↑

Properties

Label 20.0.43787482408...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 3359^{5}$
Root discriminant $34.05$
Ramified primes $2, 5, 3359$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group 20T168

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3359, 0, 17956, 0, 30671, 0, 18337, 0, 4489, 0, 381, 0, -44, 0, 10, 0, 29, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 + 29*x^16 + 10*x^14 - 44*x^12 + 381*x^10 + 4489*x^8 + 18337*x^6 + 30671*x^4 + 17956*x^2 + 3359)
 
gp: K = bnfinit(x^20 + 10*x^18 + 29*x^16 + 10*x^14 - 44*x^12 + 381*x^10 + 4489*x^8 + 18337*x^6 + 30671*x^4 + 17956*x^2 + 3359, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} + 29 x^{16} + 10 x^{14} - 44 x^{12} + 381 x^{10} + 4489 x^{8} + 18337 x^{6} + 30671 x^{4} + 17956 x^{2} + 3359 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4378748240839483381760000000000=2^{20}\cdot 5^{10}\cdot 3359^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.05$
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:  $2, 5, 3359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{121} a^{16} - \frac{3}{11} a^{14} + \frac{46}{121} a^{12} + \frac{12}{121} a^{10} + \frac{46}{121} a^{8} - \frac{29}{121} a^{6} + \frac{50}{121} a^{4} - \frac{25}{121} a^{2} + \frac{3}{121}$, $\frac{1}{121} a^{17} - \frac{3}{11} a^{15} + \frac{46}{121} a^{13} + \frac{12}{121} a^{11} + \frac{46}{121} a^{9} - \frac{29}{121} a^{7} + \frac{50}{121} a^{5} - \frac{25}{121} a^{3} + \frac{3}{121} a$, $\frac{1}{4072769798977} a^{18} + \frac{15057920899}{4072769798977} a^{16} - \frac{869019128954}{4072769798977} a^{14} + \frac{634820519068}{4072769798977} a^{12} - \frac{1730350604520}{4072769798977} a^{10} + \frac{1493023342384}{4072769798977} a^{8} + \frac{1037098584139}{4072769798977} a^{6} - \frac{1208338444377}{4072769798977} a^{4} - \frac{856985617009}{4072769798977} a^{2} - \frac{258348590041}{4072769798977}$, $\frac{1}{4072769798977} a^{19} + \frac{15057920899}{4072769798977} a^{17} - \frac{869019128954}{4072769798977} a^{15} + \frac{634820519068}{4072769798977} a^{13} - \frac{1730350604520}{4072769798977} a^{11} + \frac{1493023342384}{4072769798977} a^{9} + \frac{1037098584139}{4072769798977} a^{7} - \frac{1208338444377}{4072769798977} a^{5} - \frac{856985617009}{4072769798977} a^{3} - \frac{258348590041}{4072769798977} a$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  $9$
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:  \( 423470.438941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T168:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 800
The 44 conjugacy class representatives for t20n168
Character table for t20n168 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1343600.1, 10.6.35259003125.1

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
3359Data not computed