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Properties

Label 20.0.33626538312...2249.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $18.89$
Ramified primes $3, 7, 17$
Class number $2$
Class group $[2]$
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![289, 0, 629, 340, 424, 2241, 89, 574, 1142, -446, 723, -14, -194, 222, -113, 1, 41, -32, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 18*x^18 - 32*x^17 + 41*x^16 + x^15 - 113*x^14 + 222*x^13 - 194*x^12 - 14*x^11 + 723*x^10 - 446*x^9 + 1142*x^8 + 574*x^7 + 89*x^6 + 2241*x^5 + 424*x^4 + 340*x^3 + 629*x^2 + 289)
 
gp: K = bnfinit(x^20 - 6*x^19 + 18*x^18 - 32*x^17 + 41*x^16 + x^15 - 113*x^14 + 222*x^13 - 194*x^12 - 14*x^11 + 723*x^10 - 446*x^9 + 1142*x^8 + 574*x^7 + 89*x^6 + 2241*x^5 + 424*x^4 + 340*x^3 + 629*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 18 x^{18} - 32 x^{17} + 41 x^{16} + x^{15} - 113 x^{14} + 222 x^{13} - 194 x^{12} - 14 x^{11} + 723 x^{10} - 446 x^{9} + 1142 x^{8} + 574 x^{7} + 89 x^{6} + 2241 x^{5} + 424 x^{4} + 340 x^{3} + 629 x^{2} + 289 \)

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:  \(33626538312268515533112249=3^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.89$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} - \frac{5}{17} a^{14} + \frac{4}{17} a^{13} + \frac{4}{17} a^{12} + \frac{6}{17} a^{11} + \frac{4}{17} a^{10} + \frac{6}{17} a^{9} - \frac{5}{17} a^{8} + \frac{1}{17} a^{7} + \frac{5}{17} a^{6} + \frac{2}{17} a^{5} + \frac{8}{17} a^{4} - \frac{4}{17} a^{3} - \frac{4}{17} a^{2}$, $\frac{1}{17} a^{16} - \frac{4}{17} a^{14} + \frac{7}{17} a^{13} - \frac{8}{17} a^{12} - \frac{8}{17} a^{10} + \frac{8}{17} a^{9} - \frac{7}{17} a^{8} - \frac{7}{17} a^{7} - \frac{7}{17} a^{6} + \frac{1}{17} a^{5} + \frac{2}{17} a^{4} - \frac{7}{17} a^{3} - \frac{3}{17} a^{2}$, $\frac{1}{24497} a^{17} - \frac{21}{24497} a^{16} + \frac{128}{24497} a^{15} + \frac{177}{2227} a^{14} - \frac{766}{24497} a^{13} - \frac{1548}{24497} a^{12} + \frac{3521}{24497} a^{11} + \frac{5107}{24497} a^{10} - \frac{11606}{24497} a^{9} + \frac{7538}{24497} a^{8} + \frac{529}{1441} a^{7} + \frac{2423}{24497} a^{6} - \frac{5212}{24497} a^{5} - \frac{9822}{24497} a^{4} - \frac{7065}{24497} a^{3} - \frac{8829}{24497} a^{2} + \frac{222}{1441} a - \frac{39}{1441}$, $\frac{1}{171479} a^{18} + \frac{3}{171479} a^{17} - \frac{376}{171479} a^{16} - \frac{745}{171479} a^{15} - \frac{72200}{171479} a^{14} + \frac{78}{2227} a^{13} + \frac{41301}{171479} a^{12} + \frac{7861}{24497} a^{11} - \frac{34579}{171479} a^{10} + \frac{37368}{171479} a^{9} - \frac{26245}{171479} a^{8} + \frac{65509}{171479} a^{7} + \frac{48617}{171479} a^{6} - \frac{1409}{10087} a^{5} + \frac{5059}{171479} a^{4} - \frac{32848}{171479} a^{3} + \frac{59904}{171479} a^{2} + \frac{2407}{10087} a - \frac{2377}{10087}$, $\frac{1}{5790623817320148923028913} a^{19} + \frac{10528743612965282854}{5790623817320148923028913} a^{18} - \frac{107938294626893641912}{5790623817320148923028913} a^{17} - \frac{10392715134069980556043}{526420347029104447548083} a^{16} - \frac{165146356422209250155528}{5790623817320148923028913} a^{15} + \frac{1603315102127240882822696}{5790623817320148923028913} a^{14} + \frac{1352190730717201196617650}{5790623817320148923028913} a^{13} - \frac{2174194963212209622378989}{5790623817320148923028913} a^{12} + \frac{1937416569585088572873201}{5790623817320148923028913} a^{11} + \frac{2231639417099088234095509}{5790623817320148923028913} a^{10} - \frac{216637776538693523524466}{5790623817320148923028913} a^{9} - \frac{2457382216721756224143084}{5790623817320148923028913} a^{8} - \frac{1526232482080413422295492}{5790623817320148923028913} a^{7} + \frac{2774521608340512783795608}{5790623817320148923028913} a^{6} - \frac{100470211791246063034}{622580778122798508013} a^{5} - \frac{2118801426361907753879646}{5790623817320148923028913} a^{4} - \frac{2586646764539245385396711}{5790623817320148923028913} a^{3} - \frac{117840599515833870047884}{5790623817320148923028913} a^{2} + \frac{24870278135304882381903}{340624930430596995472289} a + \frac{43177611347046043824870}{340624930430596995472289}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16449.1582833 \)
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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{21}, \sqrt{-51})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.828405627651.1 x5, 10.2.341108199621.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$