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Properties

Label 20.0.54025471321...6721.2
Degree $20$
Signature $[0, 10]$
Discriminant $67^{4}\cdot 401^{9}$
Root discriminant $34.41$
Ramified primes $67, 401$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -216, -90, 1155, -1079, -1407, 4879, -6078, 4159, -1091, -769, 887, -171, -230, 208, -35, -20, 20, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 + 20*x^17 - 20*x^16 - 35*x^15 + 208*x^14 - 230*x^13 - 171*x^12 + 887*x^11 - 769*x^10 - 1091*x^9 + 4159*x^8 - 6078*x^7 + 4879*x^6 - 1407*x^5 - 1079*x^4 + 1155*x^3 - 90*x^2 - 216*x + 81)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^18 + 20*x^17 - 20*x^16 - 35*x^15 + 208*x^14 - 230*x^13 - 171*x^12 + 887*x^11 - 769*x^10 - 1091*x^9 + 4159*x^8 - 6078*x^7 + 4879*x^6 - 1407*x^5 - 1079*x^4 + 1155*x^3 - 90*x^2 - 216*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{18} + 20 x^{17} - 20 x^{16} - 35 x^{15} + 208 x^{14} - 230 x^{13} - 171 x^{12} + 887 x^{11} - 769 x^{10} - 1091 x^{9} + 4159 x^{8} - 6078 x^{7} + 4879 x^{6} - 1407 x^{5} - 1079 x^{4} + 1155 x^{3} - 90 x^{2} - 216 x + 81 \)

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:  \(5402547132170496407507117146721=67^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.41$
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:  $67, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{17} + \frac{1}{3} a^{15} - \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{4}{9} a^{12} - \frac{4}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{243} a^{18} - \frac{1}{81} a^{17} - \frac{2}{81} a^{16} - \frac{37}{243} a^{15} + \frac{53}{243} a^{14} - \frac{109}{243} a^{13} + \frac{80}{243} a^{12} - \frac{61}{243} a^{11} - \frac{23}{243} a^{10} - \frac{101}{243} a^{9} + \frac{37}{243} a^{8} - \frac{13}{27} a^{7} + \frac{85}{243} a^{6} + \frac{47}{243} a^{5} - \frac{52}{243} a^{4} - \frac{17}{243} a^{3} + \frac{20}{81} a^{2} + \frac{7}{27} a - \frac{4}{9}$, $\frac{1}{4732569651268059758770184001} a^{19} - \frac{5646487685224851299134196}{4732569651268059758770184001} a^{18} - \frac{19355698698605779791578126}{525841072363117750974464889} a^{17} - \frac{653397518053900946352868126}{4732569651268059758770184001} a^{16} + \frac{1620309415712769849336141631}{4732569651268059758770184001} a^{15} - \frac{577131628605672413440770746}{4732569651268059758770184001} a^{14} - \frac{1896143576226080391580790975}{4732569651268059758770184001} a^{13} + \frac{670167690763476154538294899}{4732569651268059758770184001} a^{12} + \frac{386984327931409486335452566}{1577523217089353252923394667} a^{11} - \frac{2161303800831721015856778955}{4732569651268059758770184001} a^{10} + \frac{314093993052313259024064797}{4732569651268059758770184001} a^{9} - \frac{684328804179651226788215702}{4732569651268059758770184001} a^{8} + \frac{1814186860977793004962686949}{4732569651268059758770184001} a^{7} + \frac{184547577527154622012854200}{525841072363117750974464889} a^{6} + \frac{363688229637788573175577255}{4732569651268059758770184001} a^{5} + \frac{201759001957780745774843251}{525841072363117750974464889} a^{4} - \frac{898590767301157473699097229}{4732569651268059758770184001} a^{3} + \frac{77330651181515062665836609}{1577523217089353252923394667} a^{2} - \frac{88617672362538521548572422}{525841072363117750974464889} a - \frac{436829108835098004094330}{175280357454372583658154963}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 3010209.82045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.10.116071900626889.2

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 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed