↑

Properties

Label 20.20.6659802967...8481.1
Degree $20$
Signature $[20, 0]$
Discriminant $397^{4}\cdot 401^{9}$
Root discriminant $49.11$
Ramified primes $397, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 513, -1314, -9435, 10366, 50884, -50613, -88997, 82377, 73269, -62169, -31958, 25005, 7593, -5617, -948, 694, 55, -43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 43*x^18 + 55*x^17 + 694*x^16 - 948*x^15 - 5617*x^14 + 7593*x^13 + 25005*x^12 - 31958*x^11 - 62169*x^10 + 73269*x^9 + 82377*x^8 - 88997*x^7 - 50613*x^6 + 50884*x^5 + 10366*x^4 - 9435*x^3 - 1314*x^2 + 513*x + 81)
 
gp: K = bnfinit(x^20 - x^19 - 43*x^18 + 55*x^17 + 694*x^16 - 948*x^15 - 5617*x^14 + 7593*x^13 + 25005*x^12 - 31958*x^11 - 62169*x^10 + 73269*x^9 + 82377*x^8 - 88997*x^7 - 50613*x^6 + 50884*x^5 + 10366*x^4 - 9435*x^3 - 1314*x^2 + 513*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 43 x^{18} + 55 x^{17} + 694 x^{16} - 948 x^{15} - 5617 x^{14} + 7593 x^{13} + 25005 x^{12} - 31958 x^{11} - 62169 x^{10} + 73269 x^{9} + 82377 x^{8} - 88997 x^{7} - 50613 x^{6} + 50884 x^{5} + 10366 x^{4} - 9435 x^{3} - 1314 x^{2} + 513 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6659802967827443834044914879928481=397^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.11$
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:  $397, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{17} + \frac{2}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{3} a^{13} - \frac{1}{9} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{4077411554013408265008915531} a^{19} - \frac{196686103706312697250529638}{4077411554013408265008915531} a^{18} + \frac{581148535213090554352697264}{4077411554013408265008915531} a^{17} - \frac{1675296715406891616113383067}{4077411554013408265008915531} a^{16} + \frac{919281680856465421648527979}{4077411554013408265008915531} a^{15} - \frac{435277598492792793756425069}{1359137184671136088336305177} a^{14} - \frac{1147080573712883558535043393}{4077411554013408265008915531} a^{13} - \frac{159241034947393755368686654}{453045728223712029445435059} a^{12} + \frac{43037248500022958240217049}{1359137184671136088336305177} a^{11} + \frac{1800198344952296450504782399}{4077411554013408265008915531} a^{10} + \frac{17671044295112409211839673}{151015242741237343148478353} a^{9} + \frac{86273468781904336645460387}{453045728223712029445435059} a^{8} + \frac{3907575878854489300163589}{151015242741237343148478353} a^{7} + \frac{1433374200193697197723182019}{4077411554013408265008915531} a^{6} + \frac{221134616818586194095975320}{453045728223712029445435059} a^{5} + \frac{66414488827998798395802790}{4077411554013408265008915531} a^{4} + \frac{1983569798687263187972353393}{4077411554013408265008915531} a^{3} + \frac{615728463394398596542933954}{1359137184671136088336305177} a^{2} + \frac{36911779938317271994607609}{151015242741237343148478353} a + \frac{12952173283472134233737882}{151015242741237343148478353}$

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:  $19$
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:  \( 59603781728.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.10.4075289860972009.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
397Data not computed
401Data not computed