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Properties

Label 20.20.8851788427...0649.1
Degree $20$
Signature $[20, 0]$
Discriminant $163^{10}\cdot 401^{8}$
Root discriminant $140.39$
Ramified primes $163, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-643, 11872, -51122, -70240, 330432, 132265, -712988, -120039, 703666, 66887, -357547, -23893, 99299, 5251, -15442, -653, 1325, 41, -58, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 58*x^18 + 41*x^17 + 1325*x^16 - 653*x^15 - 15442*x^14 + 5251*x^13 + 99299*x^12 - 23893*x^11 - 357547*x^10 + 66887*x^9 + 703666*x^8 - 120039*x^7 - 712988*x^6 + 132265*x^5 + 330432*x^4 - 70240*x^3 - 51122*x^2 + 11872*x - 643)
 
gp: K = bnfinit(x^20 - x^19 - 58*x^18 + 41*x^17 + 1325*x^16 - 653*x^15 - 15442*x^14 + 5251*x^13 + 99299*x^12 - 23893*x^11 - 357547*x^10 + 66887*x^9 + 703666*x^8 - 120039*x^7 - 712988*x^6 + 132265*x^5 + 330432*x^4 - 70240*x^3 - 51122*x^2 + 11872*x - 643, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 58 x^{18} + 41 x^{17} + 1325 x^{16} - 653 x^{15} - 15442 x^{14} + 5251 x^{13} + 99299 x^{12} - 23893 x^{11} - 357547 x^{10} + 66887 x^{9} + 703666 x^{8} - 120039 x^{7} - 712988 x^{6} + 132265 x^{5} + 330432 x^{4} - 70240 x^{3} - 51122 x^{2} + 11872 x - 643 \)

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:  \(8851788427171493976819614872351019223900649=163^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.39$
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:  $163, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{2}{23} a^{16} + \frac{8}{23} a^{15} + \frac{8}{23} a^{14} + \frac{3}{23} a^{13} + \frac{7}{23} a^{12} - \frac{5}{23} a^{11} - \frac{4}{23} a^{10} - \frac{2}{23} a^{9} - \frac{1}{23} a^{8} - \frac{2}{23} a^{7} - \frac{8}{23} a^{6} - \frac{9}{23} a^{5} + \frac{6}{23} a^{4} + \frac{9}{23} a^{3} - \frac{6}{23} a^{2} + \frac{3}{23} a - \frac{10}{23}$, $\frac{1}{391} a^{18} - \frac{7}{391} a^{17} - \frac{194}{391} a^{16} - \frac{133}{391} a^{15} - \frac{8}{17} a^{14} - \frac{66}{391} a^{13} + \frac{162}{391} a^{12} + \frac{64}{391} a^{11} - \frac{12}{391} a^{10} - \frac{6}{391} a^{9} - \frac{177}{391} a^{8} - \frac{105}{391} a^{7} + \frac{178}{391} a^{6} + \frac{110}{391} a^{5} - \frac{137}{391} a^{4} - \frac{87}{391} a^{3} + \frac{57}{391} a^{2} - \frac{152}{391} a + \frac{113}{391}$, $\frac{1}{23713274798225315022278090357926481963} a^{19} - \frac{58126070525709690404636188558951}{60647761632289808241120435698021693} a^{18} + \frac{489775576495689354830723163323410574}{23713274798225315022278090357926481963} a^{17} + \frac{4608395768582219352468515400516498450}{23713274798225315022278090357926481963} a^{16} - \frac{3700637083452095555865612112559093830}{23713274798225315022278090357926481963} a^{15} - \frac{2390462968205588940902983101841493755}{23713274798225315022278090357926481963} a^{14} + \frac{2826593644076875265489600471171431097}{23713274798225315022278090357926481963} a^{13} - \frac{10722032645209309846024281776130235742}{23713274798225315022278090357926481963} a^{12} - \frac{1447493190242291072259692940278375316}{23713274798225315022278090357926481963} a^{11} + \frac{7289904182157727836393880417329897911}{23713274798225315022278090357926481963} a^{10} - \frac{689996884972079424184930894532848236}{23713274798225315022278090357926481963} a^{9} + \frac{3422314348465763947307286337223497591}{23713274798225315022278090357926481963} a^{8} + \frac{4596702465245208425329497780915499586}{23713274798225315022278090357926481963} a^{7} - \frac{1828712927684826950884444002080283323}{23713274798225315022278090357926481963} a^{6} - \frac{5484105705889048260747655884977652810}{23713274798225315022278090357926481963} a^{5} - \frac{775604318534385448400527143992718371}{23713274798225315022278090357926481963} a^{4} - \frac{8390087153387331895876904596558911965}{23713274798225315022278090357926481963} a^{3} + \frac{8626274445582881048354420962005254899}{23713274798225315022278090357926481963} a^{2} - \frac{4898614565688869444234640303653321335}{23713274798225315022278090357926481963} a + \frac{8882198080006670897158936569539619276}{23713274798225315022278090357926481963}$

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:  \( 1694909520630000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.4.26569.1, 5.5.160801.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
163Data not computed
401Data not computed