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Properties

Label 20.4.82306730226...1609.2
Degree $20$
Signature $[4, 8]$
Discriminant $11^{18}\cdot 23^{6}$
Root discriminant $22.17$
Ramified primes $11, 23$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -47, 287, -726, 890, -514, -349, 615, -627, 430, -385, 406, -272, 146, -89, 55, -21, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 8*x^18 - 21*x^17 + 55*x^16 - 89*x^15 + 146*x^14 - 272*x^13 + 406*x^12 - 385*x^11 + 430*x^10 - 627*x^9 + 615*x^8 - 349*x^7 - 514*x^6 + 890*x^5 - 726*x^4 + 287*x^3 - 47*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 8*x^18 - 21*x^17 + 55*x^16 - 89*x^15 + 146*x^14 - 272*x^13 + 406*x^12 - 385*x^11 + 430*x^10 - 627*x^9 + 615*x^8 - 349*x^7 - 514*x^6 + 890*x^5 - 726*x^4 + 287*x^3 - 47*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 8 x^{18} - 21 x^{17} + 55 x^{16} - 89 x^{15} + 146 x^{14} - 272 x^{13} + 406 x^{12} - 385 x^{11} + 430 x^{10} - 627 x^{9} + 615 x^{8} - 349 x^{7} - 514 x^{6} + 890 x^{5} - 726 x^{4} + 287 x^{3} - 47 x^{2} - 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(823067302269314181883621609=11^{18}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.17$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{272426992505979031407679277} a^{19} + \frac{118854705803561947174525856}{272426992505979031407679277} a^{18} + \frac{81975142312155911578476624}{272426992505979031407679277} a^{17} - \frac{66270309643789126803891531}{272426992505979031407679277} a^{16} - \frac{18316961388663482242629976}{272426992505979031407679277} a^{15} - \frac{14442163599049707700252813}{272426992505979031407679277} a^{14} + \frac{117049779428858836430477551}{272426992505979031407679277} a^{13} + \frac{54368614855640638122157617}{272426992505979031407679277} a^{12} + \frac{107359805779412604406546297}{272426992505979031407679277} a^{11} + \frac{130817652725760468394973759}{272426992505979031407679277} a^{10} + \frac{13111761903391906161247178}{272426992505979031407679277} a^{9} + \frac{65646462976523865000057055}{272426992505979031407679277} a^{8} + \frac{128217511891270280526573451}{272426992505979031407679277} a^{7} + \frac{71326014858053902635489524}{272426992505979031407679277} a^{6} + \frac{94226898101202609559446195}{272426992505979031407679277} a^{5} - \frac{59466852255802180267272470}{272426992505979031407679277} a^{4} - \frac{453871671490114003203478}{2079595362641061308455567} a^{3} - \frac{3302585767279861760638776}{272426992505979031407679277} a^{2} - \frac{102501249504019017252915272}{272426992505979031407679277} a + \frac{111380379716063216827686771}{272426992505979031407679277}$

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:  $11$
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:  \( 142451.10279 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.54232796893.1, 10.2.28689149556397.1, 10.6.113395848049.1

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
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$