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Properties

Label 20.0.61131937356...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}\cdot 11^{18}$
Root discriminant $194.68$
Ramified primes $2, 5, 11$
Class number $19364200$ (GRH)
Class group $[2, 9682100]$ (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![714492900000, 0, 793881000000, 0, 334312110000, 0, 71939340000, 0, 8985339000, 0, 694936000, 0, 34223200, 0, 1071400, 0, 20570, 0, 220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 220*x^18 + 20570*x^16 + 1071400*x^14 + 34223200*x^12 + 694936000*x^10 + 8985339000*x^8 + 71939340000*x^6 + 334312110000*x^4 + 793881000000*x^2 + 714492900000)
 
gp: K = bnfinit(x^20 + 220*x^18 + 20570*x^16 + 1071400*x^14 + 34223200*x^12 + 694936000*x^10 + 8985339000*x^8 + 71939340000*x^6 + 334312110000*x^4 + 793881000000*x^2 + 714492900000, 1)
 

Normalized defining polynomial

\( x^{20} + 220 x^{18} + 20570 x^{16} + 1071400 x^{14} + 34223200 x^{12} + 694936000 x^{10} + 8985339000 x^{8} + 71939340000 x^{6} + 334312110000 x^{4} + 793881000000 x^{2} + 714492900000 \)

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:  \(6113193735657808322804901216256000000000000000=2^{55}\cdot 5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.68$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(880=2^{4}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(387,·)$, $\chi_{880}(641,·)$, $\chi_{880}(9,·)$, $\chi_{880}(843,·)$, $\chi_{880}(81,·)$, $\chi_{880}(787,·)$, $\chi_{880}(729,·)$, $\chi_{880}(89,·)$, $\chi_{880}(283,·)$, $\chi_{880}(523,·)$, $\chi_{880}(801,·)$, $\chi_{880}(227,·)$, $\chi_{880}(401,·)$, $\chi_{880}(489,·)$, $\chi_{880}(43,·)$, $\chi_{880}(547,·)$, $\chi_{880}(307,·)$, $\chi_{880}(169,·)$, $\chi_{880}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{10} a^{6}$, $\frac{1}{10} a^{7}$, $\frac{1}{100} a^{8}$, $\frac{1}{100} a^{9}$, $\frac{1}{1100} a^{10}$, $\frac{1}{3300} a^{11} - \frac{1}{300} a^{9} + \frac{1}{30} a^{7} - \frac{1}{30} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{99000} a^{12} + \frac{1}{2475} a^{10} - \frac{1}{450} a^{8} + \frac{1}{45} a^{6} - \frac{1}{90} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{297000} a^{13} + \frac{1}{7425} a^{11} - \frac{11}{2700} a^{9} + \frac{11}{270} a^{7} - \frac{1}{270} a^{5} - \frac{13}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{891000} a^{14} + \frac{1}{222750} a^{12} - \frac{1}{4050} a^{10} - \frac{17}{4050} a^{8} - \frac{19}{810} a^{6} + \frac{7}{405} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{2673000} a^{15} + \frac{1}{668250} a^{13} - \frac{1}{12150} a^{11} - \frac{17}{12150} a^{9} + \frac{31}{1215} a^{7} - \frac{67}{2430} a^{5} + \frac{2}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{80190000} a^{16} + \frac{1}{2004750} a^{14} - \frac{1}{200475} a^{12} + \frac{47}{400950} a^{10} + \frac{53}{72900} a^{8} - \frac{31}{729} a^{6} - \frac{1}{162} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{240570000} a^{17} + \frac{1}{6014250} a^{15} - \frac{1}{601425} a^{13} + \frac{47}{1202850} a^{11} + \frac{53}{218700} a^{9} - \frac{31}{2187} a^{7} + \frac{38}{1215} a^{5} + \frac{7}{27} a^{3}$, $\frac{1}{529706761641090000} a^{18} - \frac{597229819}{264853380820545000} a^{16} - \frac{10895752973}{26485338082054500} a^{14} - \frac{1025327927}{211882704656436} a^{12} + \frac{1914787747393}{5297067616410900} a^{10} - \frac{1126764777323}{240775800745950} a^{8} - \frac{15368076364}{535057334991} a^{6} - \frac{14073370553}{594508149990} a^{4} - \frac{2193663118}{6605646111} a^{2} - \frac{321743316}{733960679}$, $\frac{1}{1589120284923270000} a^{19} - \frac{597229819}{794560142461635000} a^{17} - \frac{10895752973}{79456014246163500} a^{15} - \frac{1025327927}{635648113969308} a^{13} + \frac{1914787747393}{15891202849232700} a^{11} + \frac{2561986460273}{1444654804475700} a^{9} - \frac{15368076364}{1605172004973} a^{7} - \frac{14073370553}{1783524449970} a^{5} - \frac{8799309229}{19816938333} a^{3} - \frac{1055703995}{2201882037} a$

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

Class group and class number

$C_{2}\times C_{9682100}$, which has order $19364200$ (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:  \( 14452469.589232503 \) (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

\(\Q(\sqrt{10}) \), 4.0.30976000.4, \(\Q(\zeta_{11})^+\), 10.10.21950349414400000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
11Data not computed