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Properties

Label 20.10.1630238666...4375.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{8}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $40.80$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, -1890, 11231, -13526, -20774, 53552, -24944, -23706, 34562, -22964, 11216, -504, -6817, 8318, -5691, 2452, -631, 54, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 24*x^18 + 54*x^17 - 631*x^16 + 2452*x^15 - 5691*x^14 + 8318*x^13 - 6817*x^12 - 504*x^11 + 11216*x^10 - 22964*x^9 + 34562*x^8 - 23706*x^7 - 24944*x^6 + 53552*x^5 - 20774*x^4 - 13526*x^3 + 11231*x^2 - 1890*x + 45)
 
gp: K = bnfinit(x^20 - 9*x^19 + 24*x^18 + 54*x^17 - 631*x^16 + 2452*x^15 - 5691*x^14 + 8318*x^13 - 6817*x^12 - 504*x^11 + 11216*x^10 - 22964*x^9 + 34562*x^8 - 23706*x^7 - 24944*x^6 + 53552*x^5 - 20774*x^4 - 13526*x^3 + 11231*x^2 - 1890*x + 45, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 24 x^{18} + 54 x^{17} - 631 x^{16} + 2452 x^{15} - 5691 x^{14} + 8318 x^{13} - 6817 x^{12} - 504 x^{11} + 11216 x^{10} - 22964 x^{9} + 34562 x^{8} - 23706 x^{7} - 24944 x^{6} + 53552 x^{5} - 20774 x^{4} - 13526 x^{3} + 11231 x^{2} - 1890 x + 45 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-163023866649059818618082021484375=-\,3^{8}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.80$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{15} a^{15} - \frac{1}{15} a^{14} + \frac{1}{15} a^{12} - \frac{7}{15} a^{11} + \frac{2}{5} a^{10} + \frac{7}{15} a^{9} - \frac{4}{15} a^{8} + \frac{4}{15} a^{7} + \frac{2}{5} a^{6} - \frac{7}{15} a^{5} + \frac{1}{15} a^{4} - \frac{1}{15} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{16} - \frac{1}{15} a^{14} + \frac{1}{15} a^{13} - \frac{7}{15} a^{11} + \frac{1}{15} a^{10} + \frac{1}{5} a^{8} + \frac{4}{15} a^{7} - \frac{7}{15} a^{6} - \frac{2}{5} a^{5} - \frac{2}{15} a^{4} - \frac{4}{15} a^{3} + \frac{1}{5} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{17} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{7}{15} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{3} a$, $\frac{1}{165} a^{18} - \frac{4}{165} a^{17} + \frac{1}{33} a^{16} - \frac{2}{165} a^{15} - \frac{3}{55} a^{14} - \frac{4}{165} a^{13} - \frac{1}{15} a^{12} - \frac{19}{55} a^{11} - \frac{17}{55} a^{10} + \frac{3}{55} a^{9} + \frac{56}{165} a^{8} + \frac{17}{55} a^{7} + \frac{5}{33} a^{6} + \frac{32}{165} a^{5} - \frac{5}{11} a^{4} - \frac{4}{33} a^{3} + \frac{13}{55} a^{2} + \frac{16}{33} a - \frac{3}{11}$, $\frac{1}{31902300414086577430972982955} a^{19} - \frac{27794049542685886584125426}{31902300414086577430972982955} a^{18} + \frac{176266344681595667788120369}{6380460082817315486194596591} a^{17} + \frac{298386507104726292997887466}{31902300414086577430972982955} a^{16} + \frac{599644774760596634892176747}{31902300414086577430972982955} a^{15} + \frac{125163715815705114206437563}{10634100138028859143657660985} a^{14} + \frac{7001009292075133153668221}{95230747504736052032755173} a^{13} - \frac{1547435032416635778552730387}{31902300414086577430972982955} a^{12} + \frac{519002917046391984103660003}{31902300414086577430972982955} a^{11} + \frac{386695941909125263811366462}{31902300414086577430972982955} a^{10} + \frac{15578672294657325282765198298}{31902300414086577430972982955} a^{9} + \frac{1339961348326743510804554419}{31902300414086577430972982955} a^{8} + \frac{3776189424935932817392539888}{10634100138028859143657660985} a^{7} - \frac{1499755179422358072419364656}{6380460082817315486194596591} a^{6} - \frac{13935967290832627370062613084}{31902300414086577430972982955} a^{5} - \frac{3821851317656747877877667638}{10634100138028859143657660985} a^{4} - \frac{200387120959065308377036862}{966736376184441740332514635} a^{3} - \frac{6010528016378288507908643041}{31902300414086577430972982955} a^{2} + \frac{2834155112173782933599794201}{6380460082817315486194596591} a - \frac{216574667185691742899865126}{2126820027605771828731532197}$

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:  $14$
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:  \( 1393819795.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.5975.1, 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$
239Data not computed