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Properties

Label 20.0.45040122374...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 31^{16}$
Root discriminant $170.88$
Ramified primes $2, 3, 5, 31$
Class number $21169600$ (GRH)
Class group $[2, 2, 2, 10, 264620]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15639805975, 2884250250, 9119322015, 778579040, 2326214329, 50123358, 370498441, -7608160, 41176558, -2250240, 3542958, -273646, 248508, -30574, 13376, -1908, 759, -74, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 - 74*x^17 + 759*x^16 - 1908*x^15 + 13376*x^14 - 30574*x^13 + 248508*x^12 - 273646*x^11 + 3542958*x^10 - 2250240*x^9 + 41176558*x^8 - 7608160*x^7 + 370498441*x^6 + 50123358*x^5 + 2326214329*x^4 + 778579040*x^3 + 9119322015*x^2 + 2884250250*x + 15639805975)
 
gp: K = bnfinit(x^20 - 6*x^19 + 15*x^18 - 74*x^17 + 759*x^16 - 1908*x^15 + 13376*x^14 - 30574*x^13 + 248508*x^12 - 273646*x^11 + 3542958*x^10 - 2250240*x^9 + 41176558*x^8 - 7608160*x^7 + 370498441*x^6 + 50123358*x^5 + 2326214329*x^4 + 778579040*x^3 + 9119322015*x^2 + 2884250250*x + 15639805975, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 15 x^{18} - 74 x^{17} + 759 x^{16} - 1908 x^{15} + 13376 x^{14} - 30574 x^{13} + 248508 x^{12} - 273646 x^{11} + 3542958 x^{10} - 2250240 x^{9} + 41176558 x^{8} - 7608160 x^{7} + 370498441 x^{6} + 50123358 x^{5} + 2326214329 x^{4} + 778579040 x^{3} + 9119322015 x^{2} + 2884250250 x + 15639805975 \)

