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Properties

Label 20.0.28450861599...000.12
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $74.59$
Ramified primes $2, 3, 5, 11$
Class number $72820$ (GRH)
Class group $[72820]$ (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![161255599, -92300794, 171978345, -75054826, 75066884, -26215344, 18813981, -5732304, 3284704, -904550, 423020, -106082, 44062, -11806, 4412, -1020, 250, -32, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 17*x^18 - 32*x^17 + 250*x^16 - 1020*x^15 + 4412*x^14 - 11806*x^13 + 44062*x^12 - 106082*x^11 + 423020*x^10 - 904550*x^9 + 3284704*x^8 - 5732304*x^7 + 18813981*x^6 - 26215344*x^5 + 75066884*x^4 - 75054826*x^3 + 171978345*x^2 - 92300794*x + 161255599)
 
gp: K = bnfinit(x^20 - 6*x^19 + 17*x^18 - 32*x^17 + 250*x^16 - 1020*x^15 + 4412*x^14 - 11806*x^13 + 44062*x^12 - 106082*x^11 + 423020*x^10 - 904550*x^9 + 3284704*x^8 - 5732304*x^7 + 18813981*x^6 - 26215344*x^5 + 75066884*x^4 - 75054826*x^3 + 171978345*x^2 - 92300794*x + 161255599, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 17 x^{18} - 32 x^{17} + 250 x^{16} - 1020 x^{15} + 4412 x^{14} - 11806 x^{13} + 44062 x^{12} - 106082 x^{11} + 423020 x^{10} - 904550 x^{9} + 3284704 x^{8} - 5732304 x^{7} + 18813981 x^{6} - 26215344 x^{5} + 75066884 x^{4} - 75054826 x^{3} + 171978345 x^{2} - 92300794 x + 161255599 \)

