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Properties

Label 20.8.41153942158...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{24}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $42.73$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-89, -7014, -33935, 133676, -165385, 48668, 113292, -186836, 158821, -91450, 33235, 302, -11393, 9874, -5312, 2090, -569, 62, 19, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 19*x^18 + 62*x^17 - 569*x^16 + 2090*x^15 - 5312*x^14 + 9874*x^13 - 11393*x^12 + 302*x^11 + 33235*x^10 - 91450*x^9 + 158821*x^8 - 186836*x^7 + 113292*x^6 + 48668*x^5 - 165385*x^4 + 133676*x^3 - 33935*x^2 - 7014*x - 89)
 
gp: K = bnfinit(x^20 - 8*x^19 + 19*x^18 + 62*x^17 - 569*x^16 + 2090*x^15 - 5312*x^14 + 9874*x^13 - 11393*x^12 + 302*x^11 + 33235*x^10 - 91450*x^9 + 158821*x^8 - 186836*x^7 + 113292*x^6 + 48668*x^5 - 165385*x^4 + 133676*x^3 - 33935*x^2 - 7014*x - 89, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 19 x^{18} + 62 x^{17} - 569 x^{16} + 2090 x^{15} - 5312 x^{14} + 9874 x^{13} - 11393 x^{12} + 302 x^{11} + 33235 x^{10} - 91450 x^{9} + 158821 x^{8} - 186836 x^{7} + 113292 x^{6} + 48668 x^{5} - 165385 x^{4} + 133676 x^{3} - 33935 x^{2} - 7014 x - 89 \)

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:  \(411539421581712055500800000000000=2^{24}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.73$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{17} - \frac{2}{5} a^{16} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{205436202852233138594930849228817719335366025} a^{19} + \frac{859264637704805817494448072534496088939459}{41087240570446627718986169845763543867073205} a^{18} - \frac{95324989432471736994111331927024148588155271}{205436202852233138594930849228817719335366025} a^{17} - \frac{15473582532888167160438769992623473605738176}{205436202852233138594930849228817719335366025} a^{16} + \frac{53834063191663923910009891605111137521490703}{205436202852233138594930849228817719335366025} a^{15} + \frac{83860250510205199502897290371738381419540024}{205436202852233138594930849228817719335366025} a^{14} - \frac{12273012647421981825009934093734946903323298}{41087240570446627718986169845763543867073205} a^{13} - \frac{34828901796483106141378721590478243195309346}{205436202852233138594930849228817719335366025} a^{12} + \frac{77403434789846385156302692557552212642993194}{205436202852233138594930849228817719335366025} a^{11} + \frac{70378530550885520936307604169980145520089534}{205436202852233138594930849228817719335366025} a^{10} + \frac{101436869268619823338840003119661589334039162}{205436202852233138594930849228817719335366025} a^{9} + \frac{22104835454476313119397024472258855912727986}{205436202852233138594930849228817719335366025} a^{8} - \frac{5057294524948728463118134671974783817849496}{205436202852233138594930849228817719335366025} a^{7} + \frac{36638522984768118023856529817230075947692326}{205436202852233138594930849228817719335366025} a^{6} + \frac{20035314426647392546880507454737207180499244}{41087240570446627718986169845763543867073205} a^{5} - \frac{26081945846363133072217250464497521858810622}{205436202852233138594930849228817719335366025} a^{4} - \frac{47392563502884903747615405011215272770290201}{205436202852233138594930849228817719335366025} a^{3} + \frac{62480200101765160974447746713921037279590348}{205436202852233138594930849228817719335366025} a^{2} + \frac{29628564232860069288469971228091191219934109}{205436202852233138594930849228817719335366025} a + \frac{236920260470320147390975370969792804044392}{2308271942159922905561020777851884486914225}$

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:  $13$
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:  \( 663413486.471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed