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Properties

Label 20.0.65416827983...000.15
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{34}\cdot 41^{18}$
Root discriminant $872.54$
Ramified primes $2, 5, 41$
Class number $66350688800$ (GRH)
Class group $[110, 603188080]$ (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![78262133223197401, 0, 44657013077067075, 0, 7168969150008775, 0, 524077724148005, 0, 20447120999480, 0, 464056945403, 0, 6428606275, 0, 55136800, 0, 285770, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 285770*x^16 + 55136800*x^14 + 6428606275*x^12 + 464056945403*x^10 + 20447120999480*x^8 + 524077724148005*x^6 + 7168969150008775*x^4 + 44657013077067075*x^2 + 78262133223197401)
 
gp: K = bnfinit(x^20 + 820*x^18 + 285770*x^16 + 55136800*x^14 + 6428606275*x^12 + 464056945403*x^10 + 20447120999480*x^8 + 524077724148005*x^6 + 7168969150008775*x^4 + 44657013077067075*x^2 + 78262133223197401, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 285770 x^{16} + 55136800 x^{14} + 6428606275 x^{12} + 464056945403 x^{10} + 20447120999480 x^{8} + 524077724148005 x^{6} + 7168969150008775 x^{4} + 44657013077067075 x^{2} + 78262133223197401 \)

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:  \(65416827983112661350313274854125976562500000000000000000000=2^{20}\cdot 5^{34}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $872.54$
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:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(3009,·)$, $\chi_{4100}(4099,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(1289,·)$, $\chi_{4100}(961,·)$, $\chi_{4100}(1281,·)$, $\chi_{4100}(3139,·)$, $\chi_{4100}(1041,·)$, $\chi_{4100}(1091,·)$, $\chi_{4100}(3059,·)$, $\chi_{4100}(4069,·)$, $\chi_{4100}(529,·)$, $\chi_{4100}(31,·)$, $\chi_{4100}(3079,·)$, $\chi_{4100}(1149,·)$, $\chi_{4100}(3571,·)$, $\chi_{4100}(2819,·)$, $\chi_{4100}(2811,·)$, $\chi_{4100}(1021,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} + \frac{1}{11} a^{4} - \frac{4}{11} a^{2} + \frac{2}{11}$, $\frac{1}{11} a^{7} + \frac{1}{11} a^{5} - \frac{4}{11} a^{3} + \frac{2}{11} a$, $\frac{1}{11} a^{8} - \frac{5}{11} a^{4} - \frac{5}{11} a^{2} - \frac{2}{11}$, $\frac{1}{11} a^{9} - \frac{5}{11} a^{5} - \frac{5}{11} a^{3} - \frac{2}{11} a$, $\frac{1}{131570681} a^{10} + \frac{10}{3209041} a^{8} + \frac{1435}{3209041} a^{6} + \frac{84050}{3209041} a^{4} - \frac{1486016}{3209041} a^{2} - \frac{313561}{3209041}$, $\frac{1}{2499842939} a^{11} + \frac{2333858}{60971779} a^{9} + \frac{2043552}{60971779} a^{7} + \frac{12920214}{60971779} a^{5} + \frac{7557645}{60971779} a^{3} - \frac{897023}{60971779} a$, $\frac{1}{2499842939} a^{12} - \frac{2}{2499842939} a^{10} - \frac{1461340}{60971779} a^{8} - \frac{2257556}{60971779} a^{6} - \frac{1690358}{60971779} a^{4} + \frac{22920142}{60971779} a^{2} + \frac{688326}{3209041}$, $\frac{1}{2499842939} a^{13} - \frac{2336513}{60971779} a^{9} + \frac{96292}{3209041} a^{7} - \frac{9107264}{60971779} a^{5} + \frac{4778098}{60971779} a^{3} + \frac{22369926}{60971779} a$, $\frac{1}{102493560499} a^{14} - \frac{7}{2499842939} a^{10} - \frac{91374}{3209041} a^{8} + \frac{973638}{60971779} a^{6} - \frac{716258241}{2499842939} a^{4} + \frac{10927828}{60971779} a^{2} + \frac{861435}{3209041}$, $\frac{1}{897741096410741} a^{15} + \frac{15}{21896124302701} a^{13} + \frac{90}{534051812261} a^{11} + \frac{1025}{48550164751} a^{9} + \frac{756450}{534051812261} a^{7} + \frac{2879986299144}{21896124302701} a^{5} - \frac{24412517077}{534051812261} a^{3} + \frac{52708399240}{534051812261} a$, $\frac{1}{9875152060518151} a^{16} - \frac{16903}{9875152060518151} a^{14} + \frac{47485}{240857367329711} a^{12} - \frac{45565}{21896124302701} a^{10} + \frac{112054985051}{5874569934871} a^{8} + \frac{2049391455852}{240857367329711} a^{6} - \frac{64599156647314}{240857367329711} a^{4} + \frac{698182338108}{5874569934871} a^{2} - \frac{6610459}{35299451}$, $\frac{1}{404881234481244191} a^{17} - \frac{5}{9875152060518151} a^{15} + \frac{26067}{240857367329711} a^{13} - \frac{2499}{21896124302701} a^{11} - \frac{28177875014}{5874569934871} a^{9} + \frac{56723534238518}{9875152060518151} a^{7} + \frac{104882035971076}{240857367329711} a^{5} - \frac{1913531055103}{5874569934871} a^{3} + \frac{1104373899657}{5874569934871} a$, $\frac{1}{404881234481244191} a^{18} + \frac{20742}{9875152060518151} a^{14} + \frac{17238}{240857367329711} a^{12} - \frac{825926}{240857367329711} a^{10} + \frac{362867473943537}{9875152060518151} a^{8} - \frac{9211383168890}{240857367329711} a^{6} - \frac{89101097523748}{240857367329711} a^{4} + \frac{504259261093}{5874569934871} a^{2} - \frac{7342721}{35299451}$, $\frac{1}{404881234481244191} a^{19} - \frac{4}{9875152060518151} a^{15} - \frac{4905}{240857367329711} a^{13} - \frac{10419}{240857367329711} a^{11} + \frac{11617439148615}{519744845290429} a^{9} - \frac{8974415432687}{240857367329711} a^{7} + \frac{110484517092849}{240857367329711} a^{5} + \frac{1467274269972}{5874569934871} a^{3} + \frac{1246861973630}{5874569934871} a$

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

Class group and class number

$C_{110}\times C_{603188080}$, which has order $66350688800$ (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:  \( 298173173992.418 \) (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

\(\Q(\sqrt{-205}) \), \(\Q(\sqrt{-41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-41})\), 5.5.1103812890625.1, 10.0.255767136245282031250000000000.4, 10.0.51153427249056406250000000000.3, 10.10.6092014487549591064453125.2

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
$41$41.10.9.5$x^{10} - 68864256$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.5$x^{10} - 68864256$$10$$1$$9$$C_{10}$$[\ ]_{10}$