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Properties

Label 20.4.18066279302...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{16}\cdot 5^{10}\cdot 73^{10}$
Root discriminant $46.01$
Ramified primes $3, 5, 73$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4061, 29682, 10583, -43179, -48995, 31415, 50275, -6192, -35399, -1076, 16307, 1166, -5377, -400, 1386, 28, -284, 42, 27, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 27*x^18 + 42*x^17 - 284*x^16 + 28*x^15 + 1386*x^14 - 400*x^13 - 5377*x^12 + 1166*x^11 + 16307*x^10 - 1076*x^9 - 35399*x^8 - 6192*x^7 + 50275*x^6 + 31415*x^5 - 48995*x^4 - 43179*x^3 + 10583*x^2 + 29682*x + 4061)
 
gp: K = bnfinit(x^20 - 10*x^19 + 27*x^18 + 42*x^17 - 284*x^16 + 28*x^15 + 1386*x^14 - 400*x^13 - 5377*x^12 + 1166*x^11 + 16307*x^10 - 1076*x^9 - 35399*x^8 - 6192*x^7 + 50275*x^6 + 31415*x^5 - 48995*x^4 - 43179*x^3 + 10583*x^2 + 29682*x + 4061, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 27 x^{18} + 42 x^{17} - 284 x^{16} + 28 x^{15} + 1386 x^{14} - 400 x^{13} - 5377 x^{12} + 1166 x^{11} + 16307 x^{10} - 1076 x^{9} - 35399 x^{8} - 6192 x^{7} + 50275 x^{6} + 31415 x^{5} - 48995 x^{4} - 43179 x^{3} + 10583 x^{2} + 29682 x + 4061 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1806627930211353113514833291015625=3^{16}\cdot 5^{10}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.01$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{6}{13} a^{13} - \frac{6}{13} a^{12} - \frac{3}{13} a^{11} + \frac{2}{13} a^{10} - \frac{2}{13} a^{9} - \frac{5}{13} a^{8} - \frac{1}{13} a^{7} + \frac{1}{13} a^{6} + \frac{3}{13} a^{5} - \frac{2}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{15} - \frac{3}{13} a^{13} - \frac{6}{13} a^{12} - \frac{6}{13} a^{11} - \frac{1}{13} a^{10} - \frac{6}{13} a^{9} + \frac{3}{13} a^{8} - \frac{6}{13} a^{7} - \frac{3}{13} a^{6} + \frac{6}{13} a^{5} + \frac{6}{13} a^{4} - \frac{3}{13} a^{3} - \frac{1}{13} a^{2} + \frac{3}{13} a + \frac{5}{13}$, $\frac{1}{39} a^{16} + \frac{1}{39} a^{15} + \frac{1}{39} a^{14} - \frac{11}{39} a^{13} + \frac{1}{13} a^{12} + \frac{7}{39} a^{11} - \frac{4}{13} a^{10} + \frac{2}{39} a^{9} + \frac{1}{13} a^{8} + \frac{1}{3} a^{7} + \frac{7}{39} a^{6} - \frac{2}{39} a^{5} + \frac{8}{39} a^{4} - \frac{2}{39} a^{3} + \frac{2}{39} a^{2} - \frac{3}{13} a + \frac{19}{39}$, $\frac{1}{39} a^{17} + \frac{8}{39} a^{13} + \frac{10}{39} a^{12} - \frac{16}{39} a^{11} - \frac{1}{39} a^{10} + \frac{16}{39} a^{9} - \frac{11}{39} a^{8} - \frac{6}{13} a^{7} + \frac{1}{13} a^{6} + \frac{7}{39} a^{5} + \frac{5}{39} a^{4} + \frac{10}{39} a^{3} - \frac{11}{39} a^{2} + \frac{16}{39} a - \frac{16}{39}$, $\frac{1}{57168576389691} a^{18} - \frac{1}{6352064043299} a^{17} - \frac{184823841409}{57168576389691} a^{16} + \frac{1478590731476}{57168576389691} a^{15} + \frac{514017113401}{57168576389691} a^{14} + \frac{9231706264780}{19056192129897} a^{13} + \frac{5729810344748}{57168576389691} a^{12} + \frac{15871930228048}{57168576389691} a^{11} + \frac{5351185389283}{57168576389691} a^{10} + \frac{25776766438277}{57168576389691} a^{9} - \frac{2443645823048}{6352064043299} a^{8} + \frac{28540723979306}{57168576389691} a^{7} + \frac{669069857439}{6352064043299} a^{6} - \frac{18567945370181}{57168576389691} a^{5} + \frac{24767170180850}{57168576389691} a^{4} + \frac{6951774458255}{19056192129897} a^{3} - \frac{891452719276}{4397582799207} a^{2} + \frac{4069354457555}{57168576389691} a - \frac{2491797656779}{57168576389691}$, $\frac{1}{7093305452703690207} a^{19} + \frac{62029}{7093305452703690207} a^{18} + \frac{59145837964001330}{7093305452703690207} a^{17} - \frac{59506749103003472}{7093305452703690207} a^{16} + \frac{12253668895755686}{2364435150901230069} a^{15} + \frac{40866835657284892}{7093305452703690207} a^{14} + \frac{545551783052527670}{7093305452703690207} a^{13} + \frac{596465268663799562}{2364435150901230069} a^{12} + \frac{237943940758351817}{7093305452703690207} a^{11} + \frac{241322217111369824}{788145050300410023} a^{10} - \frac{229946116413817312}{7093305452703690207} a^{9} + \frac{704646914764082837}{7093305452703690207} a^{8} - \frac{3353877519032483038}{7093305452703690207} a^{7} - \frac{2810930498291184782}{7093305452703690207} a^{6} + \frac{129859222594491323}{788145050300410023} a^{5} + \frac{1612464018779653541}{7093305452703690207} a^{4} + \frac{851414133038096282}{7093305452703690207} a^{3} + \frac{2223802337912157298}{7093305452703690207} a^{2} - \frac{2500827562993504181}{7093305452703690207} a + \frac{1511718079886532527}{7093305452703690207}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $11$
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:  \( 206464661.249 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.2.42504446005228125.1, 10.10.582252685003125.1, 10.2.8500889201045625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.12.10.3$x^{12} - 14527 x^{6} + 78021889$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$