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Properties

Label 20.4.19697998226...5625.2
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 61^{6}\cdot 397^{6}$
Root discriminant $46.21$
Ramified primes $5, 61, 397$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2727775, 3532675, 5045800, 1073140, 5752551, -309932, 1556367, 239947, 60753, 197277, -52647, 48731, -9767, 5543, -239, 126, 157, -38, 23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775)
 
gp: K = bnfinit(x^20 - 4*x^19 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 23 x^{18} - 38 x^{17} + 157 x^{16} + 126 x^{15} - 239 x^{14} + 5543 x^{13} - 9767 x^{12} + 48731 x^{11} - 52647 x^{10} + 197277 x^{9} + 60753 x^{8} + 239947 x^{7} + 1556367 x^{6} - 309932 x^{5} + 5752551 x^{4} + 1073140 x^{3} + 5045800 x^{2} + 3532675 x - 2727775 \)

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:  \(1969799822613689762148183291015625=5^{10}\cdot 61^{6}\cdot 397^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.21$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{3}{13} a^{15} - \frac{6}{13} a^{14} - \frac{2}{13} a^{13} - \frac{5}{13} a^{12} + \frac{3}{13} a^{11} - \frac{2}{13} a^{10} + \frac{3}{13} a^{9} - \frac{4}{13} a^{7} - \frac{2}{13} a^{6} + \frac{3}{13} a^{5} + \frac{6}{13} a^{4} + \frac{5}{13} a^{3} - \frac{1}{13} a^{2} + \frac{4}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{17} - \frac{2}{13} a^{15} + \frac{3}{13} a^{14} + \frac{1}{13} a^{13} + \frac{5}{13} a^{12} + \frac{2}{13} a^{11} - \frac{4}{13} a^{10} + \frac{4}{13} a^{9} - \frac{4}{13} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} - \frac{3}{13} a^{5} - \frac{3}{13} a^{3} - \frac{6}{13} a^{2} + \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{2023757255} a^{18} + \frac{34688361}{2023757255} a^{17} + \frac{4622583}{2023757255} a^{16} - \frac{875663403}{2023757255} a^{15} - \frac{952657138}{2023757255} a^{14} - \frac{57288609}{2023757255} a^{13} - \frac{984295349}{2023757255} a^{12} + \frac{64292786}{155673635} a^{11} + \frac{242075633}{2023757255} a^{10} + \frac{546141946}{2023757255} a^{9} + \frac{611856803}{2023757255} a^{8} - \frac{694960663}{2023757255} a^{7} + \frac{77571781}{155673635} a^{6} + \frac{2873229}{155673635} a^{5} + \frac{900648717}{2023757255} a^{4} - \frac{424063127}{2023757255} a^{3} + \frac{12439897}{155673635} a^{2} - \frac{87511968}{404751451} a + \frac{13138271}{404751451}$, $\frac{1}{14714826784147416423760653296961811890967066992350836855} a^{19} - \frac{115262748582076058677816426442371493422280717}{1131909752626724340289281022843216299305158999411602835} a^{18} - \frac{7499398341869206781238055367598025674691004606981464}{14714826784147416423760653296961811890967066992350836855} a^{17} + \frac{297447916727937862763591966583505317381711318514737416}{14714826784147416423760653296961811890967066992350836855} a^{16} + \frac{4640986778210850766421978698013137317082396029060431083}{14714826784147416423760653296961811890967066992350836855} a^{15} + \frac{540875189551073232069181603850816377067822851966716497}{14714826784147416423760653296961811890967066992350836855} a^{14} + \frac{1683880241195849454561607097165668902881898495334317039}{14714826784147416423760653296961811890967066992350836855} a^{13} + \frac{4869181410218843887337549585006701222374600920570841031}{14714826784147416423760653296961811890967066992350836855} a^{12} - \frac{6867493399615245219537766950379305322469992952133811563}{14714826784147416423760653296961811890967066992350836855} a^{11} + \frac{1070150839556452567963386332726631277327112598920801415}{2942965356829483284752130659392362378193413398470167371} a^{10} + \frac{3440202939579623804934410260893630798850972351690880171}{14714826784147416423760653296961811890967066992350836855} a^{9} + \frac{6495422780820441537294637460054031531187501598339489486}{14714826784147416423760653296961811890967066992350836855} a^{8} + \frac{5012220595751120901879782875755367248909321167057820889}{14714826784147416423760653296961811890967066992350836855} a^{7} - \frac{134636210114075083969066472248591532834926138900966498}{1131909752626724340289281022843216299305158999411602835} a^{6} - \frac{4313720780998507984575385378016174982884488383160607312}{14714826784147416423760653296961811890967066992350836855} a^{5} - \frac{3663685761802482589042162153497405894674388787885274516}{14714826784147416423760653296961811890967066992350836855} a^{4} - \frac{388971699864580116246593784501563987864894263813484291}{2942965356829483284752130659392362378193413398470167371} a^{3} - \frac{1732596499164686651873015904919849901430065559703784107}{14714826784147416423760653296961811890967066992350836855} a^{2} + \frac{1368346130573733692814369039142977729483719469107711205}{2942965356829483284752130659392362378193413398470167371} a + \frac{20468530078717797153133786252207718870144307144434895}{2942965356829483284752130659392362378193413398470167371}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 390465777.545 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1, 10.2.71011883131565.1, 10.2.8876485391445625.1

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 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
5Data not computed
61Data not computed
397Data not computed