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Properties

Label 20.4.68245483672...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{22}\cdot 17^{15}$
Root discriminant $49.17$
Ramified primes $5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1696, -8870, -25885, -58815, -85930, -85670, 35660, 231495, -126510, -237210, 304481, -118470, -9225, 24775, -9890, 1860, 0, -125, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 125*x^17 + 1860*x^15 - 9890*x^14 + 24775*x^13 - 9225*x^12 - 118470*x^11 + 304481*x^10 - 237210*x^9 - 126510*x^8 + 231495*x^7 + 35660*x^6 - 85670*x^5 - 85930*x^4 - 58815*x^3 - 25885*x^2 - 8870*x - 1696)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 125*x^17 + 1860*x^15 - 9890*x^14 + 24775*x^13 - 9225*x^12 - 118470*x^11 + 304481*x^10 - 237210*x^9 - 126510*x^8 + 231495*x^7 + 35660*x^6 - 85670*x^5 - 85930*x^4 - 58815*x^3 - 25885*x^2 - 8870*x - 1696, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 125 x^{17} + 1860 x^{15} - 9890 x^{14} + 24775 x^{13} - 9225 x^{12} - 118470 x^{11} + 304481 x^{10} - 237210 x^{9} - 126510 x^{8} + 231495 x^{7} + 35660 x^{6} - 85670 x^{5} - 85930 x^{4} - 58815 x^{3} - 25885 x^{2} - 8870 x - 1696 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6824548367285289270877838134765625=5^{22}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.17$
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:  $5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{12} a^{16} - \frac{1}{6} a^{14} - \frac{1}{4} a^{13} + \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{4} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{168} a^{17} - \frac{1}{168} a^{16} - \frac{1}{12} a^{15} - \frac{1}{168} a^{14} + \frac{17}{84} a^{13} - \frac{5}{168} a^{12} - \frac{2}{21} a^{11} + \frac{3}{14} a^{10} + \frac{2}{7} a^{9} + \frac{17}{84} a^{8} + \frac{19}{168} a^{7} - \frac{5}{168} a^{6} - \frac{19}{84} a^{5} + \frac{15}{56} a^{4} - \frac{4}{21} a^{3} + \frac{19}{56} a^{2} - \frac{5}{84} a + \frac{5}{21}$, $\frac{1}{138768} a^{18} + \frac{83}{34692} a^{17} + \frac{3797}{138768} a^{16} - \frac{22723}{138768} a^{15} + \frac{5861}{138768} a^{14} + \frac{24673}{138768} a^{13} + \frac{8689}{46256} a^{12} + \frac{101}{708} a^{11} - \frac{487}{17346} a^{10} - \frac{17597}{69384} a^{9} - \frac{58603}{138768} a^{8} + \frac{7097}{23128} a^{7} - \frac{2627}{138768} a^{6} - \frac{23753}{138768} a^{5} - \frac{2921}{46256} a^{4} - \frac{4243}{138768} a^{3} - \frac{7405}{138768} a^{2} + \frac{1679}{9912} a - \frac{1894}{8673}$, $\frac{1}{2063209665730615622761040651229650955915240576} a^{19} + \frac{175869364063465238324003074444740156883}{98248079320505505845763840534745283615011456} a^{18} - \frac{1217981409450527900657463029889840857067293}{687736555243538540920346883743216985305080192} a^{17} - \frac{429049490448836781308558528359464149307709}{11995405033317532690471166576916575325088608} a^{16} - \frac{769148283753556783380736695162143952773911}{3998468344439177563490388858972191775029536} a^{15} + \frac{26126335833422123659488654920995702608037985}{257901208216326952845130081403706369489405072} a^{14} - \frac{74193479324975114583680871572658128006848975}{343868277621769270460173441871608492652540096} a^{13} - \frac{16798343678043385090063818924932623518500291}{2063209665730615622761040651229650955915240576} a^{12} + \frac{67647288321568967453450442719544696911637883}{515802416432653905690260162807412738978810144} a^{11} - \frac{10748581269710692658151025601583151410184553}{49124039660252752922881920267372641807505728} a^{10} - \frac{745191970618091172025320714600022538034618377}{2063209665730615622761040651229650955915240576} a^{9} - \frac{970358748182628491775350808345805702433266427}{2063209665730615622761040651229650955915240576} a^{8} + \frac{359107253883681571057465736829117997911297}{1767960296255883138612716924789760887673728} a^{7} - \frac{386548233836278581579761074966969919769723553}{1031604832865307811380520325614825477957620288} a^{6} - \frac{10884215361080270364259931683730661312940007}{49124039660252752922881920267372641807505728} a^{5} - \frac{55998965476945729648685681450096168130752263}{515802416432653905690260162807412738978810144} a^{4} - \frac{92913305024514959444071806114747453320727345}{343868277621769270460173441871608492652540096} a^{3} - \frac{217033715313741782833076768018184937277667303}{687736555243538540920346883743216985305080192} a^{2} + \frac{480468822709409412587464352313478128546651419}{1031604832865307811380520325614825477957620288} a - \frac{849279318316068138481580127034194115433743}{64475302054081738211282520350926592372351268}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 7366772724.975502 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.122825.1, 5.1.903125.1, 10.2.13865791015625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
17Data not computed