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Properties

Label 20.8.50980770312...0000.7
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{5}$
Root discriminant $54.37$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6029, 0, 60290, 0, -800669, 0, -605797, 0, 16874, 0, 67830, 0, 4570, 0, -1999, 0, -197, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 - 197*x^16 - 1999*x^14 + 4570*x^12 + 67830*x^10 + 16874*x^8 - 605797*x^6 - 800669*x^4 + 60290*x^2 + 6029)
 
gp: K = bnfinit(x^20 + 9*x^18 - 197*x^16 - 1999*x^14 + 4570*x^12 + 67830*x^10 + 16874*x^8 - 605797*x^6 - 800669*x^4 + 60290*x^2 + 6029, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} - 197 x^{16} - 1999 x^{14} + 4570 x^{12} + 67830 x^{10} + 16874 x^{8} - 605797 x^{6} - 800669 x^{4} + 60290 x^{2} + 6029 \)

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:  \(50980770312185223353600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.37$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{595} a^{16} - \frac{8}{85} a^{14} - \frac{9}{119} a^{12} + \frac{142}{595} a^{10} + \frac{218}{595} a^{8} - \frac{53}{119} a^{6} - \frac{21}{85} a^{4} + \frac{233}{595} a^{2} - \frac{223}{595}$, $\frac{1}{595} a^{17} - \frac{8}{85} a^{15} - \frac{9}{119} a^{13} + \frac{142}{595} a^{11} + \frac{218}{595} a^{9} - \frac{53}{119} a^{7} - \frac{21}{85} a^{5} + \frac{233}{595} a^{3} - \frac{223}{595} a$, $\frac{1}{71783452396722421901904995} a^{18} + \frac{7744211711158210765238}{14356690479344484380380999} a^{16} + \frac{5344847442319516219963228}{71783452396722421901904995} a^{14} - \frac{17363901324769604792461334}{71783452396722421901904995} a^{12} - \frac{12692497032358852573015132}{71783452396722421901904995} a^{10} + \frac{646589195454868533026989}{5521804030517109377069615} a^{8} - \frac{1511883077658056709836344}{5521804030517109377069615} a^{6} + \frac{27908841872234028059849966}{71783452396722421901904995} a^{4} - \frac{20880157365457740863488453}{71783452396722421901904995} a^{2} + \frac{5381339177421135310882628}{71783452396722421901904995}$, $\frac{1}{71783452396722421901904995} a^{19} + \frac{7744211711158210765238}{14356690479344484380380999} a^{17} + \frac{5344847442319516219963228}{71783452396722421901904995} a^{15} - \frac{17363901324769604792461334}{71783452396722421901904995} a^{13} - \frac{12692497032358852573015132}{71783452396722421901904995} a^{11} + \frac{646589195454868533026989}{5521804030517109377069615} a^{9} - \frac{1511883077658056709836344}{5521804030517109377069615} a^{7} + \frac{27908841872234028059849966}{71783452396722421901904995} a^{5} - \frac{20880157365457740863488453}{71783452396722421901904995} a^{3} + \frac{5381339177421135310882628}{71783452396722421901904995} a$

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:  \( 7302082836.62 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 R $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
6029Data not computed