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Properties

Label 20.16.5165883938...8125.2
Degree $20$
Signature $[16, 2]$
Discriminant $3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $34.33$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79, 4077, -10753, -4181, 33336, -14280, -34369, 28511, 10737, -18357, 3172, 4187, -2333, 269, 221, -270, 81, 41, -18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 18*x^18 + 41*x^17 + 81*x^16 - 270*x^15 + 221*x^14 + 269*x^13 - 2333*x^12 + 4187*x^11 + 3172*x^10 - 18357*x^9 + 10737*x^8 + 28511*x^7 - 34369*x^6 - 14280*x^5 + 33336*x^4 - 4181*x^3 - 10753*x^2 + 4077*x - 79)
 
gp: K = bnfinit(x^20 - 2*x^19 - 18*x^18 + 41*x^17 + 81*x^16 - 270*x^15 + 221*x^14 + 269*x^13 - 2333*x^12 + 4187*x^11 + 3172*x^10 - 18357*x^9 + 10737*x^8 + 28511*x^7 - 34369*x^6 - 14280*x^5 + 33336*x^4 - 4181*x^3 - 10753*x^2 + 4077*x - 79, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 18 x^{18} + 41 x^{17} + 81 x^{16} - 270 x^{15} + 221 x^{14} + 269 x^{13} - 2333 x^{12} + 4187 x^{11} + 3172 x^{10} - 18357 x^{9} + 10737 x^{8} + 28511 x^{7} - 34369 x^{6} - 14280 x^{5} + 33336 x^{4} - 4181 x^{3} - 10753 x^{2} + 4077 x - 79 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5165883938831637970733642578125=3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.33$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15} a^{16} - \frac{4}{15} a^{15} - \frac{4}{15} a^{14} + \frac{2}{15} a^{13} - \frac{2}{5} a^{12} - \frac{7}{15} a^{10} - \frac{1}{15} a^{9} + \frac{1}{5} a^{8} + \frac{1}{15} a^{7} - \frac{7}{15} a^{6} + \frac{4}{15} a^{4} - \frac{2}{15} a^{3} + \frac{1}{15} a^{2} - \frac{1}{15} a - \frac{4}{15}$, $\frac{1}{15} a^{17} - \frac{1}{3} a^{15} + \frac{1}{15} a^{14} + \frac{2}{15} a^{13} + \frac{2}{5} a^{12} - \frac{7}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{15} a^{9} - \frac{2}{15} a^{8} - \frac{1}{5} a^{7} + \frac{2}{15} a^{6} + \frac{4}{15} a^{5} - \frac{1}{15} a^{4} - \frac{7}{15} a^{3} + \frac{1}{5} a^{2} + \frac{7}{15} a - \frac{1}{15}$, $\frac{1}{15} a^{18} - \frac{4}{15} a^{15} - \frac{1}{5} a^{14} + \frac{1}{15} a^{13} - \frac{7}{15} a^{12} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} - \frac{7}{15} a^{9} - \frac{1}{5} a^{8} + \frac{7}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{15} a^{5} - \frac{2}{15} a^{4} - \frac{7}{15} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{3}$, $\frac{1}{28409030311481365783784519925} a^{19} - \frac{29295287442053398231743491}{5681806062296273156756903985} a^{18} - \frac{15162324368956683766380283}{916420332628431154315629675} a^{17} - \frac{1872071263759643867281654}{5681806062296273156756903985} a^{16} - \frac{12346445587726519214886010994}{28409030311481365783784519925} a^{15} - \frac{11388665490329373635701862768}{28409030311481365783784519925} a^{14} - \frac{243684683904795778303142474}{631311784699585906306322665} a^{13} + \frac{13691916161205716046892660934}{28409030311481365783784519925} a^{12} - \frac{547866579250114281129180556}{1893935354098757718918967995} a^{11} - \frac{6799101974622368239467292528}{28409030311481365783784519925} a^{10} + \frac{4122739526132421026319647347}{9469676770493788594594839975} a^{9} + \frac{186371400287098269307255550}{378787070819751543783793599} a^{8} + \frac{1293592320576630279724953278}{3156558923497929531531613325} a^{7} + \frac{37758496700315418172390459}{5681806062296273156756903985} a^{6} - \frac{10672362131039840752661490749}{28409030311481365783784519925} a^{5} - \frac{4486104775420922143132849808}{28409030311481365783784519925} a^{4} - \frac{599931498591634310390322817}{5681806062296273156756903985} a^{3} + \frac{10589493973990340317063284719}{28409030311481365783784519925} a^{2} - \frac{438759409703370329773694249}{1136361212459254631351380797} a + \frac{1394995754311579993042965172}{28409030311481365783784519925}$

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:  $17$
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:  \( 366183486.951 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$