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Properties

Label 20.0.16019136114...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 47^{8}\cdot 83^{2}$
Root discriminant $16.23$
Ramified primes $5, 47, 83$
Class number $1$
Class group Trivial
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 8, 4, 19, 21, 21, 24, 49, 123, -5, 53, 96, -31, 17, -18, 9, -2, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 - 2*x^17 + 9*x^16 - 18*x^15 + 17*x^14 - 31*x^13 + 96*x^12 + 53*x^11 - 5*x^10 + 123*x^9 + 49*x^8 + 24*x^7 + 21*x^6 + 21*x^5 + 19*x^4 + 4*x^3 + 8*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - x^18 - 2*x^17 + 9*x^16 - 18*x^15 + 17*x^14 - 31*x^13 + 96*x^12 + 53*x^11 - 5*x^10 + 123*x^9 + 49*x^8 + 24*x^7 + 21*x^6 + 21*x^5 + 19*x^4 + 4*x^3 + 8*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - x^{18} - 2 x^{17} + 9 x^{16} - 18 x^{15} + 17 x^{14} - 31 x^{13} + 96 x^{12} + 53 x^{11} - 5 x^{10} + 123 x^{9} + 49 x^{8} + 24 x^{7} + 21 x^{6} + 21 x^{5} + 19 x^{4} + 4 x^{3} + 8 x^{2} + 2 x + 1 \)

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:  \(1601913611453823525390625=5^{10}\cdot 47^{8}\cdot 83^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.23$
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:  $5, 47, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} + \frac{8}{41} a^{17} - \frac{6}{41} a^{16} - \frac{4}{41} a^{15} - \frac{13}{41} a^{14} - \frac{13}{41} a^{13} + \frac{8}{41} a^{12} + \frac{6}{41} a^{11} + \frac{9}{41} a^{10} + \frac{2}{41} a^{9} - \frac{12}{41} a^{8} - \frac{3}{41} a^{7} + \frac{14}{41} a^{6} + \frac{9}{41} a^{5} - \frac{13}{41} a^{4} - \frac{13}{41} a^{3} + \frac{10}{41} a^{2} + \frac{20}{41} a + \frac{14}{41}$, $\frac{1}{34319781936588187} a^{19} + \frac{16650346630955}{2639983225891399} a^{18} - \frac{8190628941756886}{34319781936588187} a^{17} + \frac{11305024110302334}{34319781936588187} a^{16} + \frac{7684575886103377}{34319781936588187} a^{15} - \frac{6918221442977731}{34319781936588187} a^{14} - \frac{4135431978685}{12493550031521} a^{13} + \frac{6934919344599107}{34319781936588187} a^{12} - \frac{574437007742474}{2639983225891399} a^{11} + \frac{15684884379021794}{34319781936588187} a^{10} - \frac{6869584332669867}{34319781936588187} a^{9} + \frac{856209904732688}{2639983225891399} a^{8} - \frac{14348204614614705}{34319781936588187} a^{7} - \frac{14720647696634836}{34319781936588187} a^{6} - \frac{1826718376020936}{34319781936588187} a^{5} - \frac{4689361122140482}{34319781936588187} a^{4} + \frac{9568399464006815}{34319781936588187} a^{3} - \frac{16871029630453781}{34319781936588187} a^{2} + \frac{12066756353757224}{34319781936588187} a + \frac{14118728536031736}{34319781936588187}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4646.25927566 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2209.1, 10.0.1265667259375.1, 10.0.405013523.1, 10.2.15249003125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]_{2}^{4}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$