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Properties

Label 20.0.61994950383...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 401^{10}$
Root discriminant $77.56$
Ramified primes $3, 5, 401$
Class number $69696$ (GRH)
Class group $[2, 2, 2, 2, 22, 198]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![128216149, -165797355, 249064514, -220673892, 193238486, -129433389, 82262609, -43398393, 21295798, -8889480, 3378009, -1076553, 298054, -62535, 9149, 6, -310, 66, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 + 66*x^17 - 310*x^16 + 6*x^15 + 9149*x^14 - 62535*x^13 + 298054*x^12 - 1076553*x^11 + 3378009*x^10 - 8889480*x^9 + 21295798*x^8 - 43398393*x^7 + 82262609*x^6 - 129433389*x^5 + 193238486*x^4 - 220673892*x^3 + 249064514*x^2 - 165797355*x + 128216149)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 + 66*x^17 - 310*x^16 + 6*x^15 + 9149*x^14 - 62535*x^13 + 298054*x^12 - 1076553*x^11 + 3378009*x^10 - 8889480*x^9 + 21295798*x^8 - 43398393*x^7 + 82262609*x^6 - 129433389*x^5 + 193238486*x^4 - 220673892*x^3 + 249064514*x^2 - 165797355*x + 128216149, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} + 66 x^{17} - 310 x^{16} + 6 x^{15} + 9149 x^{14} - 62535 x^{13} + 298054 x^{12} - 1076553 x^{11} + 3378009 x^{10} - 8889480 x^{9} + 21295798 x^{8} - 43398393 x^{7} + 82262609 x^{6} - 129433389 x^{5} + 193238486 x^{4} - 220673892 x^{3} + 249064514 x^{2} - 165797355 x + 128216149 \)

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:  \(61994950383582739305329189990712890625=3^{10}\cdot 5^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.56$
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:  $3, 5, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{2}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} - \frac{2}{27} a^{13} + \frac{2}{27} a^{12} + \frac{4}{27} a^{11} + \frac{2}{9} a^{10} + \frac{13}{27} a^{9} + \frac{11}{27} a^{8} - \frac{13}{27} a^{7} + \frac{2}{9} a^{6} + \frac{8}{27} a^{5} + \frac{8}{27} a^{4} - \frac{13}{27} a^{3} - \frac{2}{27} a^{2} + \frac{8}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{13} - \frac{4}{27} a^{12} - \frac{4}{27} a^{11} - \frac{8}{27} a^{10} - \frac{11}{27} a^{9} - \frac{4}{9} a^{8} - \frac{2}{27} a^{7} - \frac{10}{27} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{4}{27} a^{3} - \frac{2}{27} a^{2} - \frac{4}{27} a - \frac{1}{27}$, $\frac{1}{154629} a^{18} + \frac{701}{154629} a^{17} - \frac{1526}{154629} a^{16} + \frac{5047}{154629} a^{15} + \frac{3233}{154629} a^{14} + \frac{5228}{154629} a^{13} - \frac{12661}{154629} a^{12} - \frac{15661}{51543} a^{11} + \frac{23083}{51543} a^{10} - \frac{660}{1909} a^{9} + \frac{2828}{51543} a^{8} - \frac{5555}{17181} a^{7} + \frac{71836}{154629} a^{6} + \frac{17177}{154629} a^{5} - \frac{34649}{154629} a^{4} - \frac{38708}{154629} a^{3} - \frac{64318}{154629} a^{2} - \frac{33313}{154629} a + \frac{63821}{154629}$, $\frac{1}{1598743884469877279434743703667212518966330534274081301349} a^{19} - \frac{937276192372294628867455213053579798560045395890074}{532914628156625759811581234555737506322110178091360433783} a^{18} - \frac{9477805768459211697079726030325814465020134483800370234}{532914628156625759811581234555737506322110178091360433783} a^{17} + \frac{28191106304945188490659130690015533509615039239535038171}{1598743884469877279434743703667212518966330534274081301349} a^{16} + \frac{14606864360949816774856626326312545616293875263653785086}{532914628156625759811581234555737506322110178091360433783} a^{15} - \frac{43780869883161515656342865834877410435401287780703979744}{1598743884469877279434743703667212518966330534274081301349} a^{14} + \frac{8800734096819764436168425558208944978449964378813700607}{122980298805375175341134131051324039920486964174929330873} a^{13} + \frac{7224847259348530254473657625366459652321315713453115712}{1598743884469877279434743703667212518966330534274081301349} a^{12} + \frac{28118880037215443319020243391628164038439917954920227539}{532914628156625759811581234555737506322110178091360433783} a^{11} + \frac{5621842773970762480076433456583035378754744430988437329}{177638209385541919937193744851912502107370059363786811261} a^{10} - \frac{24300393385731566031379096179943895771950210438858730359}{532914628156625759811581234555737506322110178091360433783} a^{9} - \frac{105503581856388418196946239726093765420760665853470573399}{532914628156625759811581234555737506322110178091360433783} a^{8} - \frac{96545100233716684512120998859768609485752689005819135828}{1598743884469877279434743703667212518966330534274081301349} a^{7} + \frac{4954181694394180408772615604778507532445523104740752834}{19737578820615768881910416094656944678596673262642979029} a^{6} + \frac{11640323259542871850049490973332426908611198070247381035}{59212736461847306645731248283970834035790019787928937087} a^{5} - \frac{405125918911100669313009390598747248090127399665316452085}{1598743884469877279434743703667212518966330534274081301349} a^{4} - \frac{27412160434510035905070351417869399582932768235308708544}{59212736461847306645731248283970834035790019787928937087} a^{3} + \frac{494461112431960849303131062168878709082296506664432481331}{1598743884469877279434743703667212518966330534274081301349} a^{2} - \frac{170836134851961342566640804963243852612404098968008478551}{1598743884469877279434743703667212518966330534274081301349} a - \frac{6729384387847289975395096200176746011479958444378846584}{19261974511685268426924622935749548421281090774386521703}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{198}$, which has order $69696$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-6015}) \), \(\Q(\sqrt{-15}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.19635130215759375.1 x5, 10.0.7873687216519509375.1 x5

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

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed