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Properties

Label 20.0.18564650000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{21}\cdot 13^{5}$
Root discriminant $20.58$
Ramified primes $2, 5, 13$
Class number $1$
Class group Trivial
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65, -220, -12, 748, -514, -792, 864, 162, 62, -840, 487, 8, 180, -240, 52, -10, 39, -14, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 14*x^17 + 39*x^16 - 10*x^15 + 52*x^14 - 240*x^13 + 180*x^12 + 8*x^11 + 487*x^10 - 840*x^9 + 62*x^8 + 162*x^7 + 864*x^6 - 792*x^5 - 514*x^4 + 748*x^3 - 12*x^2 - 220*x + 65)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 14*x^17 + 39*x^16 - 10*x^15 + 52*x^14 - 240*x^13 + 180*x^12 + 8*x^11 + 487*x^10 - 840*x^9 + 62*x^8 + 162*x^7 + 864*x^6 - 792*x^5 - 514*x^4 + 748*x^3 - 12*x^2 - 220*x + 65, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} - 14 x^{17} + 39 x^{16} - 10 x^{15} + 52 x^{14} - 240 x^{13} + 180 x^{12} + 8 x^{11} + 487 x^{10} - 840 x^{9} + 62 x^{8} + 162 x^{7} + 864 x^{6} - 792 x^{5} - 514 x^{4} + 748 x^{3} - 12 x^{2} - 220 x + 65 \)

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:  \(185646500000000000000000000=2^{20}\cdot 5^{21}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.58$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{75} a^{18} - \frac{1}{25} a^{17} - \frac{4}{75} a^{16} - \frac{1}{75} a^{15} + \frac{9}{25} a^{14} - \frac{14}{75} a^{13} + \frac{2}{5} a^{12} - \frac{8}{75} a^{11} + \frac{8}{75} a^{10} - \frac{26}{75} a^{9} + \frac{29}{75} a^{8} - \frac{1}{75} a^{7} - \frac{19}{75} a^{6} - \frac{7}{25} a^{5} + \frac{17}{75} a^{4} - \frac{26}{75} a^{3} + \frac{2}{25} a^{2} - \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{2246536833481411299075} a^{19} + \frac{12202576577604824419}{2246536833481411299075} a^{18} + \frac{25752767067456031159}{449307366696282259815} a^{17} - \frac{37671946269578120549}{2246536833481411299075} a^{16} + \frac{41455026221037774052}{449307366696282259815} a^{15} + \frac{126104488145918639558}{449307366696282259815} a^{14} + \frac{114310250654051581702}{2246536833481411299075} a^{13} - \frac{213261421178957030018}{2246536833481411299075} a^{12} + \frac{213281851906611340779}{748845611160470433025} a^{11} + \frac{2488861862704168054}{149769122232094086605} a^{10} - \frac{341130848243526897266}{748845611160470433025} a^{9} - \frac{182922995614053416753}{2246536833481411299075} a^{8} - \frac{105643448412154138526}{2246536833481411299075} a^{7} + \frac{408209762538878722301}{2246536833481411299075} a^{6} + \frac{47213411025921278347}{449307366696282259815} a^{5} + \frac{266748441657531214411}{748845611160470433025} a^{4} - \frac{424398026943023437061}{2246536833481411299075} a^{3} + \frac{1014009978353804459657}{2246536833481411299075} a^{2} + \frac{54448308616142900587}{149769122232094086605} a + \frac{15955392682500078479}{34562105130483250755}$

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:  \( \frac{46918790550069}{771302070956365} a^{19} - \frac{74005547606802}{771302070956365} a^{18} + \frac{14838749898896}{771302070956365} a^{17} - \frac{652349415019847}{771302070956365} a^{16} + \frac{1558247207341328}{771302070956365} a^{15} + \frac{194583893807222}{771302070956365} a^{14} + \frac{2539999859731616}{771302070956365} a^{13} - \frac{10283850521288581}{771302070956365} a^{12} + \frac{4101807103947661}{771302070956365} a^{11} + \frac{2032017742183086}{771302070956365} a^{10} + \frac{24153565772245862}{771302070956365} a^{9} - \frac{29779279830993712}{771302070956365} a^{8} - \frac{1942855519527514}{154260414191273} a^{7} + \frac{2560696630974823}{771302070956365} a^{6} + \frac{43507988368233213}{771302070956365} a^{5} - \frac{19415223169355841}{771302070956365} a^{4} - \frac{6494617607154952}{154260414191273} a^{3} + \frac{19512405120625692}{771302070956365} a^{2} + \frac{1797833509954337}{154260414191273} a - \frac{88706294870418}{11866185707021} \) (order $4$)
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:  \( 500710.067504 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.1040.1, 10.0.1690000000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.17.30$x^{10} - 15 x^{8} + 10$$10$$1$$17$$D_{10}$$[2]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$