x^16 - 9484*x^15 + 15084*x^14 + 104446*x^13 - 210818*x^12 - 689278*x^11 + 1526976*x^10 + 2757056*x^9 - 6237455*x^8 - 6302156*x^7 + 14029271*x^6 + 6647840*x^5 - 15500919*x^4 - 535552*x^3 + 6241202*x^2 - 1836338*x + 130054)
 
gp: K = bnfinit(x^20 - 10*x^19 - 21*x^18 + 474*x^17 - 319*x^16 - 9484*x^15 + 15084*x^14 + 104446*x^13 - 210818*x^12 - 689278*x^11 + 1526976*x^10 + 2757056*x^9 - 6237455*x^8 - 6302156*x^7 + 14029271*x^6 + 6647840*x^5 - 15500919*x^4 - 535552*x^3 + 6241202*x^2 - 1836338*x + 130054, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 21 x^{18} + 474 x^{17} - 319 x^{16} - 9484 x^{15} + 15084 x^{14} + 104446 x^{13} - 210818 x^{12} - 689278 x^{11} + 1526976 x^{10} + 2757056 x^{9} - 6237455 x^{8} - 6302156 x^{7} + 14029271 x^{6} + 6647840 x^{5} - 15500919 x^{4} - 535552 x^{3} + 6241202 x^{2} - 1836338 x + 130054 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(924535933252138642544025981504329875456=2^{20}\cdot 13127^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.78$
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:  $2, 13127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{654426751957206431} a^{18} - \frac{9}{654426751957206431} a^{17} - \frac{798585376827447}{654426751957206431} a^{16} + \frac{912669002088540}{93489535993886633} a^{15} - \frac{268158019307287581}{654426751957206431} a^{14} - \frac{28282296210921870}{93489535993886633} a^{13} - \frac{108565711736602651}{654426751957206431} a^{12} + \frac{34804934713239509}{93489535993886633} a^{11} - \frac{155501368770848917}{654426751957206431} a^{10} - \frac{282218965778021582}{654426751957206431} a^{9} + \frac{3687901611051213}{93489535993886633} a^{8} + \frac{266648562318073459}{654426751957206431} a^{7} + \frac{72965173754740952}{654426751957206431} a^{6} + \frac{284079264072445435}{654426751957206431} a^{5} + \frac{171555339230819687}{654426751957206431} a^{4} - \frac{120080811594060693}{654426751957206431} a^{3} - \frac{12201585541991588}{654426751957206431} a^{2} + \frac{74414244921359190}{654426751957206431} a + \frac{301984627565671319}{654426751957206431}$, $\frac{1}{207943446007650386243819} a^{19} + \frac{22695}{29706206572521483749117} a^{18} + \frac{1883719862155443511004}{207943446007650386243819} a^{17} - \frac{1285390394511312351034}{207943446007650386243819} a^{16} - \frac{2222067247934947367587}{207943446007650386243819} a^{15} + \frac{4844835109333823703074}{207943446007650386243819} a^{14} - \frac{62704573907566199069386}{207943446007650386243819} a^{13} + \frac{86638014592188779124354}{207943446007650386243819} a^{12} + \frac{50986141518558783572631}{207943446007650386243819} a^{11} + \frac{93794356492369562312487}{207943446007650386243819} a^{10} + \frac{61947359264335133732120}{207943446007650386243819} a^{9} + \frac{8752621344767883951362}{29706206572521483749117} a^{8} - \frac{56946223891826681757930}{207943446007650386243819} a^{7} - \frac{60163376977477027086374}{207943446007650386243819} a^{6} + \frac{12749337669649768204928}{29706206572521483749117} a^{5} - \frac{59879021900556536230080}{207943446007650386243819} a^{4} - \frac{14577766479114725111527}{29706206572521483749117} a^{3} - \frac{99580501001280545770683}{207943446007650386243819} a^{2} + \frac{95837162253169808368829}{207943446007650386243819} a + \frac{71433282384446670261442}{207943446007650386243819}$

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:  $19$
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:  \( 60561255281100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 20T89):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

10.10.30406182484030096384.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 10 sibling: data not computed
Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
13127Data not computed