↑

Properties

Label 20.4.17701447978...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{31}\cdot 11^{4}\cdot 19^{5}$
Root discriminant $81.73$
Ramified primes $2, 5, 11, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T168

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![969095, 0, -2663025, 0, 758075, 0, 152600, 0, -9775, 0, -24045, 0, 10740, 0, -2070, 0, 255, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 255*x^16 - 2070*x^14 + 10740*x^12 - 24045*x^10 - 9775*x^8 + 152600*x^6 + 758075*x^4 - 2663025*x^2 + 969095)
 
gp: K = bnfinit(x^20 - 20*x^18 + 255*x^16 - 2070*x^14 + 10740*x^12 - 24045*x^10 - 9775*x^8 + 152600*x^6 + 758075*x^4 - 2663025*x^2 + 969095, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 255 x^{16} - 2070 x^{14} + 10740 x^{12} - 24045 x^{10} - 9775 x^{8} + 152600 x^{6} + 758075 x^{4} - 2663025 x^{2} + 969095 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(177014479780273437500000000000000000000=2^{20}\cdot 5^{31}\cdot 11^{4}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.73$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{2}{19} a^{14} + \frac{5}{19} a^{12} + \frac{6}{19} a^{10} - \frac{7}{19} a^{8} + \frac{5}{19} a^{6} + \frac{2}{19} a^{4} + \frac{3}{19} a^{2}$, $\frac{1}{19} a^{17} - \frac{2}{19} a^{15} + \frac{5}{19} a^{13} + \frac{6}{19} a^{11} - \frac{7}{19} a^{9} + \frac{5}{19} a^{7} + \frac{2}{19} a^{5} + \frac{3}{19} a^{3}$, $\frac{1}{33812841609291863806515753563} a^{18} - \frac{781838040791070132482638820}{33812841609291863806515753563} a^{16} + \frac{11268127880006279546501268840}{33812841609291863806515753563} a^{14} - \frac{13334596360843848758658285448}{33812841609291863806515753563} a^{12} + \frac{10470973932981459468397652275}{33812841609291863806515753563} a^{10} - \frac{12866166542034763341440272386}{33812841609291863806515753563} a^{8} + \frac{16811231236716334453880571360}{33812841609291863806515753563} a^{6} + \frac{13459293830177984084169270920}{33812841609291863806515753563} a^{4} + \frac{3915069204231567452261704163}{33812841609291863806515753563} a^{2} - \frac{481080673562843908497703503}{1779623242594308621395565977}$, $\frac{1}{3415097002538478244458091109863} a^{19} - \frac{73746390987157723609700843877}{3415097002538478244458091109863} a^{17} + \frac{630577016302825679792158228836}{3415097002538478244458091109863} a^{15} + \frac{1582987452246250984633164395921}{3415097002538478244458091109863} a^{13} - \frac{1576952958461141830816447199209}{3415097002538478244458091109863} a^{11} + \frac{497885704082531810999087163013}{3415097002538478244458091109863} a^{9} + \frac{1613133279843811167845703252729}{3415097002538478244458091109863} a^{7} - \frac{267721178499722778096330153446}{3415097002538478244458091109863} a^{5} + \frac{664155292206720065990016681630}{3415097002538478244458091109863} a^{3} - \frac{73445633619929497385715908560}{179741947502025170760952163677} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 52961455987.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T168:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 800
The 44 conjugacy class representatives for t20n168
Character table for t20n168 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.38000.1, 10.2.1333038330078125.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$