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Properties

Label 20.20.3240323833...5401.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{4}\cdot 61^{2}\cdot 401^{10}$
Root discriminant $37.63$
Ramified primes $3, 61, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 19, 54, -725, -5039, -9501, 6152, 41423, 38071, -18016, -40546, -4955, 15997, 4417, -3328, -1026, 412, 104, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 30*x^18 + 104*x^17 + 412*x^16 - 1026*x^15 - 3328*x^14 + 4417*x^13 + 15997*x^12 - 4955*x^11 - 40546*x^10 - 18016*x^9 + 38071*x^8 + 41423*x^7 + 6152*x^6 - 9501*x^5 - 5039*x^4 - 725*x^3 + 54*x^2 + 19*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 30*x^18 + 104*x^17 + 412*x^16 - 1026*x^15 - 3328*x^14 + 4417*x^13 + 15997*x^12 - 4955*x^11 - 40546*x^10 - 18016*x^9 + 38071*x^8 + 41423*x^7 + 6152*x^6 - 9501*x^5 - 5039*x^4 - 725*x^3 + 54*x^2 + 19*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 30 x^{18} + 104 x^{17} + 412 x^{16} - 1026 x^{15} - 3328 x^{14} + 4417 x^{13} + 15997 x^{12} - 4955 x^{11} - 40546 x^{10} - 18016 x^{9} + 38071 x^{8} + 41423 x^{7} + 6152 x^{6} - 9501 x^{5} - 5039 x^{4} - 725 x^{3} + 54 x^{2} + 19 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32403238330091474061186973105401=3^{4}\cdot 61^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.63$
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:  $3, 61, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{5}{13} a^{16} - \frac{2}{13} a^{15} + \frac{2}{13} a^{14} + \frac{1}{13} a^{13} - \frac{2}{13} a^{11} + \frac{6}{13} a^{10} + \frac{4}{13} a^{9} - \frac{5}{13} a^{8} - \frac{2}{13} a^{7} - \frac{5}{13} a^{6} + \frac{4}{13} a^{5} - \frac{6}{13} a^{4} + \frac{6}{13} a^{3} + \frac{6}{13} a^{2} + \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{39} a^{18} - \frac{1}{39} a^{17} + \frac{7}{39} a^{16} + \frac{1}{39} a^{15} + \frac{2}{39} a^{14} + \frac{7}{39} a^{13} - \frac{2}{39} a^{12} + \frac{5}{39} a^{11} - \frac{19}{39} a^{10} - \frac{1}{13} a^{9} - \frac{11}{39} a^{8} - \frac{19}{39} a^{7} + \frac{8}{39} a^{6} - \frac{17}{39} a^{5} + \frac{16}{39} a^{4} - \frac{17}{39} a^{3} - \frac{7}{39} a^{2} - \frac{19}{39} a - \frac{7}{39}$, $\frac{1}{28285683977053188153} a^{19} + \frac{250974559150372051}{28285683977053188153} a^{18} + \frac{337803914682296471}{28285683977053188153} a^{17} + \frac{1254616896286407865}{3142853775228132017} a^{16} - \frac{5813595055562349449}{28285683977053188153} a^{15} - \frac{13519298330686120702}{28285683977053188153} a^{14} - \frac{2085156401856718538}{9428561325684396051} a^{13} + \frac{4898407043115220468}{28285683977053188153} a^{12} + \frac{666464971107869113}{9428561325684396051} a^{11} - \frac{10255595106920592581}{28285683977053188153} a^{10} + \frac{3229958930541501268}{28285683977053188153} a^{9} + \frac{10012133616730168756}{28285683977053188153} a^{8} + \frac{1291019000384884713}{3142853775228132017} a^{7} - \frac{6934802864022672016}{28285683977053188153} a^{6} - \frac{2067570176441042698}{9428561325684396051} a^{5} - \frac{2592747745838850070}{9428561325684396051} a^{4} - \frac{290196671245516760}{28285683977053188153} a^{3} + \frac{741743029429750324}{9428561325684396051} a^{2} + \frac{3513983300153442629}{9428561325684396051} a - \frac{13733370920560688330}{28285683977053188153}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2239930591.43 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

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

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.5692384239498549.1, 10.10.14195471918949.1, 10.10.10368641602001.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$61$61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed