Properties

Label 20.0.29411962588...3449.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 23^{2}$
Root discriminant $11.84$
Ramified primes $11, 23$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -72, 177, -363, 637, -973, 1307, -1556, 1649, -1556, 1307, -973, 637, -363, 177, -72, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 24 x^{18} - 72 x^{17} + 177 x^{16} - 363 x^{15} + 637 x^{14} - 973 x^{13} + 1307 x^{12} - 1556 x^{11} + 1649 x^{10} - 1556 x^{9} + 1307 x^{8} - 973 x^{7} + 637 x^{6} - 363 x^{5} + 177 x^{4} - 72 x^{3} + 24 x^{2} - 6 x + 1 \)

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:  \(2941196258837390453449=11^{18}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.84$
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]~
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109} a^{19} - \frac{7}{109} a^{18} + \frac{31}{109} a^{17} + \frac{6}{109} a^{16} - \frac{47}{109} a^{15} + \frac{11}{109} a^{14} - \frac{28}{109} a^{13} + \frac{36}{109} a^{12} - \frac{37}{109} a^{11} + \frac{7}{109} a^{10} + \frac{7}{109} a^{9} - \frac{37}{109} a^{8} + \frac{36}{109} a^{7} - \frac{28}{109} a^{6} + \frac{11}{109} a^{5} - \frac{47}{109} a^{4} + \frac{6}{109} a^{3} + \frac{31}{109} a^{2} - \frac{7}{109} a + \frac{1}{109}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{20}{109} a^{19} - \frac{140}{109} a^{18} + \frac{511}{109} a^{17} - \frac{1515}{109} a^{16} + \frac{3638}{109} a^{15} - \frac{7410}{109} a^{14} + \frac{12956}{109} a^{13} - \frac{20208}{109} a^{12} + \frac{28254}{109} a^{11} - \frac{34849}{109} a^{10} + \frac{38399}{109} a^{9} - \frac{37364}{109} a^{8} + \frac{31894}{109} a^{7} - \frac{23995}{109} a^{6} + \frac{15480}{109} a^{5} - \frac{8461}{109} a^{4} + \frac{4153}{109} a^{3} - \frac{1669}{109} a^{2} + \frac{623}{109} a - \frac{198}{109} \) (order $22$)
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:  \( 2189.13991382 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.6.54232796893.1, 10.4.4930254263.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ 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 ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$