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Properties

Label 20.8.78123008526...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{6}\cdot 5^{14}\cdot 23^{4}\cdot 89^{4}$
Root discriminant $19.71$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -45, 0, 60, 0, -165, 0, 505, 0, -301, 0, 25, 0, -25, 0, 20, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 20*x^16 - 25*x^14 + 25*x^12 - 301*x^10 + 505*x^8 - 165*x^6 + 60*x^4 - 45*x^2 + 9)
 
gp: K = bnfinit(x^20 - 5*x^18 + 20*x^16 - 25*x^14 + 25*x^12 - 301*x^10 + 505*x^8 - 165*x^6 + 60*x^4 - 45*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 20 x^{16} - 25 x^{14} + 25 x^{12} - 301 x^{10} + 505 x^{8} - 165 x^{6} + 60 x^{4} - 45 x^{2} + 9 \)

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:  \(78123008526754449462890625=3^{6}\cdot 5^{14}\cdot 23^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.71$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{30} a^{12} + \frac{1}{15} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{2}{5} a^{2} + \frac{3}{10}$, $\frac{1}{30} a^{13} + \frac{1}{15} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{2}{5} a^{3} + \frac{3}{10} a$, $\frac{1}{30} a^{14} - \frac{2}{15} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{4}{15} a^{4} - \frac{1}{2} a^{2} + \frac{2}{5}$, $\frac{1}{90} a^{15} - \frac{1}{90} a^{13} + \frac{1}{10} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{4}{9} a^{7} - \frac{11}{30} a^{5} - \frac{1}{2} a^{4} + \frac{1}{30} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a$, $\frac{1}{450} a^{16} - \frac{2}{225} a^{14} - \frac{1}{150} a^{12} - \frac{3}{50} a^{10} - \frac{19}{90} a^{8} - \frac{1}{2} a^{7} + \frac{13}{50} a^{6} - \frac{1}{2} a^{5} + \frac{59}{150} a^{4} - \frac{1}{2} a^{3} + \frac{21}{50} a^{2} + \frac{2}{25}$, $\frac{1}{450} a^{17} + \frac{1}{450} a^{15} + \frac{7}{450} a^{13} + \frac{8}{75} a^{11} - \frac{17}{45} a^{9} - \frac{1}{2} a^{8} + \frac{46}{225} a^{7} + \frac{9}{25} a^{5} - \frac{11}{75} a^{3} - \frac{1}{2} a^{2} - \frac{21}{50} a$, $\frac{1}{4402350} a^{18} - \frac{637}{1467450} a^{16} + \frac{311}{440235} a^{14} - \frac{4756}{733725} a^{12} + \frac{2972}{2201175} a^{10} - \frac{235643}{4402350} a^{8} - \frac{1}{2} a^{7} + \frac{4502}{81525} a^{6} - \frac{20611}{146745} a^{4} - \frac{1}{2} a^{3} - \frac{5806}{27175} a^{2} + \frac{117686}{244575}$, $\frac{1}{4402350} a^{19} - \frac{637}{1467450} a^{17} + \frac{311}{440235} a^{15} - \frac{4756}{733725} a^{13} + \frac{2972}{2201175} a^{11} - \frac{235643}{4402350} a^{9} - \frac{1}{2} a^{8} + \frac{4502}{81525} a^{7} - \frac{20611}{146745} a^{5} - \frac{1}{2} a^{4} - \frac{5806}{27175} a^{3} + \frac{117686}{244575} a$

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:  \( 198759.963525 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1, 10.4.8838722109375.1, 10.4.1767744421875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$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}$
23.4.0.1$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}$
23.4.2.1$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}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$