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Properties

Label 20.0.82306730226...1609.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 23^{6}$
Root discriminant $22.17$
Ramified primes $11, 23$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times C_2^4:C_5$ (as 20T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 20, -20, 110, -32, 116, 141, 101, 143, 144, 132, 127, 69, 65, 34, 33, 6, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} + 6 x^{17} + 33 x^{16} + 34 x^{15} + 65 x^{14} + 69 x^{13} + 127 x^{12} + 132 x^{11} + 144 x^{10} + 143 x^{9} + 101 x^{8} + 141 x^{7} + 116 x^{6} - 32 x^{5} + 110 x^{4} - 20 x^{3} + 20 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{16} - \frac{1}{3} a^{15} - \frac{4}{9} a^{14} + \frac{4}{9} a^{13} - \frac{2}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{2} + \frac{4}{9}$, $\frac{1}{621} a^{18} + \frac{11}{207} a^{17} + \frac{8}{621} a^{16} + \frac{56}{621} a^{15} - \frac{56}{207} a^{14} + \frac{80}{621} a^{13} - \frac{206}{621} a^{12} - \frac{40}{621} a^{11} - \frac{280}{621} a^{10} - \frac{35}{207} a^{9} + \frac{76}{621} a^{8} + \frac{53}{207} a^{7} + \frac{103}{621} a^{6} - \frac{80}{621} a^{5} - \frac{286}{621} a^{4} + \frac{116}{621} a^{3} - \frac{25}{621} a^{2} - \frac{32}{621} a + \frac{154}{621}$, $\frac{1}{98534508242933547} a^{19} + \frac{76218567797968}{98534508242933547} a^{18} + \frac{3403201604775863}{98534508242933547} a^{17} - \frac{130516148346665}{2291500191696129} a^{16} + \frac{26492564251396535}{98534508242933547} a^{15} + \frac{31358279332162802}{98534508242933547} a^{14} + \frac{2540694767321407}{10948278693659283} a^{13} + \frac{11013655113771452}{32844836080977849} a^{12} - \frac{5001907272142883}{98534508242933547} a^{11} - \frac{38711040571209409}{98534508242933547} a^{10} - \frac{16769795883076298}{98534508242933547} a^{9} + \frac{39297347127778861}{98534508242933547} a^{8} - \frac{820654432835378}{2291500191696129} a^{7} + \frac{16509570869982074}{98534508242933547} a^{6} - \frac{5857759310623187}{32844836080977849} a^{5} + \frac{42279823348683058}{98534508242933547} a^{4} + \frac{27483180390783496}{98534508242933547} a^{3} + \frac{7247361482236484}{32844836080977849} a^{2} + \frac{30019309881984896}{98534508242933547} a - \frac{10275860534242652}{98534508242933547}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( 35594.0921154 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T40):

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, 10.2.113395848049.1

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
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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ 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 ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$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}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{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}$