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:  \(450401223741795652272735907416637440000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.88$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(1349,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(3101,·)$, $\chi_{3720}(3629,·)$, $\chi_{3720}(529,·)$, $\chi_{3720}(2329,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2141,·)$, $\chi_{3720}(101,·)$, $\chi_{3720}(2209,·)$, $\chi_{3720}(869,·)$, $\chi_{3720}(1489,·)$, $\chi_{3720}(1709,·)$, $\chi_{3720}(221,·)$, $\chi_{3720}(1969,·)$, $\chi_{3720}(1589,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(3581,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} + \frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{3}{25} a^{9} + \frac{3}{25} a^{8} - \frac{3}{25} a^{7} + \frac{6}{25} a^{6} + \frac{4}{25} a^{5} + \frac{1}{25} a^{4} - \frac{1}{25} a^{3} - \frac{1}{5} a$, $\frac{1}{725} a^{15} + \frac{2}{145} a^{14} - \frac{12}{145} a^{13} - \frac{2}{145} a^{12} + \frac{27}{725} a^{11} + \frac{6}{145} a^{10} - \frac{11}{145} a^{9} - \frac{66}{145} a^{8} - \frac{82}{725} a^{7} - \frac{48}{145} a^{6} - \frac{61}{145} a^{5} + \frac{18}{145} a^{4} - \frac{96}{725} a^{3} - \frac{1}{29} a^{2} + \frac{24}{145} a - \frac{11}{29}$, $\frac{1}{725} a^{16} + \frac{14}{725} a^{14} - \frac{19}{725} a^{13} + \frac{11}{725} a^{12} + \frac{21}{725} a^{11} + \frac{22}{725} a^{10} - \frac{157}{725} a^{9} - \frac{6}{145} a^{8} - \frac{3}{25} a^{7} + \frac{239}{725} a^{6} - \frac{224}{725} a^{5} + \frac{338}{725} a^{4} - \frac{254}{725} a^{3} - \frac{71}{145} a^{2} - \frac{34}{145} a - \frac{6}{29}$, $\frac{1}{725} a^{17} - \frac{14}{725} a^{14} - \frac{19}{725} a^{13} + \frac{16}{725} a^{12} - \frac{66}{725} a^{11} + \frac{3}{725} a^{10} + \frac{61}{145} a^{9} - \frac{252}{725} a^{8} + \frac{82}{725} a^{7} + \frac{91}{725} a^{6} + \frac{113}{725} a^{5} - \frac{209}{725} a^{4} - \frac{171}{725} a^{3} + \frac{13}{29} a^{2} + \frac{69}{145} a + \frac{9}{29}$, $\frac{1}{68476656943076129786875} a^{18} - \frac{13947851332807082127}{68476656943076129786875} a^{17} - \frac{14009959155060119852}{68476656943076129786875} a^{16} + \frac{45478096686779136086}{68476656943076129786875} a^{15} + \frac{1250359231537372271121}{68476656943076129786875} a^{14} + \frac{6203014814946687557452}{68476656943076129786875} a^{13} + \frac{339105527075250762354}{13695331388615225957375} a^{12} - \frac{5989087214351244263587}{68476656943076129786875} a^{11} + \frac{1157870110253502982611}{13695331388615225957375} a^{10} - \frac{1226344821600829186468}{68476656943076129786875} a^{9} + \frac{20322983390348456276541}{68476656943076129786875} a^{8} - \frac{8467296228779796147214}{68476656943076129786875} a^{7} + \frac{6784093535665372534133}{68476656943076129786875} a^{6} - \frac{24184579676485011986002}{68476656943076129786875} a^{5} + \frac{19817027062168044666961}{68476656943076129786875} a^{4} - \frac{2057230288115167417216}{13695331388615225957375} a^{3} + \frac{5447861813971434462227}{13695331388615225957375} a^{2} + \frac{461034192506754475446}{2739066277723045191475} a + \frac{5389907267305791574}{94450561300794661775}$, $\frac{1}{3419624771582171595970592373348326706457616350406875} a^{19} + \frac{4113750550169167700043419581}{683924954316434319194118474669665341291523270081375} a^{18} + \frac{830293396411217040540688256759233485693696002334}{3419624771582171595970592373348326706457616350406875} a^{17} - \frac{1743047660536625196421457429636506820599407989678}{3419624771582171595970592373348326706457616350406875} a^{16} - \frac{762522543380731982431654328783596329565507461052}{3419624771582171595970592373348326706457616350406875} a^{15} + \frac{32509972374365684895039708769868328304010814304}{18892954539128019867240841841703462466616664919375} a^{14} + \frac{98898346966251211705359543895055090599378518504534}{3419624771582171595970592373348326706457616350406875} a^{13} + \frac{225534863475819131018825539291131569751931480892253}{3419624771582171595970592373348326706457616350406875} a^{12} - \frac{331677347014739226690211964723328203230968700361029}{3419624771582171595970592373348326706457616350406875} a^{11} + \frac{240745337013793201098666288378871072142514559206642}{3419624771582171595970592373348326706457616350406875} a^{10} + \frac{2497495727883465291222429136343422101147721602968}{683924954316434319194118474669665341291523270081375} a^{9} + \frac{867648383537238363552348512215267546804838153167448}{3419624771582171595970592373348326706457616350406875} a^{8} - \frac{235739001522759032006138166771596211004577620250168}{683924954316434319194118474669665341291523270081375} a^{7} + \frac{1362479925057887610409242637130607431168372154150704}{3419624771582171595970592373348326706457616350406875} a^{6} - \frac{1307269111616200562783242120427240824086523155357053}{3419624771582171595970592373348326706457616350406875} a^{5} + \frac{280004288977795601916299617849775216603663579119247}{3419624771582171595970592373348326706457616350406875} a^{4} + \frac{48153299239887570457894613720078384051787949761587}{136784990863286863838823694933933068258304654016275} a^{3} - \frac{127162461281291536613053724314346427297827349458406}{683924954316434319194118474669665341291523270081375} a^{2} - \frac{44767636607395435082777289874068768653476412106287}{136784990863286863838823694933933068258304654016275} a + \frac{35252017563771714422193785355380815529098747598147}{136784990863286863838823694933933068258304654016275}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{264620}$, which has order $21169600$ (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:  \( 24173706.832424585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), 5.5.923521.1, 10.10.2665284492003125.1, 10.0.6791250644112605184.1, 10.0.21222658262851891200000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$