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Properties

Label 20.16.2545661450...4649.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{10}\cdot 401^{11}$
Root discriminant $46.80$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1521, -25974, -45039, 92079, 220924, -92982, -383278, 10081, 340575, 42618, -169371, -28139, 50016, 7482, -8930, -954, 929, 55, -50, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 50*x^18 + 55*x^17 + 929*x^16 - 954*x^15 - 8930*x^14 + 7482*x^13 + 50016*x^12 - 28139*x^11 - 169371*x^10 + 42618*x^9 + 340575*x^8 + 10081*x^7 - 383278*x^6 - 92982*x^5 + 220924*x^4 + 92079*x^3 - 45039*x^2 - 25974*x - 1521)
 
gp: K = bnfinit(x^20 - x^19 - 50*x^18 + 55*x^17 + 929*x^16 - 954*x^15 - 8930*x^14 + 7482*x^13 + 50016*x^12 - 28139*x^11 - 169371*x^10 + 42618*x^9 + 340575*x^8 + 10081*x^7 - 383278*x^6 - 92982*x^5 + 220924*x^4 + 92079*x^3 - 45039*x^2 - 25974*x - 1521, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 50 x^{18} + 55 x^{17} + 929 x^{16} - 954 x^{15} - 8930 x^{14} + 7482 x^{13} + 50016 x^{12} - 28139 x^{11} - 169371 x^{10} + 42618 x^{9} + 340575 x^{8} + 10081 x^{7} - 383278 x^{6} - 92982 x^{5} + 220924 x^{4} + 92079 x^{3} - 45039 x^{2} - 25974 x - 1521 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2545661450630827874451149331074649=3^{10}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.80$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} + \frac{1}{3} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{40318043098279997476933984488882} a^{19} - \frac{45201808237184462669630686210}{6719673849713332912822330748147} a^{18} - \frac{430772980522730569672457642852}{6719673849713332912822330748147} a^{17} + \frac{105372070412659808199638096207}{20159021549139998738466992244441} a^{16} + \frac{544868423211991824417935727746}{6719673849713332912822330748147} a^{15} - \frac{987723874287923319650860393484}{6719673849713332912822330748147} a^{14} - \frac{5310399399266314238446260015}{253572598102389921238578518798} a^{13} - \frac{869688669392848318570905912787}{13439347699426665825644661496294} a^{12} + \frac{604681653645751865762894417566}{20159021549139998738466992244441} a^{11} + \frac{8012093679612136560297593200189}{20159021549139998738466992244441} a^{10} - \frac{6297248150060062004791777095895}{20159021549139998738466992244441} a^{9} - \frac{19470872376675639140657202262973}{40318043098279997476933984488882} a^{8} + \frac{141611500865486551551425368043}{380358897153584881857867778197} a^{7} + \frac{2155762881777471478299194337646}{6719673849713332912822330748147} a^{6} + \frac{113114214990790085763447176135}{40318043098279997476933984488882} a^{5} - \frac{203458075213240793687472088429}{760717794307169763715735556394} a^{4} - \frac{3211330440325576371097220670587}{40318043098279997476933984488882} a^{3} + \frac{376714880371098242665301214661}{3101387930636922882841075729914} a^{2} - \frac{2232955128909926922936193354322}{6719673849713332912822330748147} a + \frac{168336277776937782476211911949}{516897988439487147140179288319}$

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:  $17$
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:  \( 10258760614.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

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 $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed