↑

Properties

Label 20.8.48960645842...8608.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 13^{14}\cdot 17^{9}$
Root discriminant $43.10$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-299, 5122, 19227, 7332, -42341, -45500, 18342, 39476, -2467, -20880, -1905, 7250, 1035, -1462, -148, 86, 7, 26, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 5*x^18 + 26*x^17 + 7*x^16 + 86*x^15 - 148*x^14 - 1462*x^13 + 1035*x^12 + 7250*x^11 - 1905*x^10 - 20880*x^9 - 2467*x^8 + 39476*x^7 + 18342*x^6 - 45500*x^5 - 42341*x^4 + 7332*x^3 + 19227*x^2 + 5122*x - 299)
 
gp: K = bnfinit(x^20 - 4*x^19 - 5*x^18 + 26*x^17 + 7*x^16 + 86*x^15 - 148*x^14 - 1462*x^13 + 1035*x^12 + 7250*x^11 - 1905*x^10 - 20880*x^9 - 2467*x^8 + 39476*x^7 + 18342*x^6 - 45500*x^5 - 42341*x^4 + 7332*x^3 + 19227*x^2 + 5122*x - 299, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 5 x^{18} + 26 x^{17} + 7 x^{16} + 86 x^{15} - 148 x^{14} - 1462 x^{13} + 1035 x^{12} + 7250 x^{11} - 1905 x^{10} - 20880 x^{9} - 2467 x^{8} + 39476 x^{7} + 18342 x^{6} - 45500 x^{5} - 42341 x^{4} + 7332 x^{3} + 19227 x^{2} + 5122 x - 299 \)

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:  \(489606458428108592552566464708608=2^{20}\cdot 13^{14}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.10$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{295} a^{18} - \frac{67}{295} a^{17} + \frac{113}{295} a^{16} - \frac{52}{295} a^{15} + \frac{139}{295} a^{14} - \frac{9}{59} a^{13} - \frac{10}{59} a^{12} - \frac{117}{295} a^{11} - \frac{24}{295} a^{10} - \frac{2}{295} a^{9} - \frac{67}{295} a^{8} + \frac{102}{295} a^{7} - \frac{82}{295} a^{6} - \frac{99}{295} a^{5} - \frac{11}{59} a^{4} + \frac{132}{295} a^{3} + \frac{73}{295} a^{2} + \frac{102}{295} a + \frac{22}{295}$, $\frac{1}{159383250388491768953775115018804205} a^{19} - \frac{249007551070075852345323357423711}{159383250388491768953775115018804205} a^{18} + \frac{30897927652791482775715625439774811}{159383250388491768953775115018804205} a^{17} + \frac{7886929648302204344151533834270846}{159383250388491768953775115018804205} a^{16} - \frac{10146637625336888791374513624578118}{159383250388491768953775115018804205} a^{15} + \frac{36765689245766334985618229116194614}{159383250388491768953775115018804205} a^{14} - \frac{3323336367177804540472428546695130}{31876650077698353790755023003760841} a^{13} + \frac{77236360251393399167210942076716893}{159383250388491768953775115018804205} a^{12} + \frac{75826228604062399463027345580008734}{159383250388491768953775115018804205} a^{11} + \frac{74340604463314728902343962752295579}{159383250388491768953775115018804205} a^{10} + \frac{8841986212782267162318389934291951}{159383250388491768953775115018804205} a^{9} + \frac{2317375613226385845658262039366058}{31876650077698353790755023003760841} a^{8} - \frac{2163616738449802935390141339263262}{31876650077698353790755023003760841} a^{7} - \frac{6599392486779421562437451673404351}{159383250388491768953775115018804205} a^{6} + \frac{76933573200830132445776155180037926}{159383250388491768953775115018804205} a^{5} - \frac{42083446120600533336490477403077393}{159383250388491768953775115018804205} a^{4} - \frac{9383509548100044127349344299681102}{31876650077698353790755023003760841} a^{3} - \frac{4556256725582101387953929202546884}{31876650077698353790755023003760841} a^{2} - \frac{59478303684536484601032016967898701}{159383250388491768953775115018804205} a - \frac{75483947941726175893370269917349968}{159383250388491768953775115018804205}$

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:  \( 918439849.453 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$