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Properties

Label 20.20.1348081107...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{14}\cdot 5^{18}\cdot 43^{14}$
Root discriminant $127.79$
Ramified primes $3, 5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2:Q_8$ (as 20T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6117451, 25430025, 16874650, -48627275, -49287350, 45162920, 44354270, -27345215, -18164180, 10037010, 3803273, -2157325, -405450, 269325, 17675, -18970, 255, 690, -45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 45*x^18 + 690*x^17 + 255*x^16 - 18970*x^15 + 17675*x^14 + 269325*x^13 - 405450*x^12 - 2157325*x^11 + 3803273*x^10 + 10037010*x^9 - 18164180*x^8 - 27345215*x^7 + 44354270*x^6 + 45162920*x^5 - 49287350*x^4 - 48627275*x^3 + 16874650*x^2 + 25430025*x + 6117451)
 
gp: K = bnfinit(x^20 - 10*x^19 - 45*x^18 + 690*x^17 + 255*x^16 - 18970*x^15 + 17675*x^14 + 269325*x^13 - 405450*x^12 - 2157325*x^11 + 3803273*x^10 + 10037010*x^9 - 18164180*x^8 - 27345215*x^7 + 44354270*x^6 + 45162920*x^5 - 49287350*x^4 - 48627275*x^3 + 16874650*x^2 + 25430025*x + 6117451, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 45 x^{18} + 690 x^{17} + 255 x^{16} - 18970 x^{15} + 17675 x^{14} + 269325 x^{13} - 405450 x^{12} - 2157325 x^{11} + 3803273 x^{10} + 10037010 x^{9} - 18164180 x^{8} - 27345215 x^{7} + 44354270 x^{6} + 45162920 x^{5} - 49287350 x^{4} - 48627275 x^{3} + 16874650 x^{2} + 25430025 x + 6117451 \)

