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Properties

Label 20.0.16259651371...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{18}\cdot 5^{10}\cdot 73^{10}$
Root discriminant $51.35$
Ramified primes $3, 5, 73$
Class number $180$ (GRH)
Class group $[180]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1120149, -3496365, 4716981, -3483729, 1515159, -340038, -40776, 51663, 3953, 1575, -9528, 2796, -540, 762, 115, -312, 72, 4, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 9*x^18 + 4*x^17 + 72*x^16 - 312*x^15 + 115*x^14 + 762*x^13 - 540*x^12 + 2796*x^11 - 9528*x^10 + 1575*x^9 + 3953*x^8 + 51663*x^7 - 40776*x^6 - 340038*x^5 + 1515159*x^4 - 3483729*x^3 + 4716981*x^2 - 3496365*x + 1120149)
 
gp: K = bnfinit(x^20 - 6*x^19 + 9*x^18 + 4*x^17 + 72*x^16 - 312*x^15 + 115*x^14 + 762*x^13 - 540*x^12 + 2796*x^11 - 9528*x^10 + 1575*x^9 + 3953*x^8 + 51663*x^7 - 40776*x^6 - 340038*x^5 + 1515159*x^4 - 3483729*x^3 + 4716981*x^2 - 3496365*x + 1120149, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 9 x^{18} + 4 x^{17} + 72 x^{16} - 312 x^{15} + 115 x^{14} + 762 x^{13} - 540 x^{12} + 2796 x^{11} - 9528 x^{10} + 1575 x^{9} + 3953 x^{8} + 51663 x^{7} - 40776 x^{6} - 340038 x^{5} + 1515159 x^{4} - 3483729 x^{3} + 4716981 x^{2} - 3496365 x + 1120149 \)

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:  \(16259651371902178021633499619140625=3^{18}\cdot 5^{10}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.35$
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:  $3, 5, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{17} - \frac{1}{9} a^{15} + \frac{4}{27} a^{14} - \frac{1}{9} a^{13} - \frac{2}{27} a^{11} + \frac{4}{9} a^{10} - \frac{4}{9} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{7}{27} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{7965} a^{18} - \frac{32}{7965} a^{17} + \frac{89}{2655} a^{16} - \frac{133}{1593} a^{15} - \frac{491}{7965} a^{14} + \frac{113}{2655} a^{13} - \frac{184}{1593} a^{12} - \frac{869}{7965} a^{11} + \frac{61}{885} a^{10} + \frac{586}{2655} a^{9} + \frac{247}{885} a^{8} + \frac{251}{531} a^{7} - \frac{3499}{7965} a^{6} + \frac{2867}{7965} a^{5} + \frac{31}{2655} a^{4} + \frac{947}{2655} a^{3} - \frac{57}{295} a^{2} - \frac{181}{885} a - \frac{67}{295}$, $\frac{1}{423552205155982371457406999646524149492472816262675} a^{19} + \frac{4101155588789086678892612485269388259761097212}{141184068385327457152468999882174716497490938754225} a^{18} - \frac{873224194388413384566062074685407660249552523923}{141184068385327457152468999882174716497490938754225} a^{17} + \frac{2606366883252226588374513607228531264207770599161}{423552205155982371457406999646524149492472816262675} a^{16} + \frac{5709184675773324342196288797825875589498965853331}{47061356128442485717489666627391572165830312918075} a^{15} + \frac{2907393267575793022618953072108269489477598047359}{47061356128442485717489666627391572165830312918075} a^{14} - \frac{12588899063141258510372233433638529353207784031368}{423552205155982371457406999646524149492472816262675} a^{13} - \frac{14503139425855910140027570360351987903299959978143}{141184068385327457152468999882174716497490938754225} a^{12} + \frac{6642390169283645487644760588398915628437782968164}{141184068385327457152468999882174716497490938754225} a^{11} - \frac{4546780865854779473163962940348585632912760770312}{28236813677065491430493799976434943299498187750845} a^{10} - \frac{18482137159962565625157091667516828338345551504191}{141184068385327457152468999882174716497490938754225} a^{9} - \frac{23229971593965522009073261505514147660252140504629}{47061356128442485717489666627391572165830312918075} a^{8} + \frac{178689184999133263669901505203326193174989389432821}{423552205155982371457406999646524149492472816262675} a^{7} + \frac{4785133001879862305815597577226365874149276107448}{28236813677065491430493799976434943299498187750845} a^{6} - \frac{234779248533682328565694217743198956145687436138}{2392950311615719612753711862409740957584592182275} a^{5} + \frac{76176617835279119611697425023907864675869363787}{9412271225688497143497933325478314433166062583615} a^{4} + \frac{1368699125185771551879855102691986094863233169341}{47061356128442485717489666627391572165830312918075} a^{3} + \frac{7423971552011562528268390171382676303106562471852}{15687118709480828572496555542463857388610104306025} a^{2} + \frac{1481260099509268137290776992698151203120305891989}{5229039569826942857498851847487952462870034768675} a + \frac{11243937611730824737070685586571194801067090633}{5229039569826942857498851847487952462870034768675}$

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

Class group and class number

$C_{180}$, which has order $180$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6959926.34763 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.0.25502667603136875.1, 10.10.582252685003125.1, 10.0.127513338015684375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.12.10.3$x^{12} - 14527 x^{6} + 78021889$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$