↑

Properties

Label 20.0.32917742720...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{18}\cdot 29^{15}$
Root discriminant $53.20$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288749, 547813, -1262, -352434, 463258, 868195, 129604, -212999, 82491, 75647, -44264, 5065, 12038, -6638, 507, 1072, -407, 1, 38, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 38*x^18 + x^17 - 407*x^16 + 1072*x^15 + 507*x^14 - 6638*x^13 + 12038*x^12 + 5065*x^11 - 44264*x^10 + 75647*x^9 + 82491*x^8 - 212999*x^7 + 129604*x^6 + 868195*x^5 + 463258*x^4 - 352434*x^3 - 1262*x^2 + 547813*x + 288749)
 
gp: K = bnfinit(x^20 - 10*x^19 + 38*x^18 + x^17 - 407*x^16 + 1072*x^15 + 507*x^14 - 6638*x^13 + 12038*x^12 + 5065*x^11 - 44264*x^10 + 75647*x^9 + 82491*x^8 - 212999*x^7 + 129604*x^6 + 868195*x^5 + 463258*x^4 - 352434*x^3 - 1262*x^2 + 547813*x + 288749, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 38 x^{18} + x^{17} - 407 x^{16} + 1072 x^{15} + 507 x^{14} - 6638 x^{13} + 12038 x^{12} + 5065 x^{11} - 44264 x^{10} + 75647 x^{9} + 82491 x^{8} - 212999 x^{7} + 129604 x^{6} + 868195 x^{5} + 463258 x^{4} - 352434 x^{3} - 1262 x^{2} + 547813 x + 288749 \)

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:  \(32917742720024812473102569580078125=5^{18}\cdot 29^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{65} a^{16} - \frac{6}{65} a^{15} + \frac{8}{65} a^{14} + \frac{8}{65} a^{13} + \frac{7}{65} a^{12} + \frac{11}{65} a^{11} + \frac{1}{5} a^{10} + \frac{11}{65} a^{9} - \frac{23}{65} a^{8} + \frac{21}{65} a^{7} - \frac{32}{65} a^{6} + \frac{7}{65} a^{5} + \frac{21}{65} a^{4} + \frac{29}{65} a^{3} + \frac{1}{13} a^{2} + \frac{17}{65} a + \frac{32}{65}$, $\frac{1}{59995} a^{17} - \frac{4}{11999} a^{16} + \frac{1100}{11999} a^{15} + \frac{62}{4615} a^{14} + \frac{21774}{59995} a^{13} + \frac{22156}{59995} a^{12} + \frac{353}{59995} a^{11} - \frac{13509}{59995} a^{10} - \frac{4831}{59995} a^{9} - \frac{23278}{59995} a^{8} + \frac{7474}{59995} a^{7} + \frac{356}{4615} a^{6} - \frac{4451}{11999} a^{5} + \frac{22758}{59995} a^{4} + \frac{2564}{11999} a^{3} + \frac{3892}{11999} a^{2} - \frac{27298}{59995} a + \frac{16556}{59995}$, $\frac{1}{3899675} a^{18} - \frac{21}{3899675} a^{17} + \frac{1104}{779935} a^{16} + \frac{14691}{155987} a^{15} + \frac{140958}{3899675} a^{14} - \frac{383586}{3899675} a^{13} + \frac{1310086}{3899675} a^{12} + \frac{65680}{155987} a^{11} + \frac{1496554}{3899675} a^{10} - \frac{31878}{299975} a^{9} - \frac{473206}{3899675} a^{8} - \frac{302821}{3899675} a^{7} + \frac{1844961}{3899675} a^{6} + \frac{33014}{3899675} a^{5} + \frac{841991}{3899675} a^{4} + \frac{1110548}{3899675} a^{3} + \frac{1525111}{3899675} a^{2} + \frac{415823}{3899675} a + \frac{1759296}{3899675}$, $\frac{1}{422474563361096707631697530216599915021219025} a^{19} + \frac{835338797574516312270063589644029253}{84494912672219341526339506043319983004243805} a^{18} + \frac{2042336730957140094810601792217418389369}{422474563361096707631697530216599915021219025} a^{17} + \frac{317944664228834232430971132808377504259866}{84494912672219341526339506043319983004243805} a^{16} - \frac{451763850061833153349965058752397495048444}{32498043335468977510130579247430762693939925} a^{15} + \frac{48090932374415185816464895650222666990448762}{422474563361096707631697530216599915021219025} a^{14} - \frac{34726122650889392332810928245512629363579179}{84494912672219341526339506043319983004243805} a^{13} + \frac{158012423403896038696801490712324979851303851}{422474563361096707631697530216599915021219025} a^{12} - \frac{25918731280679476727899060943961146874532256}{422474563361096707631697530216599915021219025} a^{11} + \frac{32352270194480618292890582341991581967337007}{84494912672219341526339506043319983004243805} a^{10} + \frac{24383694843168324621326028381031516208578216}{84494912672219341526339506043319983004243805} a^{9} - \frac{92141028474028766602163054100112619376869767}{422474563361096707631697530216599915021219025} a^{8} - \frac{1380947108966187084494745477052887476777246}{6499608667093795502026115849486152538787985} a^{7} + \frac{17923143799514212937421247176977294011610182}{84494912672219341526339506043319983004243805} a^{6} + \frac{1850420625326213533306120229169475791984779}{6499608667093795502026115849486152538787985} a^{5} + \frac{178996759290140612733014339907105941623193429}{422474563361096707631697530216599915021219025} a^{4} + \frac{118117554186970563200245635647620630704225664}{422474563361096707631697530216599915021219025} a^{3} + \frac{114587039368168602020075715592873775774021774}{422474563361096707631697530216599915021219025} a^{2} + \frac{99998910114494049928374502204850743198497884}{422474563361096707631697530216599915021219025} a - \frac{121169415433053581291378299104698851590484929}{422474563361096707631697530216599915021219025}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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

\(\Q(\sqrt{29}) \), 4.0.609725.2, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ $20$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.10.11.8$x^{10} + 20 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
29Data not computed