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Properties

Label 20.4.10237178602...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{15}\cdot 13^{15}$
Root discriminant $39.86$
Ramified primes $2, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20736, -27648, -8064, 25728, -35648, 21312, 27328, -29104, 14928, 808, -12604, 7060, -200, -572, 412, -275, -25, 64, -6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 6*x^18 + 64*x^17 - 25*x^16 - 275*x^15 + 412*x^14 - 572*x^13 - 200*x^12 + 7060*x^11 - 12604*x^10 + 808*x^9 + 14928*x^8 - 29104*x^7 + 27328*x^6 + 21312*x^5 - 35648*x^4 + 25728*x^3 - 8064*x^2 - 27648*x + 20736)
 
gp: K = bnfinit(x^20 - 5*x^19 - 6*x^18 + 64*x^17 - 25*x^16 - 275*x^15 + 412*x^14 - 572*x^13 - 200*x^12 + 7060*x^11 - 12604*x^10 + 808*x^9 + 14928*x^8 - 29104*x^7 + 27328*x^6 + 21312*x^5 - 35648*x^4 + 25728*x^3 - 8064*x^2 - 27648*x + 20736, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 6 x^{18} + 64 x^{17} - 25 x^{16} - 275 x^{15} + 412 x^{14} - 572 x^{13} - 200 x^{12} + 7060 x^{11} - 12604 x^{10} + 808 x^{9} + 14928 x^{8} - 29104 x^{7} + 27328 x^{6} + 21312 x^{5} - 35648 x^{4} + 25728 x^{3} - 8064 x^{2} - 27648 x + 20736 \)

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:  \(102371786028181514000000000000000=2^{16}\cdot 5^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.86$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} + \frac{1}{8} a^{7} - \frac{1}{6} a^{6} - \frac{1}{12} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} + \frac{1}{24} a^{12} - \frac{1}{12} a^{11} + \frac{1}{16} a^{10} + \frac{5}{48} a^{9} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} + \frac{5}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{96} a^{15} - \frac{1}{96} a^{14} - \frac{1}{48} a^{13} - \frac{1}{24} a^{12} - \frac{5}{96} a^{11} + \frac{1}{96} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{7}{24} a^{7} + \frac{1}{24} a^{6} + \frac{5}{24} a^{5} - \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{96} a^{16} - \frac{1}{96} a^{14} - \frac{5}{96} a^{12} + \frac{1}{24} a^{11} - \frac{1}{96} a^{10} - \frac{7}{48} a^{9} - \frac{5}{24} a^{8} + \frac{5}{12} a^{7} + \frac{1}{3} a^{6} + \frac{1}{24} a^{5} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{192} a^{17} - \frac{1}{192} a^{16} - \frac{1}{96} a^{14} + \frac{1}{64} a^{13} + \frac{1}{192} a^{12} + \frac{7}{96} a^{11} + \frac{7}{96} a^{10} + \frac{1}{8} a^{9} + \frac{5}{48} a^{8} + \frac{23}{48} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{1152} a^{18} + \frac{1}{1152} a^{17} + \frac{1}{288} a^{15} + \frac{11}{1152} a^{14} - \frac{17}{1152} a^{13} - \frac{25}{576} a^{12} - \frac{11}{288} a^{11} + \frac{1}{288} a^{10} + \frac{49}{288} a^{9} - \frac{43}{288} a^{8} - \frac{5}{72} a^{7} - \frac{1}{3} a^{6} - \frac{17}{36} a^{5} + \frac{2}{9} a^{4} + \frac{5}{12} a^{3} + \frac{7}{18} a^{2}$, $\frac{1}{20278495558828447130556991337894784} a^{19} + \frac{2039838204131713151128344979297}{5069623889707111782639247834473696} a^{18} + \frac{7079714828014359843369162471067}{6759498519609482376852330445964928} a^{17} - \frac{9883888239673654886406410217787}{10139247779414223565278495668947392} a^{16} + \frac{56469819220576713437272538346299}{20278495558828447130556991337894784} a^{15} + \frac{13280984845015199829144938936677}{5069623889707111782639247834473696} a^{14} - \frac{398732949219332828631383952674699}{20278495558828447130556991337894784} a^{13} + \frac{133653031910840656792961217372767}{2534811944853555891319623917236848} a^{12} - \frac{620691904315163613262512681433}{158425746553347243207476494827303} a^{11} - \frac{279316353977510379269620112947061}{5069623889707111782639247834473696} a^{10} - \frac{68565573196410859937096126749699}{316851493106694486414952989654606} a^{9} + \frac{473803389132920412343847169007369}{5069623889707111782639247834473696} a^{8} - \frac{86721756415069048592955635166761}{844937314951185297106541305745616} a^{7} - \frac{473018761428484151254349820241477}{1267405972426777945659811958618424} a^{6} + \frac{459003282841721120861535838996813}{1267405972426777945659811958618424} a^{5} - \frac{10434410064023378374236076352615}{23470480970866258252959480715156} a^{4} + \frac{268865416494370877195993712961907}{633702986213388972829905979309212} a^{3} + \frac{23159042094389962721889246290129}{105617164368898162138317663218202} a^{2} - \frac{6596886514970731535313282746864}{17602860728149693689719610536367} a - \frac{1382279781557240600920215031758}{5867620242716564563239870178789}$

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:  \( 823000471.8315867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{65}) \), 4.4.274625.1, 5.1.338000.1, 10.2.7425860000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed
13Data not computed