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Properties

Label 20.20.2384639853...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 23^{8}\cdot 89^{10}$
Root discriminant $73.94$
Ramified primes $5, 23, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_5\wr C_2$ (as 20T55)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, -2310, 38629, -135903, 38154, 712238, -1649718, 1375989, 78546, -917700, 488062, 97727, -162275, 27381, 17832, -6513, -416, 439, -33, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 33*x^18 + 439*x^17 - 416*x^16 - 6513*x^15 + 17832*x^14 + 27381*x^13 - 162275*x^12 + 97727*x^11 + 488062*x^10 - 917700*x^9 + 78546*x^8 + 1375989*x^7 - 1649718*x^6 + 712238*x^5 + 38154*x^4 - 135903*x^3 + 38629*x^2 - 2310*x - 121)
 
gp: K = bnfinit(x^20 - 8*x^19 - 33*x^18 + 439*x^17 - 416*x^16 - 6513*x^15 + 17832*x^14 + 27381*x^13 - 162275*x^12 + 97727*x^11 + 488062*x^10 - 917700*x^9 + 78546*x^8 + 1375989*x^7 - 1649718*x^6 + 712238*x^5 + 38154*x^4 - 135903*x^3 + 38629*x^2 - 2310*x - 121, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 33 x^{18} + 439 x^{17} - 416 x^{16} - 6513 x^{15} + 17832 x^{14} + 27381 x^{13} - 162275 x^{12} + 97727 x^{11} + 488062 x^{10} - 917700 x^{9} + 78546 x^{8} + 1375989 x^{7} - 1649718 x^{6} + 712238 x^{5} + 38154 x^{4} - 135903 x^{3} + 38629 x^{2} - 2310 x - 121 \)

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:  \(23846398539764122605523970477353515625=5^{10}\cdot 23^{8}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.94$
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:  $5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{2}{11} a^{15} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{2}{11} a^{12} - \frac{1}{11} a^{11} - \frac{4}{11} a^{10} - \frac{4}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{3}{11} a^{6} + \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{4}{11} a^{3} - \frac{4}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{11} a^{17} + \frac{2}{11} a^{15} - \frac{2}{11} a^{14} + \frac{3}{11} a^{12} - \frac{2}{11} a^{11} + \frac{4}{11} a^{10} - \frac{2}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} + \frac{5}{11} a$, $\frac{1}{605} a^{18} + \frac{23}{605} a^{17} - \frac{24}{605} a^{16} + \frac{278}{605} a^{15} + \frac{238}{605} a^{14} + \frac{128}{605} a^{13} + \frac{53}{605} a^{12} + \frac{72}{605} a^{11} + \frac{19}{121} a^{10} + \frac{14}{55} a^{9} + \frac{201}{605} a^{8} + \frac{18}{121} a^{7} + \frac{167}{605} a^{6} - \frac{269}{605} a^{5} - \frac{45}{121} a^{4} - \frac{101}{605} a^{3} + \frac{208}{605} a^{2} - \frac{106}{605} a - \frac{14}{55}$, $\frac{1}{43761384866925207176226968359825} a^{19} - \frac{3269467663619536295403927422}{8752276973385041435245393671965} a^{18} - \frac{1982234146925487233468699168743}{43761384866925207176226968359825} a^{17} - \frac{6008233186236662559123640433}{8752276973385041435245393671965} a^{16} + \frac{11293478213218218193215172832159}{43761384866925207176226968359825} a^{15} + \frac{11433850489678872469277368717404}{43761384866925207176226968359825} a^{14} + \frac{8872502414202738881839013025484}{43761384866925207176226968359825} a^{13} - \frac{1337554418187417950578984952627}{43761384866925207176226968359825} a^{12} + \frac{17541564665385367305334956427164}{43761384866925207176226968359825} a^{11} + \frac{6189503502239965454778705754364}{43761384866925207176226968359825} a^{10} - \frac{114183230834854266745732847266}{43761384866925207176226968359825} a^{9} + \frac{10168743525659637653127872669162}{43761384866925207176226968359825} a^{8} - \frac{7504543971128728784287519600808}{43761384866925207176226968359825} a^{7} - \frac{262053420704078965402904706644}{8752276973385041435245393671965} a^{6} - \frac{4380809453213947802392967553963}{43761384866925207176226968359825} a^{5} - \frac{9282210868518431899797080747316}{43761384866925207176226968359825} a^{4} + \frac{6039974209693234031532961084036}{43761384866925207176226968359825} a^{3} - \frac{3813864024678370376440754238404}{8752276973385041435245393671965} a^{2} - \frac{8643609340497646666829704821571}{43761384866925207176226968359825} a - \frac{1780823028362169637945581693258}{3978307715175018834202451669075}$

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:  $19$
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:  \( 2155949009410 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T55):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{445}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 10.10.4883277438336278125.1

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 25 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.10.8.1$x^{10} - 23 x^{5} + 3703$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
89Data not computed