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Properties

Label 20.10.2014844636...6875.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 19^{5}\cdot 1699^{4}$
Root discriminant $20.66$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 8, 91, -177, 66, -133, 756, -811, 45, 620, -720, 354, 49, -198, 139, -42, -6, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 42*x^16 + 139*x^15 - 198*x^14 + 49*x^13 + 354*x^12 - 720*x^11 + 620*x^10 + 45*x^9 - 811*x^8 + 756*x^7 - 133*x^6 + 66*x^5 - 177*x^4 + 91*x^3 + 8*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 42*x^16 + 139*x^15 - 198*x^14 + 49*x^13 + 354*x^12 - 720*x^11 + 620*x^10 + 45*x^9 - 811*x^8 + 756*x^7 - 133*x^6 + 66*x^5 - 177*x^4 + 91*x^3 + 8*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 42 x^{16} + 139 x^{15} - 198 x^{14} + 49 x^{13} + 354 x^{12} - 720 x^{11} + 620 x^{10} + 45 x^{9} - 811 x^{8} + 756 x^{7} - 133 x^{6} + 66 x^{5} - 177 x^{4} + 91 x^{3} + 8 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-201484463662828739248046875=-\,5^{10}\cdot 19^{5}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.66$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{8}{19} a^{17} + \frac{5}{19} a^{16} - \frac{3}{19} a^{15} + \frac{5}{19} a^{13} + \frac{5}{19} a^{12} - \frac{6}{19} a^{11} - \frac{7}{19} a^{10} - \frac{3}{19} a^{9} - \frac{7}{19} a^{8} + \frac{3}{19} a^{7} - \frac{5}{19} a^{6} + \frac{3}{19} a^{5} - \frac{4}{19} a^{4} - \frac{8}{19} a^{3} + \frac{1}{19} a^{2} + \frac{1}{19} a - \frac{3}{19}$, $\frac{1}{21541906056010547} a^{19} + \frac{386421590329578}{21541906056010547} a^{18} + \frac{1918601513773065}{21541906056010547} a^{17} - \frac{7787622041634622}{21541906056010547} a^{16} + \frac{5951999106420232}{21541906056010547} a^{15} - \frac{445185490480032}{21541906056010547} a^{14} + \frac{9873978578161439}{21541906056010547} a^{13} + \frac{5155400957837170}{21541906056010547} a^{12} + \frac{2958502610838272}{21541906056010547} a^{11} + \frac{1330814536816476}{21541906056010547} a^{10} - \frac{1733553416023454}{21541906056010547} a^{9} - \frac{5260142032798852}{21541906056010547} a^{8} + \frac{134233391217753}{1133784529263713} a^{7} - \frac{7061156471910177}{21541906056010547} a^{6} + \frac{4721569525091195}{21541906056010547} a^{5} + \frac{959321497859726}{21541906056010547} a^{4} + \frac{697076247124125}{21541906056010547} a^{3} - \frac{5432104518664560}{21541906056010547} a^{2} + \frac{10523500696059674}{21541906056010547} a - \frac{7529023465563206}{21541906056010547}$

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:  $14$
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:  \( 496664.119055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 230 conjugacy class representatives for t20n876 are not computed
Character table for t20n876 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
1699Data not computed