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Properties

Label 20.20.4841421802...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{18}\cdot 5^{12}\cdot 13^{15}$
Root discriminant $48.33$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (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![26139, 116496, 65157, -310989, -292176, 385218, 388827, -298191, -269151, 159198, 109138, -59264, -26247, 14856, 3459, -2352, -171, 210, -9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 9*x^18 + 210*x^17 - 171*x^16 - 2352*x^15 + 3459*x^14 + 14856*x^13 - 26247*x^12 - 59264*x^11 + 109138*x^10 + 159198*x^9 - 269151*x^8 - 298191*x^7 + 388827*x^6 + 385218*x^5 - 292176*x^4 - 310989*x^3 + 65157*x^2 + 116496*x + 26139)
 
gp: K = bnfinit(x^20 - 8*x^19 - 9*x^18 + 210*x^17 - 171*x^16 - 2352*x^15 + 3459*x^14 + 14856*x^13 - 26247*x^12 - 59264*x^11 + 109138*x^10 + 159198*x^9 - 269151*x^8 - 298191*x^7 + 388827*x^6 + 385218*x^5 - 292176*x^4 - 310989*x^3 + 65157*x^2 + 116496*x + 26139, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 9 x^{18} + 210 x^{17} - 171 x^{16} - 2352 x^{15} + 3459 x^{14} + 14856 x^{13} - 26247 x^{12} - 59264 x^{11} + 109138 x^{10} + 159198 x^{9} - 269151 x^{8} - 298191 x^{7} + 388827 x^{6} + 385218 x^{5} - 292176 x^{4} - 310989 x^{3} + 65157 x^{2} + 116496 x + 26139 \)

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:  \(4841421802104669181474651611328125=3^{18}\cdot 5^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.33$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{14} + \frac{1}{3} a^{11}$, $\frac{1}{237} a^{18} - \frac{1}{237} a^{17} + \frac{2}{79} a^{16} - \frac{7}{237} a^{15} + \frac{70}{237} a^{14} + \frac{39}{79} a^{13} - \frac{107}{237} a^{12} - \frac{67}{237} a^{11} + \frac{27}{79} a^{10} + \frac{9}{79} a^{9} + \frac{38}{79} a^{8} - \frac{32}{79} a^{7} + \frac{7}{79} a^{6} - \frac{38}{79} a^{5} + \frac{16}{79} a^{4} + \frac{18}{79} a^{3} + \frac{19}{79} a^{2} - \frac{39}{79} a - \frac{19}{79}$, $\frac{1}{4032795994825263363} a^{19} - \frac{1200306562204892}{1344265331608421121} a^{18} - \frac{54934566489341407}{1344265331608421121} a^{17} + \frac{18151430855062079}{1344265331608421121} a^{16} + \frac{118476959342061703}{1344265331608421121} a^{15} + \frac{536725726068754313}{1344265331608421121} a^{14} + \frac{514152799990894633}{1344265331608421121} a^{13} - \frac{509637688711770487}{1344265331608421121} a^{12} + \frac{314817036997000126}{1344265331608421121} a^{11} - \frac{1597080898363271891}{4032795994825263363} a^{10} - \frac{396125463963723574}{1344265331608421121} a^{9} - \frac{47368563447132554}{448088443869473707} a^{8} - \frac{4181370144868444}{58446318765583527} a^{7} + \frac{281970700416238138}{1344265331608421121} a^{6} - \frac{180772060874307562}{1344265331608421121} a^{5} - \frac{562608520400505446}{1344265331608421121} a^{4} - \frac{291504872473259860}{1344265331608421121} a^{3} + \frac{167429265701242828}{448088443869473707} a^{2} - \frac{324788260368927337}{1344265331608421121} a + \frac{17991523925434270}{1344265331608421121}$

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:  \( 59031185556.1 \) (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

\(\Q(\sqrt{13}) \), 4.4.19773.1, 5.5.22244625.1, 10.10.6432703438078125.1

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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed
13Data not computed