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Properties

Label 20.0.11784448085...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}\cdot 41^{18}$
Root discriminant $636.16$
Ramified primes $2, 5, 41$
Class number $1371710288$ (GRH)
Class group $[2, 2, 342927572]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![804017088900000, 0, 1647738725400000, 0, 893242030590000, 0, 115706894580000, 0, 6734386899000, 0, 212532192000, 0, 3900904000, 0, 42172600, 0, 259530, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 259530*x^16 + 42172600*x^14 + 3900904000*x^12 + 212532192000*x^10 + 6734386899000*x^8 + 115706894580000*x^6 + 893242030590000*x^4 + 1647738725400000*x^2 + 804017088900000)
 
gp: K = bnfinit(x^20 + 820*x^18 + 259530*x^16 + 42172600*x^14 + 3900904000*x^12 + 212532192000*x^10 + 6734386899000*x^8 + 115706894580000*x^6 + 893242030590000*x^4 + 1647738725400000*x^2 + 804017088900000, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 259530 x^{16} + 42172600 x^{14} + 3900904000 x^{12} + 212532192000 x^{10} + 6734386899000 x^{8} + 115706894580000 x^{6} + 893242030590000 x^{4} + 1647738725400000 x^{2} + 804017088900000 \)

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:  \(117844480851402406822671795818334357815296000000000000000=2^{55}\cdot 5^{15}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $636.16$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3280=2^{4}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3280}(1,·)$, $\chi_{3280}(961,·)$, $\chi_{3280}(1289,·)$, $\chi_{3280}(1281,·)$, $\chi_{3280}(1357,·)$, $\chi_{3280}(1041,·)$, $\chi_{3280}(1173,·)$, $\chi_{3280}(933,·)$, $\chi_{3280}(1369,·)$, $\chi_{3280}(329,·)$, $\chi_{3280}(2213,·)$, $\chi_{3280}(1253,·)$, $\chi_{3280}(1917,·)$, $\chi_{3280}(2157,·)$, $\chi_{3280}(2237,·)$, $\chi_{3280}(1841,·)$, $\chi_{3280}(373,·)$, $\chi_{3280}(1609,·)$, $\chi_{3280}(2169,·)$, $\chi_{3280}(3197,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{30} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{30} a^{5} - \frac{1}{3} a$, $\frac{1}{90} a^{6} - \frac{1}{90} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{90} a^{7} - \frac{1}{90} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{900} a^{8} + \frac{1}{9} a^{2}$, $\frac{1}{8100} a^{9} + \frac{1}{270} a^{7} + \frac{1}{270} a^{5} - \frac{8}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{332100} a^{10} - \frac{1}{2700} a^{8} - \frac{1}{270} a^{6} + \frac{11}{810} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{332100} a^{11} - \frac{1}{270} a^{7} + \frac{1}{405} a^{5} - \frac{2}{27} a^{3}$, $\frac{1}{9963000} a^{12} + \frac{1}{996300} a^{10} - \frac{1}{4050} a^{8} - \frac{1}{2430} a^{6} - \frac{13}{2430} a^{4} - \frac{7}{27} a^{2}$, $\frac{1}{29889000} a^{13} + \frac{1}{2988900} a^{11} + \frac{1}{24300} a^{9} - \frac{14}{3645} a^{7} - \frac{47}{3645} a^{5} + \frac{2}{27} a^{3} - \frac{2}{9} a$, $\frac{1}{269001000} a^{14} + \frac{1}{26900100} a^{12} + \frac{11}{8966700} a^{10} - \frac{43}{164025} a^{8} - \frac{31}{6561} a^{6} - \frac{52}{3645} a^{4} + \frac{26}{81} a^{2}$, $\frac{1}{807003000} a^{15} + \frac{1}{80700300} a^{13} + \frac{19}{13450050} a^{11} + \frac{71}{1968300} a^{9} + \frac{181}{39366} a^{7} - \frac{43}{10935} a^{5} + \frac{5}{243} a^{3} + \frac{2}{9} a$, $\frac{1}{15906029130000} a^{16} + \frac{869}{795301456500} a^{14} + \frac{16439}{530200971000} a^{12} + \frac{36011}{79530145650} a^{10} + \frac{788351}{3879519300} a^{8} - \frac{4844}{7184295} a^{6} - \frac{8611}{1596510} a^{4} - \frac{24178}{53217} a^{2} - \frac{64}{657}$, $\frac{1}{47718087390000} a^{17} + \frac{869}{2385904369500} a^{15} + \frac{16439}{1590602913000} a^{13} + \frac{36011}{238590436950} a^{11} - \frac{162127}{2909639475} a^{9} - \frac{169339}{43105770} a^{7} - \frac{79567}{4789530} a^{5} - \frac{26149}{159651} a^{3} - \frac{283}{1971} a$, $\frac{1}{94589363110616877805212870000} a^{18} - \frac{1263240188856193}{47294681555308438902606435000} a^{16} + \frac{419583035468795699}{788244692588473981710107250} a^{14} - \frac{23264800403607764231}{945893631106168778052128700} a^{12} - \frac{1228301756308777506581}{945893631106168778052128700} a^{10} - \frac{41086630056992383}{2563397374271460103122300} a^{8} + \frac{22987438995744528329}{4747032174576777968745} a^{6} - \frac{25835585596590062653}{3164688116384518645830} a^{4} + \frac{16536013463509667939}{35163201293161318287} a^{2} + \frac{120288821680312682}{434113596211868127}$, $\frac{1}{283768089331850633415638610000} a^{19} - \frac{1263240188856193}{141884044665925316707819305000} a^{17} + \frac{419583035468795699}{2364734077765421945130321750} a^{15} - \frac{23264800403607764231}{2837680893318506334156386100} a^{13} - \frac{1019130265263711071957}{709420223329626583539096525} a^{11} - \frac{41086630056992383}{7690192122814380309366900} a^{9} - \frac{129841128474317534777}{28482193047460667812470} a^{7} + \frac{7335354451583789527}{1898812869830711187498} a^{5} - \frac{12115483886473628443}{105489603879483954861} a^{3} + \frac{409697885821558100}{1302340788635604381} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{342927572}$, which has order $1371710288$ (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:  \( 126615231021.771 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{10}) \), 4.0.430336000.10, 5.5.2825761.1, 10.10.817656343461990400000.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
2Data not computed
5Data not computed
$41$41.10.9.8$x^{10} + 318816$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.8$x^{10} + 318816$$10$$1$$9$$C_{10}$$[\ ]_{10}$