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Properties

Label 20.12.8230673022...1609.1
Degree $20$
Signature $[12, 4]$
Discriminant $11^{18}\cdot 23^{6}$
Root discriminant $22.17$
Ramified primes $11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, 262, -537, -1760, 909, 4485, -162, -5522, -402, 3939, -276, -2154, 493, 814, -293, -184, 96, 22, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 16*x^18 + 22*x^17 + 96*x^16 - 184*x^15 - 293*x^14 + 814*x^13 + 493*x^12 - 2154*x^11 - 276*x^10 + 3939*x^9 - 402*x^8 - 5522*x^7 - 162*x^6 + 4485*x^5 + 909*x^4 - 1760*x^3 - 537*x^2 + 262*x + 89)
 
gp: K = bnfinit(x^20 - x^19 - 16*x^18 + 22*x^17 + 96*x^16 - 184*x^15 - 293*x^14 + 814*x^13 + 493*x^12 - 2154*x^11 - 276*x^10 + 3939*x^9 - 402*x^8 - 5522*x^7 - 162*x^6 + 4485*x^5 + 909*x^4 - 1760*x^3 - 537*x^2 + 262*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 16 x^{18} + 22 x^{17} + 96 x^{16} - 184 x^{15} - 293 x^{14} + 814 x^{13} + 493 x^{12} - 2154 x^{11} - 276 x^{10} + 3939 x^{9} - 402 x^{8} - 5522 x^{7} - 162 x^{6} + 4485 x^{5} + 909 x^{4} - 1760 x^{3} - 537 x^{2} + 262 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{8}{23} a^{15} + \frac{10}{23} a^{14} - \frac{11}{23} a^{13} - \frac{11}{23} a^{12} - \frac{1}{23} a^{11} + \frac{8}{23} a^{10} - \frac{10}{23} a^{9} + \frac{2}{23} a^{8} - \frac{6}{23} a^{7} - \frac{1}{23} a^{6} + \frac{9}{23} a^{5} - \frac{2}{23} a^{4} - \frac{7}{23} a^{2} - \frac{7}{23} a - \frac{2}{23}$, $\frac{1}{23} a^{17} - \frac{8}{23} a^{15} - \frac{7}{23} a^{13} + \frac{3}{23} a^{12} + \frac{8}{23} a^{10} - \frac{9}{23} a^{9} + \frac{10}{23} a^{8} - \frac{3}{23} a^{7} + \frac{1}{23} a^{6} + \frac{1}{23} a^{5} + \frac{7}{23} a^{4} - \frac{7}{23} a^{3} + \frac{6}{23} a^{2} + \frac{11}{23} a + \frac{7}{23}$, $\frac{1}{23} a^{18} + \frac{5}{23} a^{15} + \frac{4}{23} a^{14} + \frac{7}{23} a^{13} + \frac{4}{23} a^{12} + \frac{9}{23} a^{10} - \frac{1}{23} a^{9} - \frac{10}{23} a^{8} - \frac{1}{23} a^{7} - \frac{7}{23} a^{6} + \frac{10}{23} a^{5} + \frac{6}{23} a^{3} + \frac{1}{23} a^{2} - \frac{3}{23} a + \frac{7}{23}$, $\frac{1}{566200248020573} a^{19} - \frac{8110047217559}{566200248020573} a^{18} + \frac{9494254580893}{566200248020573} a^{17} - \frac{10270135857797}{566200248020573} a^{16} - \frac{280260572239839}{566200248020573} a^{15} + \frac{41855530800754}{566200248020573} a^{14} + \frac{241052227037496}{566200248020573} a^{13} + \frac{273959707010027}{566200248020573} a^{12} + \frac{230038781487439}{566200248020573} a^{11} - \frac{232822716631192}{566200248020573} a^{10} - \frac{13323706624767}{566200248020573} a^{9} + \frac{121275369482952}{566200248020573} a^{8} - \frac{181201233070494}{566200248020573} a^{7} + \frac{173159464958790}{566200248020573} a^{6} + \frac{276221814358130}{566200248020573} a^{5} + \frac{57297238934315}{566200248020573} a^{4} + \frac{7496319427074}{24617402087851} a^{3} + \frac{8254553770689}{566200248020573} a^{2} - \frac{250527669312567}{566200248020573} a - \frac{1314753605574}{6361800539557}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 1478677.49853 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.1247354328539.1, 10.6.54232796893.1, 10.8.2608104505127.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ 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
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$