↑

Properties

Label 20.20.3165599696...5281.2
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 43^{4}\cdot 67^{4}$
Root discriminant $33.50$
Ramified primes $11, 43, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -55, 2, 625, -173, -3142, 480, 8089, 385, -10603, -2024, 6862, 1592, -2298, -470, 399, 54, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 33 x^{18} + 54 x^{17} + 399 x^{16} - 470 x^{15} - 2298 x^{14} + 1592 x^{13} + 6862 x^{12} - 2024 x^{11} - 10603 x^{10} + 385 x^{9} + 8089 x^{8} + 480 x^{7} - 3142 x^{6} - 173 x^{5} + 625 x^{4} + 2 x^{3} - 55 x^{2} + 3 x + 1 \)

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:  \(3165599696740582502041142585281=11^{16}\cdot 43^{4}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.50$
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:  $11, 43, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{5}{11} a^{14} - \frac{4}{11} a^{13} + \frac{3}{11} a^{11} + \frac{4}{11} a^{10} - \frac{4}{11} a^{9} + \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{5}{11} a^{6} + \frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{4}{11} a^{14} + \frac{2}{11} a^{13} + \frac{3}{11} a^{12} - \frac{3}{11} a^{11} + \frac{5}{11} a^{10} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} - \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{4}{11} a^{5} + \frac{1}{11} a^{4} + \frac{2}{11} a^{3} + \frac{1}{11} a^{2} + \frac{5}{11}$, $\frac{1}{11} a^{17} - \frac{3}{11} a^{13} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} + \frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} - \frac{4}{11} a^{5} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{18} - \frac{3}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{2}{11} a^{9} - \frac{2}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{3}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{599891102963471} a^{19} - \frac{11953905355039}{599891102963471} a^{18} - \frac{10702609768043}{599891102963471} a^{17} - \frac{1837671988623}{54535554814861} a^{16} + \frac{121813648770}{54535554814861} a^{15} + \frac{130915099735718}{599891102963471} a^{14} + \frac{83998874557407}{599891102963471} a^{13} + \frac{256678055273520}{599891102963471} a^{12} - \frac{121289523549966}{599891102963471} a^{11} - \frac{103954886844602}{599891102963471} a^{10} + \frac{283123979059233}{599891102963471} a^{9} + \frac{161477848611038}{599891102963471} a^{8} + \frac{273201634975738}{599891102963471} a^{7} - \frac{178400656581219}{599891102963471} a^{6} - \frac{1172042131983}{26082221867977} a^{5} + \frac{233838651208639}{599891102963471} a^{4} - \frac{85132545774436}{599891102963471} a^{3} + \frac{93402393057284}{599891102963471} a^{2} - \frac{190462426347679}{599891102963471} a - \frac{146367845238720}{599891102963471}$

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:  \( 500606346.877 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.617567936161.1 x2, 10.10.1779213224079841.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$