↑

Properties

Label 20.0.22634291258...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{13}\cdot 199^{2}\cdot 1471^{4}$
Root discriminant $20.78$
Ramified primes $5, 199, 1471$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1439, 595, -3418, -1202, 4773, 1597, -4142, -1524, 2215, 1294, -970, -500, 217, 17, 119, -104, 47, -19, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 19*x^17 + 47*x^16 - 104*x^15 + 119*x^14 + 17*x^13 + 217*x^12 - 500*x^11 - 970*x^10 + 1294*x^9 + 2215*x^8 - 1524*x^7 - 4142*x^6 + 1597*x^5 + 4773*x^4 - 1202*x^3 - 3418*x^2 + 595*x + 1439)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 19*x^17 + 47*x^16 - 104*x^15 + 119*x^14 + 17*x^13 + 217*x^12 - 500*x^11 - 970*x^10 + 1294*x^9 + 2215*x^8 - 1524*x^7 - 4142*x^6 + 1597*x^5 + 4773*x^4 - 1202*x^3 - 3418*x^2 + 595*x + 1439, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 19 x^{17} + 47 x^{16} - 104 x^{15} + 119 x^{14} + 17 x^{13} + 217 x^{12} - 500 x^{11} - 970 x^{10} + 1294 x^{9} + 2215 x^{8} - 1524 x^{7} - 4142 x^{6} + 1597 x^{5} + 4773 x^{4} - 1202 x^{3} - 3418 x^{2} + 595 x + 1439 \)

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:  \(226342912585206153564453125=5^{13}\cdot 199^{2}\cdot 1471^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.78$
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:  $5, 199, 1471$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{207869921337926738886050614819307897} a^{19} + \frac{66203897540905689091579068463387982}{207869921337926738886050614819307897} a^{18} - \frac{58315180826775685539463909375803346}{207869921337926738886050614819307897} a^{17} + \frac{18045179749682759731500645145223224}{207869921337926738886050614819307897} a^{16} + \frac{6058678292608331089136697292861784}{207869921337926738886050614819307897} a^{15} + \frac{82342755607015834244145512159811629}{207869921337926738886050614819307897} a^{14} + \frac{92648930985738286127673144851187010}{207869921337926738886050614819307897} a^{13} - \frac{92588624847275688679330450189834746}{207869921337926738886050614819307897} a^{12} - \frac{103124889228210881188532126116607415}{207869921337926738886050614819307897} a^{11} - \frac{86834512802809856207236767184489137}{207869921337926738886050614819307897} a^{10} - \frac{64637677837641310500454843197688871}{207869921337926738886050614819307897} a^{9} + \frac{89111255038346696695931676326895430}{207869921337926738886050614819307897} a^{8} + \frac{66001841412008629213546873233404914}{207869921337926738886050614819307897} a^{7} + \frac{53101782343170061922051578564931602}{207869921337926738886050614819307897} a^{6} + \frac{54543040718611829752556645097004863}{207869921337926738886050614819307897} a^{5} + \frac{101130313111934452084255383644242753}{207869921337926738886050614819307897} a^{4} + \frac{90115975119389201692444135521433372}{207869921337926738886050614819307897} a^{3} + \frac{16489079740817473606547600251157902}{207869921337926738886050614819307897} a^{2} + \frac{17777454361521782640196820106007130}{207869921337926738886050614819307897} a - \frac{80820258777043765061024339177998855}{207869921337926738886050614819307897}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 56811.6092328 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.914778125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
199.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1471Data not computed