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:  \(28450861599572073223437680640000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.59$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(301,·)$, $\chi_{1320}(1169,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(1049,·)$, $\chi_{1320}(89,·)$, $\chi_{1320}(929,·)$, $\chi_{1320}(421,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(449,·)$, $\chi_{1320}(181,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(1021,·)$, $\chi_{1320}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{7}{23} a^{11} - \frac{5}{23} a^{10} + \frac{6}{23} a^{9} + \frac{10}{23} a^{8} - \frac{8}{23} a^{7} - \frac{10}{23} a^{6} + \frac{7}{23} a^{5} + \frac{10}{23} a^{4} + \frac{7}{23} a^{3} + \frac{11}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{23} a^{13} - \frac{8}{23} a^{11} - \frac{6}{23} a^{10} + \frac{6}{23} a^{9} - \frac{7}{23} a^{8} + \frac{3}{23} a^{7} + \frac{6}{23} a^{6} - \frac{10}{23} a^{5} + \frac{8}{23} a^{4} - \frac{9}{23} a^{3} - \frac{3}{23} a^{2} - \frac{8}{23} a$, $\frac{1}{23} a^{14} + \frac{7}{23} a^{11} - \frac{11}{23} a^{10} - \frac{5}{23} a^{9} - \frac{9}{23} a^{8} + \frac{11}{23} a^{7} + \frac{2}{23} a^{6} - \frac{5}{23} a^{5} + \frac{2}{23} a^{4} + \frac{7}{23} a^{3} + \frac{11}{23} a^{2} + \frac{4}{23} a$, $\frac{1}{23} a^{15} - \frac{8}{23} a^{11} + \frac{7}{23} a^{10} - \frac{5}{23} a^{9} + \frac{10}{23} a^{8} - \frac{11}{23} a^{7} - \frac{4}{23} a^{6} - \frac{1}{23} a^{5} + \frac{6}{23} a^{4} + \frac{8}{23} a^{3} - \frac{4}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{23} a^{16} - \frac{3}{23} a^{11} + \frac{1}{23} a^{10} - \frac{11}{23} a^{9} + \frac{1}{23} a^{7} + \frac{11}{23} a^{6} - \frac{7}{23} a^{5} - \frac{4}{23} a^{4} + \frac{6}{23} a^{3} + \frac{4}{23} a^{2} + \frac{4}{23} a$, $\frac{1}{4577} a^{17} + \frac{82}{4577} a^{16} + \frac{81}{4577} a^{15} - \frac{73}{4577} a^{14} + \frac{66}{4577} a^{13} + \frac{20}{4577} a^{12} - \frac{71}{199} a^{11} - \frac{450}{4577} a^{10} - \frac{1880}{4577} a^{9} - \frac{558}{4577} a^{8} + \frac{9}{199} a^{7} - \frac{651}{4577} a^{6} - \frac{34}{4577} a^{5} + \frac{1075}{4577} a^{4} + \frac{660}{4577} a^{3} + \frac{1445}{4577} a^{2} - \frac{1845}{4577} a - \frac{24}{199}$, $\frac{1}{2533956268726661263} a^{18} + \frac{97735065721134}{2533956268726661263} a^{17} + \frac{17365585790413140}{2533956268726661263} a^{16} - \frac{22748953271067094}{2533956268726661263} a^{15} - \frac{46121779409509521}{2533956268726661263} a^{14} + \frac{47621338309037061}{2533956268726661263} a^{13} + \frac{14738223461292451}{2533956268726661263} a^{12} - \frac{348480470757442262}{2533956268726661263} a^{11} - \frac{440706259092881726}{2533956268726661263} a^{10} + \frac{238602804780800820}{2533956268726661263} a^{9} - \frac{304454409209284758}{2533956268726661263} a^{8} - \frac{1227500465823411976}{2533956268726661263} a^{7} - \frac{357329832408629451}{2533956268726661263} a^{6} - \frac{764462743984039702}{2533956268726661263} a^{5} + \frac{1067989812199807225}{2533956268726661263} a^{4} + \frac{43515123573000608}{110172011683767881} a^{3} + \frac{476994493890629234}{2533956268726661263} a^{2} - \frac{36393440867150208}{110172011683767881} a - \frac{177832994292175}{4790087464511647}$, $\frac{1}{4485812817263734260981538117680827426904431} a^{19} + \frac{170267395869157194165082}{4485812817263734260981538117680827426904431} a^{18} + \frac{20288857765401304209394242625890758082}{4485812817263734260981538117680827426904431} a^{17} + \frac{12016398997916772564303669892850946105503}{4485812817263734260981538117680827426904431} a^{16} + \frac{30515806845932421805839048903017916544555}{4485812817263734260981538117680827426904431} a^{15} + \frac{55127148639115830139195591293435592129076}{4485812817263734260981538117680827426904431} a^{14} - \frac{330924379019962160561183927273870171099}{22541772951074041512470040792365966969369} a^{13} + \frac{63540652516027951683180959096931089733231}{4485812817263734260981538117680827426904431} a^{12} + \frac{1420884668027327855773469382164798134563265}{4485812817263734260981538117680827426904431} a^{11} - \frac{209959859364610611400103891428483889607770}{4485812817263734260981538117680827426904431} a^{10} + \frac{1251315983252916244756751124307538800786295}{4485812817263734260981538117680827426904431} a^{9} + \frac{1743838455819008818691153773697367924060044}{4485812817263734260981538117680827426904431} a^{8} + \frac{295596957295540537675523616178922783359003}{4485812817263734260981538117680827426904431} a^{7} - \frac{1427500642439508715935749494069918956814146}{4485812817263734260981538117680827426904431} a^{6} - \frac{1307757967491418152618218646512568028594188}{4485812817263734260981538117680827426904431} a^{5} - \frac{426622500574277591704264963600793433312561}{4485812817263734260981538117680827426904431} a^{4} - \frac{1787460231946888650281083910539325685431120}{4485812817263734260981538117680827426904431} a^{3} - \frac{15817171280358285437097095409481728127665}{50402391205210497314399304693043004796679} a^{2} - \frac{24435382550349131259567801847385582582480}{195035339881031924390501657290470757691497} a + \frac{700674603764281678475343699001645431664}{8479797386131822799587028577846554682239}$

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

Class group and class number

$C_{72820}$, which has order $72820$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.0.5333934907699200000.1, 10.0.162778775259375.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
3Data not computed
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$