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Properties

Label 20.8.21799845600...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 60662149^{3}$
Root discriminant $32.88$
Ramified primes $5, 60662149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-244, 100, 4377, -12445, 3528, 19668, -11511, -10810, 6819, 1894, -963, -156, -617, 520, 211, -278, 16, 55, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 12*x^18 + 55*x^17 + 16*x^16 - 278*x^15 + 211*x^14 + 520*x^13 - 617*x^12 - 156*x^11 - 963*x^10 + 1894*x^9 + 6819*x^8 - 10810*x^7 - 11511*x^6 + 19668*x^5 + 3528*x^4 - 12445*x^3 + 4377*x^2 + 100*x - 244)
 
gp: K = bnfinit(x^20 - 4*x^19 - 12*x^18 + 55*x^17 + 16*x^16 - 278*x^15 + 211*x^14 + 520*x^13 - 617*x^12 - 156*x^11 - 963*x^10 + 1894*x^9 + 6819*x^8 - 10810*x^7 - 11511*x^6 + 19668*x^5 + 3528*x^4 - 12445*x^3 + 4377*x^2 + 100*x - 244, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 12 x^{18} + 55 x^{17} + 16 x^{16} - 278 x^{15} + 211 x^{14} + 520 x^{13} - 617 x^{12} - 156 x^{11} - 963 x^{10} + 1894 x^{9} + 6819 x^{8} - 10810 x^{7} - 11511 x^{6} + 19668 x^{5} + 3528 x^{4} - 12445 x^{3} + 4377 x^{2} + 100 x - 244 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2179984560030696704042470703125=5^{10}\cdot 60662149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.88$
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:  $5, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17929698774117156629577724641386775240116} a^{19} + \frac{3847918440697848122542109732243256296525}{8964849387058578314788862320693387620058} a^{18} + \frac{1261327373565126189876771012922318021920}{4482424693529289157394431160346693810029} a^{17} - \frac{4960766147300093960355998146598684019401}{17929698774117156629577724641386775240116} a^{16} - \frac{1418969414184202691602893584356236693663}{8964849387058578314788862320693387620058} a^{15} + \frac{342827113018225400683967146692437943829}{689603799004506024214527870822568278466} a^{14} - \frac{7376762194535990714759915024610189900309}{17929698774117156629577724641386775240116} a^{13} - \frac{1794190439353001948590301920479578397331}{8964849387058578314788862320693387620058} a^{12} - \frac{5800441023809480240834791871757323810953}{17929698774117156629577724641386775240116} a^{11} + \frac{2029104054011639360615539394773016632537}{8964849387058578314788862320693387620058} a^{10} - \frac{6533008382205610430723394326867276387443}{17929698774117156629577724641386775240116} a^{9} + \frac{1795100000974352747636039129459577855307}{4482424693529289157394431160346693810029} a^{8} + \frac{5916067828757808598830271234978237913415}{17929698774117156629577724641386775240116} a^{7} + \frac{2125132403180911812093615429758862166541}{4482424693529289157394431160346693810029} a^{6} - \frac{4581322819392634732872751254647226156683}{17929698774117156629577724641386775240116} a^{5} + \frac{2052836432997793944845278595502643231381}{8964849387058578314788862320693387620058} a^{4} - \frac{1260789324164535760065794283849981099447}{4482424693529289157394431160346693810029} a^{3} - \frac{390283355055752836931419126993436821249}{1379207598009012048429055741645136556932} a^{2} + \frac{2607201854579754012468559784139950948723}{17929698774117156629577724641386775240116} a - \frac{1266070024811623711170489729335344915607}{8964849387058578314788862320693387620058}$

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:  $13$
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:  \( 49462582.2378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.189569215625.1

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 32 sibling: 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 $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
60662149Data not computed