↑

Properties

Label 20.12.1983378117...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{6}\cdot 5^{15}\cdot 6329^{5}$
Root discriminant $36.72$
Ramified primes $2, 5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-109, -293, 7458, 17125, 8862, 6667, -4137, -13731, -10397, -3792, 4264, 5058, 1053, -881, -867, -123, 197, 45, -22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 22*x^18 + 45*x^17 + 197*x^16 - 123*x^15 - 867*x^14 - 881*x^13 + 1053*x^12 + 5058*x^11 + 4264*x^10 - 3792*x^9 - 10397*x^8 - 13731*x^7 - 4137*x^6 + 6667*x^5 + 8862*x^4 + 17125*x^3 + 7458*x^2 - 293*x - 109)
 
gp: K = bnfinit(x^20 - 3*x^19 - 22*x^18 + 45*x^17 + 197*x^16 - 123*x^15 - 867*x^14 - 881*x^13 + 1053*x^12 + 5058*x^11 + 4264*x^10 - 3792*x^9 - 10397*x^8 - 13731*x^7 - 4137*x^6 + 6667*x^5 + 8862*x^4 + 17125*x^3 + 7458*x^2 - 293*x - 109, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 22 x^{18} + 45 x^{17} + 197 x^{16} - 123 x^{15} - 867 x^{14} - 881 x^{13} + 1053 x^{12} + 5058 x^{11} + 4264 x^{10} - 3792 x^{9} - 10397 x^{8} - 13731 x^{7} - 4137 x^{6} + 6667 x^{5} + 8862 x^{4} + 17125 x^{3} + 7458 x^{2} - 293 x - 109 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19833781175906177048828125000000=2^{6}\cdot 5^{15}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.72$
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:  $2, 5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5}$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{15} + \frac{1}{25} a^{14} + \frac{2}{25} a^{13} + \frac{2}{25} a^{12} + \frac{1}{25} a^{11} + \frac{12}{25} a^{10} - \frac{8}{25} a^{9} - \frac{4}{25} a^{8} - \frac{8}{25} a^{7} + \frac{12}{25} a^{6} - \frac{9}{25} a^{5} - \frac{3}{25} a^{4} - \frac{8}{25} a^{3} - \frac{4}{25} a^{2} + \frac{1}{25} a + \frac{11}{25}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{14} - \frac{1}{25} a^{12} + \frac{11}{25} a^{11} + \frac{1}{5} a^{10} + \frac{4}{25} a^{9} - \frac{4}{25} a^{8} - \frac{1}{5} a^{7} + \frac{4}{25} a^{6} + \frac{6}{25} a^{5} - \frac{1}{5} a^{4} + \frac{4}{25} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{11}{25}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{15} - \frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{1}{5} a^{11} + \frac{4}{25} a^{10} + \frac{11}{25} a^{9} - \frac{1}{5} a^{8} + \frac{4}{25} a^{7} - \frac{4}{25} a^{6} - \frac{1}{5} a^{5} + \frac{4}{25} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{11}{25} a - \frac{2}{5}$, $\frac{1}{8082067507528606452216915153475} a^{19} - \frac{548212125336360157298847943}{1616413501505721290443383030695} a^{18} - \frac{27130754787938824364783916211}{8082067507528606452216915153475} a^{17} - \frac{51974931011749808236407218102}{8082067507528606452216915153475} a^{16} - \frac{511609786019946249235111722533}{8082067507528606452216915153475} a^{15} + \frac{69629895629979253195993124883}{1616413501505721290443383030695} a^{14} - \frac{27263140705434452951840959248}{323282700301144258088676606139} a^{13} + \frac{30659872240896519720522779379}{323282700301144258088676606139} a^{12} - \frac{167596580234020748130647507438}{1616413501505721290443383030695} a^{11} - \frac{53229865234043983923876000972}{323282700301144258088676606139} a^{10} + \frac{709949370192882937121565557558}{1616413501505721290443383030695} a^{9} - \frac{230930526998748567645467087547}{1616413501505721290443383030695} a^{8} - \frac{326356022218589449273709886639}{1616413501505721290443383030695} a^{7} + \frac{129445717151450079835650722351}{1616413501505721290443383030695} a^{6} + \frac{389114627683357994524462696174}{1616413501505721290443383030695} a^{5} - \frac{192327708582071379358289533586}{8082067507528606452216915153475} a^{4} - \frac{449555164469581285027247023341}{1616413501505721290443383030695} a^{3} - \frac{1971440980335686158819834107524}{8082067507528606452216915153475} a^{2} - \frac{1161794630454620236393979016918}{8082067507528606452216915153475} a - \frac{184680013809117093913634372817}{8082067507528606452216915153475}$

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:  $15$
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:  \( 357382719.867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
6329Data not computed