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:  \(1348081107066867047182181694339752197265625=3^{14}\cdot 5^{18}\cdot 43^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.79$
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:  $3, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{2}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} + \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{100} a^{12} - \frac{1}{25} a^{11} - \frac{3}{100} a^{10} + \frac{1}{50} a^{9} + \frac{2}{25} a^{8} + \frac{1}{50} a^{7} - \frac{21}{50} a^{6} + \frac{17}{100} a^{5} - \frac{27}{100} a^{4} - \frac{3}{100} a^{3} - \frac{9}{50} a^{2} + \frac{11}{100} a - \frac{19}{100}$, $\frac{1}{100} a^{13} + \frac{1}{100} a^{11} + \frac{3}{50} a^{9} + \frac{1}{25} a^{8} + \frac{3}{50} a^{7} - \frac{21}{100} a^{6} + \frac{11}{100} a^{5} + \frac{49}{100} a^{4} - \frac{1}{2} a^{3} + \frac{49}{100} a^{2} + \frac{1}{4} a - \frac{3}{50}$, $\frac{1}{200} a^{14} - \frac{1}{200} a^{13} - \frac{7}{200} a^{11} + \frac{9}{200} a^{10} + \frac{2}{25} a^{9} - \frac{2}{25} a^{8} - \frac{19}{200} a^{7} - \frac{23}{100} a^{6} - \frac{89}{200} a^{5} + \frac{1}{25} a^{4} + \frac{3}{50} a^{3} + \frac{21}{50} a^{2} + \frac{11}{25} a - \frac{13}{40}$, $\frac{1}{200} a^{15} - \frac{1}{200} a^{13} - \frac{1}{200} a^{12} - \frac{1}{100} a^{11} + \frac{7}{200} a^{10} + \frac{3}{50} a^{9} - \frac{7}{200} a^{8} + \frac{7}{200} a^{7} + \frac{93}{200} a^{6} - \frac{79}{200} a^{5} + \frac{29}{100} a^{4} + \frac{49}{100} a^{3} + \frac{1}{50} a^{2} - \frac{11}{200} a - \frac{79}{200}$, $\frac{1}{6000} a^{16} + \frac{1}{1000} a^{15} + \frac{1}{600} a^{13} + \frac{7}{3000} a^{11} - \frac{1}{6000} a^{10} - \frac{11}{240} a^{9} - \frac{13}{400} a^{8} + \frac{1}{30} a^{7} + \frac{253}{2000} a^{6} - \frac{691}{6000} a^{5} + \frac{1}{3} a^{4} - \frac{13}{75} a^{3} - \frac{109}{240} a^{2} + \frac{1201}{6000} a - \frac{2039}{6000}$, $\frac{1}{6000} a^{17} - \frac{1}{1000} a^{15} + \frac{1}{600} a^{14} - \frac{1}{200} a^{13} - \frac{1}{375} a^{12} - \frac{17}{1200} a^{11} - \frac{59}{6000} a^{10} - \frac{3}{80} a^{9} + \frac{1}{30} a^{8} + \frac{43}{2000} a^{7} + \frac{457}{1200} a^{6} - \frac{91}{1500} a^{5} + \frac{1}{150} a^{4} - \frac{269}{1200} a^{3} + \frac{2611}{6000} a^{2} - \frac{83}{240} a + \frac{48}{125}$, $\frac{1}{60000} a^{18} + \frac{1}{20000} a^{17} + \frac{1}{20000} a^{16} - \frac{11}{15000} a^{15} - \frac{3}{2000} a^{14} - \frac{17}{7500} a^{13} - \frac{103}{60000} a^{12} - \frac{1219}{30000} a^{11} - \frac{157}{20000} a^{10} + \frac{221}{6000} a^{9} + \frac{97}{5000} a^{8} - \frac{371}{7500} a^{7} + \frac{589}{7500} a^{6} - \frac{1871}{60000} a^{5} + \frac{4831}{12000} a^{4} - \frac{3163}{7500} a^{3} - \frac{15707}{60000} a^{2} - \frac{667}{10000} a - \frac{7113}{20000}$, $\frac{1}{2397086056599693812700000} a^{19} - \frac{2436672134074020323}{1198543028299846906350000} a^{18} - \frac{11761094488660433897}{1198543028299846906350000} a^{17} - \frac{114177273587040827491}{2397086056599693812700000} a^{16} - \frac{1137527374055225624117}{1198543028299846906350000} a^{15} + \frac{396340220750999542187}{1198543028299846906350000} a^{14} + \frac{6374232108744300778261}{2397086056599693812700000} a^{13} + \frac{1392935951007131256203}{799028685533231270900000} a^{12} - \frac{71680006725845466374359}{2397086056599693812700000} a^{11} + \frac{10008984592174015438313}{799028685533231270900000} a^{10} - \frac{28007911456174835133797}{299635757074961726587500} a^{9} - \frac{4525140727680908288123}{49939292845826954431250} a^{8} + \frac{57826214290446732077647}{1198543028299846906350000} a^{7} - \frac{276133567383007330300903}{799028685533231270900000} a^{6} + \frac{283648129967639080351067}{1198543028299846906350000} a^{5} - \frac{75485530530391260877833}{799028685533231270900000} a^{4} - \frac{1089179725726314733919761}{2397086056599693812700000} a^{3} - \frac{231768390535645502105203}{799028685533231270900000} a^{2} - \frac{226581748392336989380797}{799028685533231270900000} a - \frac{86419601908264825405289}{2397086056599693812700000}$

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:  \( 2777762864880000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:Q_8$ (as 20T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 8 conjugacy class representatives for $C_5^2:Q_8$
Character table for $C_5^2:Q_8$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{129}) \), \(\Q(\sqrt{645}) \), \(\Q(\sqrt{5}, \sqrt{129})\), 10.10.232213790035550390625.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.8.6.2$x^{8} + 215 x^{4} + 16641$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
43.8.6.2$x^{8} + 215 x^{4} + 16641$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$