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Properties

Label 20.8.45534314555...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 419^{4}\cdot 695771^{2}$
Root discriminant $42.95$
Ramified primes $5, 419, 695771$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, -10493, 57603, 111707, 12935, 20653, 91210, 43229, 22346, 29987, 13632, 6397, 3744, 980, 412, 68, -50, -4, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 3*x^18 - 4*x^17 - 50*x^16 + 68*x^15 + 412*x^14 + 980*x^13 + 3744*x^12 + 6397*x^11 + 13632*x^10 + 29987*x^9 + 22346*x^8 + 43229*x^7 + 91210*x^6 + 20653*x^5 + 12935*x^4 + 111707*x^3 + 57603*x^2 - 10493*x - 29)
 
gp: K = bnfinit(x^20 - 4*x^19 - 3*x^18 - 4*x^17 - 50*x^16 + 68*x^15 + 412*x^14 + 980*x^13 + 3744*x^12 + 6397*x^11 + 13632*x^10 + 29987*x^9 + 22346*x^8 + 43229*x^7 + 91210*x^6 + 20653*x^5 + 12935*x^4 + 111707*x^3 + 57603*x^2 - 10493*x - 29, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 3 x^{18} - 4 x^{17} - 50 x^{16} + 68 x^{15} + 412 x^{14} + 980 x^{13} + 3744 x^{12} + 6397 x^{11} + 13632 x^{10} + 29987 x^{9} + 22346 x^{8} + 43229 x^{7} + 91210 x^{6} + 20653 x^{5} + 12935 x^{4} + 111707 x^{3} + 57603 x^{2} - 10493 x - 29 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(455343145550142575255157470703125=5^{15}\cdot 419^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.95$
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:  $5, 419, 695771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{15} + \frac{2}{5} a^{12} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{4183327963786284332749686323820367550091456765955} a^{19} - \frac{207909065342765380147388197887536826222552298793}{4183327963786284332749686323820367550091456765955} a^{18} - \frac{174483817596332244001124367398697443131659118246}{4183327963786284332749686323820367550091456765955} a^{17} + \frac{167366939335193639129791786465978750201517963604}{4183327963786284332749686323820367550091456765955} a^{16} + \frac{115701473724027054136388755264341556414945048568}{836665592757256866549937264764073510018291353191} a^{15} - \frac{1917787451656147176309327638724574486457786691442}{4183327963786284332749686323820367550091456765955} a^{14} + \frac{1956460265832077138068294779574517625064040697458}{4183327963786284332749686323820367550091456765955} a^{13} - \frac{263988183412024815523401977645973646497262163549}{836665592757256866549937264764073510018291353191} a^{12} + \frac{1614714596661152692104116069563756845031296804499}{4183327963786284332749686323820367550091456765955} a^{11} + \frac{51167809265691225382413796544673536323615981527}{4183327963786284332749686323820367550091456765955} a^{10} + \frac{2075420364601455733892115458812166570398424166103}{4183327963786284332749686323820367550091456765955} a^{9} + \frac{226753676610078654754036790610543496639167466477}{4183327963786284332749686323820367550091456765955} a^{8} - \frac{619495935196541737956919069794328950220050056437}{4183327963786284332749686323820367550091456765955} a^{7} + \frac{1631297229113857038961233162597887083920652376704}{4183327963786284332749686323820367550091456765955} a^{6} - \frac{1676508500512295642286849600793455610382020856087}{4183327963786284332749686323820367550091456765955} a^{5} + \frac{1202841383720841442867979838280383862665132327457}{4183327963786284332749686323820367550091456765955} a^{4} - \frac{1512419120449616123333927819130023843688520919176}{4183327963786284332749686323820367550091456765955} a^{3} - \frac{1631300890431929099335931839261875162522649816476}{4183327963786284332749686323820367550091456765955} a^{2} + \frac{209426557225639511458313496778260726136769166643}{4183327963786284332749686323820367550091456765955} a - \frac{129260591329275894622723344359940232001760021431}{4183327963786284332749686323820367550091456765955}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 223065882.192 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
419Data not computed
695771Data not computed