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Properties

Label 20.0.15891083381...4912.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{45}\cdot 461^{4}$
Root discriminant $16.22$
Ramified primes $2, 461$
Class number $1$
Class group Trivial
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 20, -4, -45, 4, 130, -84, 23, -124, 42, 24, 49, -12, -2, -16, 5, -4, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - 4*x^17 + 5*x^16 - 16*x^15 - 2*x^14 - 12*x^13 + 49*x^12 + 24*x^11 + 42*x^10 - 124*x^9 + 23*x^8 - 84*x^7 + 130*x^6 + 4*x^5 - 45*x^4 - 4*x^3 + 20*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 + 4*x^18 - 4*x^17 + 5*x^16 - 16*x^15 - 2*x^14 - 12*x^13 + 49*x^12 + 24*x^11 + 42*x^10 - 124*x^9 + 23*x^8 - 84*x^7 + 130*x^6 + 4*x^5 - 45*x^4 - 4*x^3 + 20*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - 4 x^{17} + 5 x^{16} - 16 x^{15} - 2 x^{14} - 12 x^{13} + 49 x^{12} + 24 x^{11} + 42 x^{10} - 124 x^{9} + 23 x^{8} - 84 x^{7} + 130 x^{6} + 4 x^{5} - 45 x^{4} - 4 x^{3} + 20 x^{2} - 8 x + 1 \)

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:  \(1589108338173520916774912=2^{45}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.22$
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:  $2, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{2} a^{14} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{46895817899716312} a^{19} - \frac{1136817240572701}{46895817899716312} a^{18} - \frac{11680393400180197}{46895817899716312} a^{17} + \frac{14002944101783885}{46895817899716312} a^{16} - \frac{2097594943658153}{23447908949858156} a^{15} - \frac{9624027428452619}{23447908949858156} a^{14} + \frac{2850938018024528}{5861977237464539} a^{13} + \frac{2982992592980635}{11723954474929078} a^{12} + \frac{16594261776754181}{46895817899716312} a^{11} + \frac{20144745806515647}{46895817899716312} a^{10} + \frac{11627886098065853}{46895817899716312} a^{9} + \frac{19331787176180851}{46895817899716312} a^{8} + \frac{6856117140131399}{23447908949858156} a^{7} + \frac{4531346847807367}{23447908949858156} a^{6} + \frac{784563039736659}{5861977237464539} a^{5} + \frac{271480004145355}{11723954474929078} a^{4} - \frac{5568266266658481}{46895817899716312} a^{3} - \frac{20333031107146287}{46895817899716312} a^{2} - \frac{10766183825391783}{46895817899716312} a - \frac{4419779838453173}{46895817899716312}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{66494568449}{8156174566} a^{19} - \frac{10980525226}{4078087283} a^{18} - \frac{272458565409}{8156174566} a^{17} + \frac{88392083412}{4078087283} a^{16} - \frac{135240627521}{4078087283} a^{15} + \frac{487524434979}{4078087283} a^{14} + \frac{229016824782}{4078087283} a^{13} + \frac{470307197387}{4078087283} a^{12} - \frac{2959470092965}{8156174566} a^{11} - \frac{1295086992267}{4078087283} a^{10} - \frac{3625738580255}{8156174566} a^{9} + \frac{3547532976764}{4078087283} a^{8} + \frac{441208173189}{4078087283} a^{7} + \frac{2919709775833}{4078087283} a^{6} - \frac{3379007415662}{4078087283} a^{5} - \frac{1287702880566}{4078087283} a^{4} + \frac{2164187984969}{8156174566} a^{3} + \frac{521119064164}{4078087283} a^{2} - \frac{993128960883}{8156174566} a + \frac{93910485806}{4078087283} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 27370.3106615 \)
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

\(\Q(\sqrt{-1}) \), 5.1.29504.1, 10.0.55711105024.3

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $20$

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
461Data not computed