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Properties

Label 20.0.32473988377...0464.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -256, 872, -1032, -428, 1441, -503, 464, -1072, -198, 1673, -1657, 1462, -1307, 694, -102, -68, 23, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 10*x^18 + 23*x^17 - 68*x^16 - 102*x^15 + 694*x^14 - 1307*x^13 + 1462*x^12 - 1657*x^11 + 1673*x^10 - 198*x^9 - 1072*x^8 + 464*x^7 - 503*x^6 + 1441*x^5 - 428*x^4 - 1032*x^3 + 872*x^2 - 256*x + 32)
 
gp: K = bnfinit(x^20 - 6*x^19 + 10*x^18 + 23*x^17 - 68*x^16 - 102*x^15 + 694*x^14 - 1307*x^13 + 1462*x^12 - 1657*x^11 + 1673*x^10 - 198*x^9 - 1072*x^8 + 464*x^7 - 503*x^6 + 1441*x^5 - 428*x^4 - 1032*x^3 + 872*x^2 - 256*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 10 x^{18} + 23 x^{17} - 68 x^{16} - 102 x^{15} + 694 x^{14} - 1307 x^{13} + 1462 x^{12} - 1657 x^{11} + 1673 x^{10} - 198 x^{9} - 1072 x^{8} + 464 x^{7} - 503 x^{6} + 1441 x^{5} - 428 x^{4} - 1032 x^{3} + 872 x^{2} - 256 x + 32 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{16} + \frac{3}{8} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} + \frac{3}{8} a^{9} + \frac{3}{8} a^{8} - \frac{1}{2} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{244250875964038416118127490935152} a^{19} - \frac{2829306715717725014013054734281}{122125437982019208059063745467576} a^{18} + \frac{3799995396678697280853458895831}{122125437982019208059063745467576} a^{17} - \frac{30291880287535761053971575987257}{244250875964038416118127490935152} a^{16} - \frac{4683815599857985341452646286042}{15265679747752401007382968183447} a^{15} + \frac{11021231709071025890379236442383}{122125437982019208059063745467576} a^{14} + \frac{48446087836500833303850990132455}{122125437982019208059063745467576} a^{13} - \frac{36544016002991930623196850138699}{244250875964038416118127490935152} a^{12} + \frac{21224564795732043781220574421801}{122125437982019208059063745467576} a^{11} - \frac{81637333261817728401571534718709}{244250875964038416118127490935152} a^{10} + \frac{15306813227735005688231060752461}{244250875964038416118127490935152} a^{9} + \frac{44703563143514141652104224701161}{122125437982019208059063745467576} a^{8} - \frac{23702972652940557705323396796151}{61062718991009604029531872733788} a^{7} + \frac{9733651899400450977582197936723}{30531359495504802014765936366894} a^{6} + \frac{74023366445473563743864403860185}{244250875964038416118127490935152} a^{5} + \frac{32022270447841062878654213805941}{244250875964038416118127490935152} a^{4} - \frac{27099500408072105259494394491027}{61062718991009604029531872733788} a^{3} - \frac{9257086020220826924326108308127}{61062718991009604029531872733788} a^{2} + \frac{9167761357141574796776061755719}{30531359495504802014765936366894} a - \frac{1398286173927825616324481576233}{15265679747752401007382968183447}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 382865.542588 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 R $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$