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Properties

Label 20.0.13167097088...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{16}\cdot 29^{15}$
Root discriminant $45.29$
Ramified primes $5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20809, -98180, 286080, -409350, 367525, -26045, -19525, 54080, -50850, 21900, -2406, -2315, 775, -1225, 550, -9, 95, -35, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 35*x^17 + 95*x^16 - 9*x^15 + 550*x^14 - 1225*x^13 + 775*x^12 - 2315*x^11 - 2406*x^10 + 21900*x^9 - 50850*x^8 + 54080*x^7 - 19525*x^6 - 26045*x^5 + 367525*x^4 - 409350*x^3 + 286080*x^2 - 98180*x + 20809)
 
gp: K = bnfinit(x^20 - 35*x^17 + 95*x^16 - 9*x^15 + 550*x^14 - 1225*x^13 + 775*x^12 - 2315*x^11 - 2406*x^10 + 21900*x^9 - 50850*x^8 + 54080*x^7 - 19525*x^6 - 26045*x^5 + 367525*x^4 - 409350*x^3 + 286080*x^2 - 98180*x + 20809, 1)
 

Normalized defining polynomial

\( x^{20} - 35 x^{17} + 95 x^{16} - 9 x^{15} + 550 x^{14} - 1225 x^{13} + 775 x^{12} - 2315 x^{11} - 2406 x^{10} + 21900 x^{9} - 50850 x^{8} + 54080 x^{7} - 19525 x^{6} - 26045 x^{5} + 367525 x^{4} - 409350 x^{3} + 286080 x^{2} - 98180 x + 20809 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1316709708800992498924102783203125=5^{16}\cdot 29^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.29$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{11} + \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{12} + \frac{1}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{18} + \frac{2}{25} a^{17} - \frac{2}{25} a^{16} - \frac{1}{25} a^{15} + \frac{2}{25} a^{13} + \frac{4}{25} a^{12} - \frac{4}{25} a^{11} - \frac{7}{25} a^{10} + \frac{2}{5} a^{9} - \frac{9}{25} a^{8} - \frac{3}{25} a^{7} - \frac{2}{25} a^{6} - \frac{6}{25} a^{5} - \frac{1}{5} a^{4} + \frac{11}{25} a^{3} + \frac{2}{25} a^{2} - \frac{12}{25} a - \frac{6}{25}$, $\frac{1}{1495236939257165170082638611192071850362325224842625} a^{19} + \frac{17219174913407198264549762821584702886748764379996}{1495236939257165170082638611192071850362325224842625} a^{18} - \frac{49296282390390510187237176564415681514598470850759}{1495236939257165170082638611192071850362325224842625} a^{17} + \frac{93028091498683053009772163975173087548951504207151}{1495236939257165170082638611192071850362325224842625} a^{16} + \frac{59903444965128652215103244666572794536181343317816}{1495236939257165170082638611192071850362325224842625} a^{15} + \frac{30518827856081610960704785181567595778109556930977}{1495236939257165170082638611192071850362325224842625} a^{14} - \frac{90294331749651825961600046160032474651035437740694}{213605277036737881440376944456010264337475032120375} a^{13} + \frac{57822435460043073811986536534814295578051230400576}{213605277036737881440376944456010264337475032120375} a^{12} + \frac{747183473124718736168499631949761646704091585177947}{1495236939257165170082638611192071850362325224842625} a^{11} - \frac{300418712329309807012287692431411859326921819058953}{1495236939257165170082638611192071850362325224842625} a^{10} - \frac{469331276360680923528375112748929221948173663242094}{1495236939257165170082638611192071850362325224842625} a^{9} - \frac{462966730418562127221219893030480341995625490037224}{1495236939257165170082638611192071850362325224842625} a^{8} + \frac{5879305538931357829249108924473331957927342945021}{1495236939257165170082638611192071850362325224842625} a^{7} - \frac{597419437283659723666747498058554207899719844312229}{1495236939257165170082638611192071850362325224842625} a^{6} - \frac{50640932911524388110612825165501900843453456347362}{213605277036737881440376944456010264337475032120375} a^{5} + \frac{143624257360256364160178010081362354602348983431466}{1495236939257165170082638611192071850362325224842625} a^{4} + \frac{169308837799341875098601854133518875309917613260661}{1495236939257165170082638611192071850362325224842625} a^{3} - \frac{129021950535991897772163662794424749694515316644019}{1495236939257165170082638611192071850362325224842625} a^{2} + \frac{186163420959090732818050430656359511833692475245556}{1495236939257165170082638611192071850362325224842625} a + \frac{92803211367727990188158316472745522642660881920421}{1495236939257165170082638611192071850362325224842625}$

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:  $9$
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:  \( 318100225.529 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_5.Q_8$ (as 20T105):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_5:D_5.Q_8$
Character table for $C_5:D_5.Q_8$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 10.2.8012167578125.1

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

Sibling fields

Degree 20 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ $20$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
29Data not computed