↑

Properties

Label 20.0.26233125242...2352.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{15}\cdot 11491^{4}$
Root discriminant $29.58$
Ramified primes $2, 3, 11491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2187, 0, -8424, 0, 14218, 0, -13104, 0, 6806, 0, -1719, 0, 21, 0, 78, 0, -1, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - x^16 + 78*x^14 + 21*x^12 - 1719*x^10 + 6806*x^8 - 13104*x^6 + 14218*x^4 - 8424*x^2 + 2187)
 
gp: K = bnfinit(x^20 - 6*x^18 - x^16 + 78*x^14 + 21*x^12 - 1719*x^10 + 6806*x^8 - 13104*x^6 + 14218*x^4 - 8424*x^2 + 2187, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - x^{16} + 78 x^{14} + 21 x^{12} - 1719 x^{10} + 6806 x^{8} - 13104 x^{6} + 14218 x^{4} - 8424 x^{2} + 2187 \)

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:  \(262331252425064668558343012352=2^{20}\cdot 3^{15}\cdot 11491^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.58$
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:  $2, 3, 11491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} - \frac{1}{5} a^{12} - \frac{2}{5} a^{10} + \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{15} a^{17} + \frac{1}{5} a^{15} - \frac{1}{15} a^{13} + \frac{1}{5} a^{11} - \frac{4}{15} a^{5} + \frac{1}{15} a$, $\frac{1}{218382234705} a^{18} - \frac{768610928}{72794078235} a^{16} - \frac{14181665167}{218382234705} a^{14} + \frac{2175678061}{14558815647} a^{12} + \frac{11019069358}{72794078235} a^{10} + \frac{2263607135}{4852938549} a^{8} - \frac{102254435854}{218382234705} a^{6} + \frac{6832646267}{24264692745} a^{4} + \frac{67480168651}{218382234705} a^{2} - \frac{1207290696}{8088230915}$, $\frac{1}{1965440112345} a^{19} + \frac{13790204719}{655146704115} a^{17} + \frac{771994379771}{1965440112345} a^{15} - \frac{149268581812}{655146704115} a^{13} - \frac{236480796641}{655146704115} a^{11} - \frac{17148147061}{43676446941} a^{9} + \frac{771274502966}{1965440112345} a^{7} - \frac{85373186164}{218382234705} a^{5} - \frac{369284300759}{1965440112345} a^{3} - \frac{5255368781}{24264692745} a$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{224954276}{43676446941} a^{18} - \frac{358200253}{14558815647} a^{16} - \frac{1481965196}{43676446941} a^{14} + \frac{5180781739}{14558815647} a^{12} + \frac{7713034937}{14558815647} a^{10} - \frac{39485394505}{4852938549} a^{8} + \frac{1105656620554}{43676446941} a^{6} - \frac{185429458148}{4852938549} a^{4} + \frac{1345331949551}{43676446941} a^{2} - \frac{16775571868}{1617646183} \) (order $6$)
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:  \( 31226279.5737 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.310257.1, 10.0.288778218147.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11491Data not computed