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Properties

Label 20.0.12300947894...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 23^{2}\cdot 47^{8}$
Root discriminant $14.27$
Ramified primes $5, 23, 47$
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, 1, 5, 6, 26, 21, 71, 30, 112, 42, 159, -42, 112, -30, 71, -21, 26, -6, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 6*x^17 + 26*x^16 - 21*x^15 + 71*x^14 - 30*x^13 + 112*x^12 - 42*x^11 + 159*x^10 + 42*x^9 + 112*x^8 + 30*x^7 + 71*x^6 + 21*x^5 + 26*x^4 + 6*x^3 + 5*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 6*x^17 + 26*x^16 - 21*x^15 + 71*x^14 - 30*x^13 + 112*x^12 - 42*x^11 + 159*x^10 + 42*x^9 + 112*x^8 + 30*x^7 + 71*x^6 + 21*x^5 + 26*x^4 + 6*x^3 + 5*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 26 x^{16} - 21 x^{15} + 71 x^{14} - 30 x^{13} + 112 x^{12} - 42 x^{11} + 159 x^{10} + 42 x^{9} + 112 x^{8} + 30 x^{7} + 71 x^{6} + 21 x^{5} + 26 x^{4} + 6 x^{3} + 5 x^{2} + x + 1 \)

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:  \(123009478946011416015625=5^{10}\cdot 23^{2}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.27$
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:  $5, 23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{244755995} a^{18} - \frac{1182074}{48951199} a^{17} - \frac{87003844}{244755995} a^{16} + \frac{11697781}{48951199} a^{15} + \frac{2894534}{10641565} a^{14} + \frac{34736646}{244755995} a^{13} - \frac{105382121}{244755995} a^{12} + \frac{6393021}{48951199} a^{11} + \frac{12948776}{244755995} a^{10} + \frac{5515284}{244755995} a^{9} - \frac{12948776}{244755995} a^{8} + \frac{6393021}{48951199} a^{7} + \frac{105382121}{244755995} a^{6} + \frac{34736646}{244755995} a^{5} - \frac{2894534}{10641565} a^{4} + \frac{11697781}{48951199} a^{3} + \frac{87003844}{244755995} a^{2} - \frac{1182074}{48951199} a - \frac{1}{244755995}$, $\frac{1}{3181827935} a^{19} + \frac{4}{3181827935} a^{18} + \frac{704716141}{3181827935} a^{17} + \frac{44678538}{138340345} a^{16} + \frac{694952697}{3181827935} a^{15} + \frac{636116289}{3181827935} a^{14} - \frac{738766397}{3181827935} a^{13} + \frac{976361981}{3181827935} a^{12} + \frac{788711501}{3181827935} a^{11} - \frac{1371827027}{3181827935} a^{10} - \frac{285602123}{636365587} a^{9} + \frac{83617766}{289257085} a^{8} - \frac{832147119}{3181827935} a^{7} + \frac{13872750}{636365587} a^{6} + \frac{1258089402}{3181827935} a^{5} + \frac{1415157222}{3181827935} a^{4} + \frac{1449650244}{3181827935} a^{3} + \frac{1227776116}{3181827935} a^{2} + \frac{4096743}{12576395} a - \frac{495422364}{3181827935}$

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:  \( 1492.37830154 \)
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, 10.0.70145414375.1, 10.2.15249003125.1, 10.0.561163315.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ 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$$[\ ]_{2}^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{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}$