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Properties

Label 20.0.32289424279...7849.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{18}\cdot 11^{6}\cdot 19^{6}$
Root discriminant $13.35$
Ramified primes $3, 11, 19$
Class number $1$
Class group Trivial
Galois group $C_2\times S_5$ (as 20T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -27, 36, -27, 18, -9, 0, 9, -9, 0, 6, 0, -6, 0, 9, -3, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^15 + 9*x^14 - 6*x^12 + 6*x^10 - 9*x^8 + 9*x^7 - 9*x^5 + 18*x^4 - 27*x^3 + 36*x^2 - 27*x + 9)
 
gp: K = bnfinit(x^20 - 3*x^15 + 9*x^14 - 6*x^12 + 6*x^10 - 9*x^8 + 9*x^7 - 9*x^5 + 18*x^4 - 27*x^3 + 36*x^2 - 27*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{15} + 9 x^{14} - 6 x^{12} + 6 x^{10} - 9 x^{8} + 9 x^{7} - 9 x^{5} + 18 x^{4} - 27 x^{3} + 36 x^{2} - 27 x + 9 \)

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:  \(32289424279912035447849=3^{18}\cdot 11^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.35$
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:  $3, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{3} a^{13}$, $\frac{1}{3} a^{14}$, $\frac{1}{3} a^{15}$, $\frac{1}{3} a^{16}$, $\frac{1}{33} a^{17} + \frac{2}{33} a^{16} + \frac{2}{33} a^{15} - \frac{5}{33} a^{14} - \frac{2}{33} a^{13} + \frac{4}{33} a^{12} + \frac{2}{33} a^{11} - \frac{5}{33} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} + \frac{2}{11} a^{7} + \frac{3}{11} a^{6} + \frac{4}{11} a^{5} - \frac{5}{11} a^{4} - \frac{2}{11} a^{2} + \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{4059} a^{18} - \frac{2}{451} a^{17} - \frac{58}{451} a^{16} + \frac{172}{1353} a^{15} - \frac{16}{451} a^{14} - \frac{5}{41} a^{13} - \frac{125}{1353} a^{12} + \frac{62}{1353} a^{11} - \frac{1}{11} a^{10} + \frac{146}{451} a^{9} + \frac{14}{123} a^{8} - \frac{5}{451} a^{7} - \frac{125}{451} a^{6} - \frac{120}{451} a^{5} - \frac{84}{451} a^{4} + \frac{80}{451} a^{3} + \frac{213}{451} a^{2} + \frac{192}{451} a + \frac{194}{451}$, $\frac{1}{3705867} a^{19} - \frac{433}{3705867} a^{18} - \frac{7196}{1235289} a^{17} + \frac{140401}{1235289} a^{16} + \frac{66947}{1235289} a^{15} - \frac{115340}{1235289} a^{14} + \frac{10253}{1235289} a^{13} - \frac{187708}{1235289} a^{12} - \frac{95389}{1235289} a^{11} + \frac{38727}{411763} a^{10} + \frac{265612}{1235289} a^{9} + \frac{493634}{1235289} a^{8} + \frac{72675}{411763} a^{7} - \frac{177025}{411763} a^{6} - \frac{79434}{411763} a^{5} + \frac{13896}{37433} a^{4} + \frac{113588}{411763} a^{3} - \frac{2517}{37433} a^{2} - \frac{202814}{411763} a + \frac{173198}{411763}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{7514}{30129} a^{19} - \frac{1645}{10043} a^{18} - \frac{768}{10043} a^{17} + \frac{212}{10043} a^{16} + \frac{3077}{30129} a^{15} + \frac{8640}{10043} a^{14} - \frac{16710}{10043} a^{13} - \frac{11964}{10043} a^{12} + \frac{7774}{10043} a^{11} + \frac{9219}{10043} a^{10} - \frac{3350}{10043} a^{9} - \frac{2946}{10043} a^{8} + \frac{19047}{10043} a^{7} - \frac{9069}{10043} a^{6} - \frac{6327}{10043} a^{5} + \frac{1605}{913} a^{4} - \frac{28716}{10043} a^{3} + \frac{4311}{913} a^{2} - \frac{59049}{10043} a + \frac{32584}{10043} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2749.61967806 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.2.59897527569.1

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 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$