↑

Properties

Label 20.20.1585743500...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 11^{7}\cdot 941011^{3}$
Root discriminant $40.74$
Ramified primes $5, 11, 941011$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T540

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 5, 137, -283, -541, 1731, 579, -4387, 896, 5461, -2645, -3328, 2321, 836, -856, -24, 125, -13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 13*x^18 + 125*x^17 - 24*x^16 - 856*x^15 + 836*x^14 + 2321*x^13 - 3328*x^12 - 2645*x^11 + 5461*x^10 + 896*x^9 - 4387*x^8 + 579*x^7 + 1731*x^6 - 541*x^5 - 283*x^4 + 137*x^3 + 5*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 - 13*x^18 + 125*x^17 - 24*x^16 - 856*x^15 + 836*x^14 + 2321*x^13 - 3328*x^12 - 2645*x^11 + 5461*x^10 + 896*x^9 - 4387*x^8 + 579*x^7 + 1731*x^6 - 541*x^5 - 283*x^4 + 137*x^3 + 5*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 13 x^{18} + 125 x^{17} - 24 x^{16} - 856 x^{15} + 836 x^{14} + 2321 x^{13} - 3328 x^{12} - 2645 x^{11} + 5461 x^{10} + 896 x^{9} - 4387 x^{8} + 579 x^{7} + 1731 x^{6} - 541 x^{5} - 283 x^{4} + 137 x^{3} + 5 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(158574350027984605001148447265625=5^{10}\cdot 11^{7}\cdot 941011^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.74$
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:  $5, 11, 941011$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11} a^{17} + \frac{4}{11} a^{16} + \frac{1}{11} a^{15} + \frac{2}{11} a^{13} + \frac{2}{11} a^{12} + \frac{1}{11} a^{11} - \frac{5}{11} a^{10} - \frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{18} - \frac{4}{11} a^{16} - \frac{4}{11} a^{15} + \frac{2}{11} a^{14} + \frac{5}{11} a^{13} + \frac{4}{11} a^{12} + \frac{2}{11} a^{11} - \frac{3}{11} a^{10} - \frac{3}{11} a^{8} + \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{3230579} a^{19} + \frac{54396}{3230579} a^{18} + \frac{40815}{3230579} a^{17} + \frac{1009982}{3230579} a^{16} - \frac{646892}{3230579} a^{15} - \frac{1522393}{3230579} a^{14} + \frac{1130673}{3230579} a^{13} + \frac{650707}{3230579} a^{12} - \frac{925796}{3230579} a^{11} - \frac{430027}{3230579} a^{10} + \frac{1529725}{3230579} a^{9} + \frac{385306}{3230579} a^{8} + \frac{1416073}{3230579} a^{7} + \frac{74281}{293689} a^{6} - \frac{1380727}{3230579} a^{5} - \frac{118466}{3230579} a^{4} + \frac{217490}{3230579} a^{3} + \frac{1510819}{3230579} a^{2} - \frac{815675}{3230579} a + \frac{881785}{3230579}$

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:  $19$
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:  \( 4389436210.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T540:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 35 conjugacy class representatives for t20n540
Character table for t20n540 is not computed

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
941011Data not computed