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Properties

Label 20.0.54383102575...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{6}\cdot 9931^{6}$
Root discriminant $217.16$
Ramified primes $2, 5, 11, 9931$
Class number $170824256$ (GRH)
Class group $[2, 2, 2, 21353032]$ (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59667980405, 0, 375696185150, 0, 310756387810, 0, 108465111260, 0, 19835671615, 0, 2038773950, 0, 118265156, 0, 3687046, 0, 55991, 0, 391, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 391*x^18 + 55991*x^16 + 3687046*x^14 + 118265156*x^12 + 2038773950*x^10 + 19835671615*x^8 + 108465111260*x^6 + 310756387810*x^4 + 375696185150*x^2 + 59667980405)
 
gp: K = bnfinit(x^20 + 391*x^18 + 55991*x^16 + 3687046*x^14 + 118265156*x^12 + 2038773950*x^10 + 19835671615*x^8 + 108465111260*x^6 + 310756387810*x^4 + 375696185150*x^2 + 59667980405, 1)
 

Normalized defining polynomial

\( x^{20} + 391 x^{18} + 55991 x^{16} + 3687046 x^{14} + 118265156 x^{12} + 2038773950 x^{10} + 19835671615 x^{8} + 108465111260 x^{6} + 310756387810 x^{4} + 375696185150 x^{2} + 59667980405 \)

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:  \(54383102575415196991694675182112000000000000000=2^{20}\cdot 5^{15}\cdot 11^{6}\cdot 9931^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $217.16$
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:  $2, 5, 11, 9931$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{55} a^{16} - \frac{27}{55} a^{14} + \frac{1}{55} a^{12} + \frac{4}{11} a^{8} + \frac{3}{11} a^{6} - \frac{2}{11} a^{4}$, $\frac{1}{55} a^{17} - \frac{27}{55} a^{15} + \frac{1}{55} a^{13} + \frac{4}{11} a^{9} + \frac{3}{11} a^{7} - \frac{2}{11} a^{5}$, $\frac{1}{149948921694620036962829941073954306814100138034645605} a^{18} - \frac{340807840089527996062279609365587070933384718079783}{149948921694620036962829941073954306814100138034645605} a^{16} + \frac{58307002996267540327871056554891254347066337828094778}{149948921694620036962829941073954306814100138034645605} a^{14} + \frac{69506174100952036519458970337201734002451036098488599}{149948921694620036962829941073954306814100138034645605} a^{12} - \frac{2834213881295299878707451613772344423668351241761259}{29989784338924007392565988214790861362820027606929121} a^{10} - \frac{8623740228558297469537677871867476014965646488630599}{29989784338924007392565988214790861362820027606929121} a^{8} + \frac{3015430448640311023307295134885458584830805890512770}{29989784338924007392565988214790861362820027606929121} a^{6} - \frac{14042931732858114058054470547640199753392537489317857}{29989784338924007392565988214790861362820027606929121} a^{4} + \frac{1212392825126044100412306817136095643858449240556282}{2726344030811273399324180746799169214801820691539011} a^{2} - \frac{111677741391928626800935663328328423759925919549}{274528650771450347328988092518293144174989496681}$, $\frac{1}{149948921694620036962829941073954306814100138034645605} a^{19} - \frac{340807840089527996062279609365587070933384718079783}{149948921694620036962829941073954306814100138034645605} a^{17} + \frac{58307002996267540327871056554891254347066337828094778}{149948921694620036962829941073954306814100138034645605} a^{15} + \frac{69506174100952036519458970337201734002451036098488599}{149948921694620036962829941073954306814100138034645605} a^{13} - \frac{2834213881295299878707451613772344423668351241761259}{29989784338924007392565988214790861362820027606929121} a^{11} - \frac{8623740228558297469537677871867476014965646488630599}{29989784338924007392565988214790861362820027606929121} a^{9} + \frac{3015430448640311023307295134885458584830805890512770}{29989784338924007392565988214790861362820027606929121} a^{7} - \frac{14042931732858114058054470547640199753392537489317857}{29989784338924007392565988214790861362820027606929121} a^{5} + \frac{1212392825126044100412306817136095643858449240556282}{2726344030811273399324180746799169214801820691539011} a^{3} - \frac{111677741391928626800935663328328423759925919549}{274528650771450347328988092518293144174989496681} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{21353032}$, which has order $170824256$ (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:  \( -1 \) (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:  \( 8467206.71416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

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 $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
9931Data not computed