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Properties

Label 20.20.6083576961...7689.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 163^{10}$
Root discriminant $86.94$
Ramified primes $11, 163$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times A_4$ (as 20T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -3172, -18671, 17687, 161071, -2135, -425102, -35258, 479629, 38670, -265853, -18184, 78424, 4472, -12841, -590, 1162, 39, -54, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 54*x^18 + 39*x^17 + 1162*x^16 - 590*x^15 - 12841*x^14 + 4472*x^13 + 78424*x^12 - 18184*x^11 - 265853*x^10 + 38670*x^9 + 479629*x^8 - 35258*x^7 - 425102*x^6 - 2135*x^5 + 161071*x^4 + 17687*x^3 - 18671*x^2 - 3172*x + 89)
 
gp: K = bnfinit(x^20 - x^19 - 54*x^18 + 39*x^17 + 1162*x^16 - 590*x^15 - 12841*x^14 + 4472*x^13 + 78424*x^12 - 18184*x^11 - 265853*x^10 + 38670*x^9 + 479629*x^8 - 35258*x^7 - 425102*x^6 - 2135*x^5 + 161071*x^4 + 17687*x^3 - 18671*x^2 - 3172*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 54 x^{18} + 39 x^{17} + 1162 x^{16} - 590 x^{15} - 12841 x^{14} + 4472 x^{13} + 78424 x^{12} - 18184 x^{11} - 265853 x^{10} + 38670 x^{9} + 479629 x^{8} - 35258 x^{7} - 425102 x^{6} - 2135 x^{5} + 161071 x^{4} + 17687 x^{3} - 18671 x^{2} - 3172 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(608357696176518435903681459502978907689=11^{16}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.94$
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:  $11, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{551932968057681518140619002341281534488813} a^{19} + \frac{218612748667953628858408702947984158012608}{551932968057681518140619002341281534488813} a^{18} - \frac{19221955399933173964463404442269339876907}{551932968057681518140619002341281534488813} a^{17} - \frac{68572281768143445292747372765644335590218}{551932968057681518140619002341281534488813} a^{16} - \frac{134344085606399440650159157913301788091606}{551932968057681518140619002341281534488813} a^{15} + \frac{102203671790568222773539372248529371859848}{551932968057681518140619002341281534488813} a^{14} + \frac{156181114238798382990418210399979612095199}{551932968057681518140619002341281534488813} a^{13} + \frac{111676470496538323242443849275029541275943}{551932968057681518140619002341281534488813} a^{12} + \frac{19229565918952721814672508504130808198011}{551932968057681518140619002341281534488813} a^{11} - \frac{177231534320837981045669119873216577355570}{551932968057681518140619002341281534488813} a^{10} - \frac{226166982322525966495584647108452612680874}{551932968057681518140619002341281534488813} a^{9} + \frac{149786798847405384661130962454003668378904}{551932968057681518140619002341281534488813} a^{8} - \frac{266430567656405065454402992263178754254956}{551932968057681518140619002341281534488813} a^{7} - \frac{242079944359498975563015561003172644968153}{551932968057681518140619002341281534488813} a^{6} + \frac{5624096360990773104470905955541014075274}{551932968057681518140619002341281534488813} a^{5} + \frac{109762033876991411604536667051997796960129}{551932968057681518140619002341281534488813} a^{4} + \frac{50802222094256231260527378446276247246409}{551932968057681518140619002341281534488813} a^{3} + \frac{43180003528541873592219059154075332648766}{551932968057681518140619002341281534488813} a^{2} + \frac{18733490950925870881476379689131947590735}{551932968057681518140619002341281534488813} a - \frac{9348379700671609310168072212915025381928}{551932968057681518140619002341281534488813}$

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:  \( 8339520574340 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times A_4$ (as 20T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 60
The 20 conjugacy class representatives for $C_5\times A_4$
Character table for $C_5\times A_4$

Intermediate fields

4.4.26569.1, \(\Q(\zeta_{11})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 sibling: 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
163Data not computed