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Properties

Label 20.18.1630238666...4375.1
Degree $20$
Signature $[18, 1]$
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 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 320, 635, -4002, -7548, 12631, 28572, -11191, -40347, 2633, 27547, -804, -10548, 1292, 2315, -638, -246, 121, 3, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 3*x^18 + 121*x^17 - 246*x^16 - 638*x^15 + 2315*x^14 + 1292*x^13 - 10548*x^12 - 804*x^11 + 27547*x^10 + 2633*x^9 - 40347*x^8 - 11191*x^7 + 28572*x^6 + 12631*x^5 - 7548*x^4 - 4002*x^3 + 635*x^2 + 320*x - 29)
 
gp: K = bnfinit(x^20 - 8*x^19 + 3*x^18 + 121*x^17 - 246*x^16 - 638*x^15 + 2315*x^14 + 1292*x^13 - 10548*x^12 - 804*x^11 + 27547*x^10 + 2633*x^9 - 40347*x^8 - 11191*x^7 + 28572*x^6 + 12631*x^5 - 7548*x^4 - 4002*x^3 + 635*x^2 + 320*x - 29, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 3 x^{18} + 121 x^{17} - 246 x^{16} - 638 x^{15} + 2315 x^{14} + 1292 x^{13} - 10548 x^{12} - 804 x^{11} + 27547 x^{10} + 2633 x^{9} - 40347 x^{8} - 11191 x^{7} + 28572 x^{6} + 12631 x^{5} - 7548 x^{4} - 4002 x^{3} + 635 x^{2} + 320 x - 29 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 1]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{165} a^{18} + \frac{7}{55} a^{17} - \frac{10}{33} a^{16} - \frac{17}{55} a^{15} + \frac{5}{33} a^{14} + \frac{8}{55} a^{13} - \frac{3}{55} a^{12} - \frac{7}{165} a^{11} - \frac{8}{165} a^{10} - \frac{32}{165} a^{9} + \frac{14}{33} a^{8} - \frac{58}{165} a^{7} + \frac{71}{165} a^{6} + \frac{59}{165} a^{5} - \frac{18}{55} a^{4} + \frac{19}{55} a^{3} - \frac{4}{55} a^{2} - \frac{3}{55} a + \frac{68}{165}$, $\frac{1}{4987306021309116965782317105} a^{19} + \frac{458841559213990910590117}{997461204261823393156463421} a^{18} + \frac{81665632484991517232100368}{216839392230831172425318135} a^{17} - \frac{202146435816235371451012681}{4987306021309116965782317105} a^{16} + \frac{54944558255337636425585776}{4987306021309116965782317105} a^{15} - \frac{816787254278893601965282441}{4987306021309116965782317105} a^{14} + \frac{515394342224096220502613019}{1662435340436372321927439035} a^{13} - \frac{1511630963991164167596108868}{4987306021309116965782317105} a^{12} + \frac{2096909619410861179854428834}{4987306021309116965782317105} a^{11} - \frac{429272233940810514443066428}{1662435340436372321927439035} a^{10} + \frac{307396989420622163120697899}{1662435340436372321927439035} a^{9} + \frac{2450843765975878345397372662}{4987306021309116965782317105} a^{8} + \frac{559830927158422282624515213}{1662435340436372321927439035} a^{7} - \frac{735318596393396834551808544}{1662435340436372321927439035} a^{6} + \frac{61097640058072580362953149}{216839392230831172425318135} a^{5} + \frac{369317469949110568641926312}{1662435340436372321927439035} a^{4} + \frac{48234816486932346573806227}{151130485494215665629767185} a^{3} - \frac{319715357418901544416682679}{1662435340436372321927439035} a^{2} + \frac{2482014039077821770376218092}{4987306021309116965782317105} a + \frac{2044700975336784417615423847}{4987306021309116965782317105}$

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:  $18$
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:  \( 3731575983.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$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$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed