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Properties

Label 20.20.7303816416...9296.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-473, 22, 22770, -87604, 54989, 263660, -485786, 15436, 641159, -525816, -83864, 328090, -161406, 2950, 21734, -5658, -557, 398, -26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 26*x^18 + 398*x^17 - 557*x^16 - 5658*x^15 + 21734*x^14 + 2950*x^13 - 161406*x^12 + 328090*x^11 - 83864*x^10 - 525816*x^9 + 641159*x^8 + 15436*x^7 - 485786*x^6 + 263660*x^5 + 54989*x^4 - 87604*x^3 + 22770*x^2 + 22*x - 473)
 
gp: K = bnfinit(x^20 - 8*x^19 - 26*x^18 + 398*x^17 - 557*x^16 - 5658*x^15 + 21734*x^14 + 2950*x^13 - 161406*x^12 + 328090*x^11 - 83864*x^10 - 525816*x^9 + 641159*x^8 + 15436*x^7 - 485786*x^6 + 263660*x^5 + 54989*x^4 - 87604*x^3 + 22770*x^2 + 22*x - 473, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 26 x^{18} + 398 x^{17} - 557 x^{16} - 5658 x^{15} + 21734 x^{14} + 2950 x^{13} - 161406 x^{12} + 328090 x^{11} - 83864 x^{10} - 525816 x^{9} + 641159 x^{8} + 15436 x^{7} - 485786 x^{6} + 263660 x^{5} + 54989 x^{4} - 87604 x^{3} + 22770 x^{2} + 22 x - 473 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12699975804795718979740937620201} a^{19} - \frac{5223035623297877825260330707162}{12699975804795718979740937620201} a^{18} + \frac{2331279866565439290138544672536}{12699975804795718979740937620201} a^{17} + \frac{801139249184023087244780229494}{12699975804795718979740937620201} a^{16} + \frac{5528488344755790796913097954185}{12699975804795718979740937620201} a^{15} - \frac{2888567521171695326138888947248}{12699975804795718979740937620201} a^{14} - \frac{1163226654720915906643912478240}{12699975804795718979740937620201} a^{13} + \frac{829339045538970782415544329951}{12699975804795718979740937620201} a^{12} - \frac{1548702756607172970362314879405}{12699975804795718979740937620201} a^{11} - \frac{527160486649069825072939145974}{12699975804795718979740937620201} a^{10} + \frac{456878018815759304128650876108}{12699975804795718979740937620201} a^{9} + \frac{1713310291708465101344575942633}{12699975804795718979740937620201} a^{8} + \frac{5953484698117104875721545432141}{12699975804795718979740937620201} a^{7} - \frac{1002863546049966456339070538984}{12699975804795718979740937620201} a^{6} - \frac{2710324400229786377729168915857}{12699975804795718979740937620201} a^{5} + \frac{5428842733037230228763193385701}{12699975804795718979740937620201} a^{4} - \frac{478939548947722493872654731267}{12699975804795718979740937620201} a^{3} - \frac{3234274105674841010884320631657}{12699975804795718979740937620201} a^{2} + \frac{314247818952592827431369227400}{12699975804795718979740937620201} a - \frac{3974301028237824472342377946596}{12699975804795718979740937620201}$

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:  $19$
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:  \( 43619173869.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5$ (as 20T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2 x2, 10.10.116117348402176.1

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 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$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$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$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}$
23.2.1.1$x^{2} - 23$$2$$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}$