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Properties

Label 20.0.62618189972...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 19^{4}\cdot 1699^{4}$
Root discriminant $30.89$
Ramified primes $3, 5, 19, 1699$
Class number $31$ (GRH)
Class group $[31]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T100)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 53, -148, 393, -217, 1026, -391, 1546, -259, 1424, -261, 835, -131, 334, -62, 84, -12, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 13*x^18 - 12*x^17 + 84*x^16 - 62*x^15 + 334*x^14 - 131*x^13 + 835*x^12 - 261*x^11 + 1424*x^10 - 259*x^9 + 1546*x^8 - 391*x^7 + 1026*x^6 - 217*x^5 + 393*x^4 - 148*x^3 + 53*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 13*x^18 - 12*x^17 + 84*x^16 - 62*x^15 + 334*x^14 - 131*x^13 + 835*x^12 - 261*x^11 + 1424*x^10 - 259*x^9 + 1546*x^8 - 391*x^7 + 1026*x^6 - 217*x^5 + 393*x^4 - 148*x^3 + 53*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 13 x^{18} - 12 x^{17} + 84 x^{16} - 62 x^{15} + 334 x^{14} - 131 x^{13} + 835 x^{12} - 261 x^{11} + 1424 x^{10} - 259 x^{9} + 1546 x^{8} - 391 x^{7} + 1026 x^{6} - 217 x^{5} + 393 x^{4} - 148 x^{3} + 53 x^{2} - 8 x + 1 \)

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:  \(626181899727703906518837890625=3^{10}\cdot 5^{10}\cdot 19^{4}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.89$
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:  $3, 5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{32281} a^{18} - \frac{7057}{32281} a^{17} + \frac{2845}{32281} a^{16} - \frac{8859}{32281} a^{15} + \frac{3845}{32281} a^{14} - \frac{439}{32281} a^{13} + \frac{2012}{32281} a^{12} + \frac{15593}{32281} a^{11} - \frac{5628}{32281} a^{10} + \frac{7398}{32281} a^{9} - \frac{6562}{32281} a^{8} - \frac{10977}{32281} a^{7} + \frac{6861}{32281} a^{6} + \frac{5670}{32281} a^{5} - \frac{4398}{32281} a^{4} + \frac{15592}{32281} a^{3} + \frac{6805}{32281} a^{2} + \frac{7174}{32281} a + \frac{9042}{32281}$, $\frac{1}{46233739719000293} a^{19} - \frac{57517685107}{46233739719000293} a^{18} - \frac{491141727803487}{2433354722052647} a^{17} - \frac{12219458365572879}{46233739719000293} a^{16} - \frac{22129964734650595}{46233739719000293} a^{15} - \frac{22484070062905301}{46233739719000293} a^{14} + \frac{5414170351347896}{46233739719000293} a^{13} - \frac{23504473693254}{2433354722052647} a^{12} - \frac{3695431510719062}{46233739719000293} a^{11} + \frac{544913584295067}{2433354722052647} a^{10} - \frac{20072431248245036}{46233739719000293} a^{9} + \frac{3461345774137186}{46233739719000293} a^{8} + \frac{14978310317575641}{46233739719000293} a^{7} - \frac{3075434014886573}{46233739719000293} a^{6} + \frac{16765572826119122}{46233739719000293} a^{5} - \frac{4630776773835344}{46233739719000293} a^{4} - \frac{1286077094050494}{46233739719000293} a^{3} - \frac{19375250473254905}{46233739719000293} a^{2} - \frac{7869787312611392}{46233739719000293} a - \frac{302546129091893}{46233739719000293}$

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

Class group and class number

$C_{31}$, which has order $31$ (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:  \( -\frac{6140469551760967}{46233739719000293} a^{19} + \frac{12041133273761351}{46233739719000293} a^{18} - \frac{79747152442969007}{46233739719000293} a^{17} + \frac{71275317646564587}{46233739719000293} a^{16} - \frac{517827133234206935}{46233739719000293} a^{15} + \frac{363905742148279267}{46233739719000293} a^{14} - \frac{2067140043822169222}{46233739719000293} a^{13} + \frac{739849691625530459}{46233739719000293} a^{12} - \frac{5215077144063729038}{46233739719000293} a^{11} + \frac{1415996519061374678}{46233739719000293} a^{10} - \frac{8970679599615864647}{46233739719000293} a^{9} + \frac{1259424950078825444}{46233739719000293} a^{8} - \frac{9902186172725148888}{46233739719000293} a^{7} + \frac{1963753166845278761}{46233739719000293} a^{6} - \frac{6685547166356904043}{46233739719000293} a^{5} + \frac{1060600226570998709}{46233739719000293} a^{4} - \frac{2632632099092964580}{46233739719000293} a^{3} + \frac{726570397914715064}{46233739719000293} a^{2} - \frac{372695086362260451}{46233739719000293} a + \frac{56630305275134303}{46233739719000293} \) (order $6$)
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:  \( 351148.02443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T100):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
3Data not computed
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
1699Data not computed