↑

Properties

Label 20.8.82969775896...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 11^{16}\cdot 43^{2}$
Root discriminant $22.18$
Ramified primes $5, 11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -4, -20, -54, -69, -31, 59, 99, 92, -33, -92, 99, -59, -31, 69, -54, 20, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 4*x^18 + 20*x^17 - 54*x^16 + 69*x^15 - 31*x^14 - 59*x^13 + 99*x^12 - 92*x^11 - 33*x^10 + 92*x^9 + 99*x^8 + 59*x^7 - 31*x^6 - 69*x^5 - 54*x^4 - 20*x^3 - 4*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 4*x^18 + 20*x^17 - 54*x^16 + 69*x^15 - 31*x^14 - 59*x^13 + 99*x^12 - 92*x^11 - 33*x^10 + 92*x^9 + 99*x^8 + 59*x^7 - 31*x^6 - 69*x^5 - 54*x^4 - 20*x^3 - 4*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 4 x^{18} + 20 x^{17} - 54 x^{16} + 69 x^{15} - 31 x^{14} - 59 x^{13} + 99 x^{12} - 92 x^{11} - 33 x^{10} + 92 x^{9} + 99 x^{8} + 59 x^{7} - 31 x^{6} - 69 x^{5} - 54 x^{4} - 20 x^{3} - 4 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(829697758962352789931640625=5^{10}\cdot 11^{16}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.18$
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:  $5, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{989} a^{16} + \frac{94}{989} a^{15} + \frac{175}{989} a^{14} + \frac{10}{43} a^{13} + \frac{54}{989} a^{12} + \frac{250}{989} a^{11} - \frac{9}{43} a^{10} - \frac{466}{989} a^{9} - \frac{174}{989} a^{8} + \frac{466}{989} a^{7} - \frac{9}{43} a^{6} - \frac{250}{989} a^{5} + \frac{54}{989} a^{4} - \frac{10}{43} a^{3} + \frac{175}{989} a^{2} - \frac{94}{989} a + \frac{1}{989}$, $\frac{1}{989} a^{17} + \frac{240}{989} a^{15} - \frac{396}{989} a^{14} + \frac{192}{989} a^{13} + \frac{119}{989} a^{12} + \frac{29}{989} a^{11} + \frac{201}{989} a^{10} + \frac{114}{989} a^{9} + \frac{9}{989} a^{8} + \frac{494}{989} a^{7} + \frac{417}{989} a^{6} - \frac{182}{989} a^{5} - \frac{361}{989} a^{4} + \frac{37}{989} a^{3} + \frac{269}{989} a^{2} - \frac{64}{989} a - \frac{94}{989}$, $\frac{1}{107801} a^{18} + \frac{47}{107801} a^{17} + \frac{11}{107801} a^{16} - \frac{21521}{107801} a^{15} + \frac{31504}{107801} a^{14} + \frac{35593}{107801} a^{13} - \frac{45315}{107801} a^{12} + \frac{10577}{107801} a^{11} - \frac{45892}{107801} a^{10} - \frac{28357}{107801} a^{9} - \frac{39346}{107801} a^{8} - \frac{39563}{107801} a^{7} - \frac{44937}{107801} a^{6} - \frac{44631}{107801} a^{5} + \frac{21143}{107801} a^{4} + \frac{53689}{107801} a^{3} + \frac{15031}{107801} a^{2} - \frac{30037}{107801} a - \frac{50141}{107801}$, $\frac{1}{21236797} a^{19} + \frac{12}{21236797} a^{18} - \frac{2397}{21236797} a^{17} - \frac{3594}{21236797} a^{16} + \frac{3185355}{21236797} a^{15} - \frac{3381553}{21236797} a^{14} + \frac{1372781}{21236797} a^{13} + \frac{6806693}{21236797} a^{12} + \frac{1983766}{21236797} a^{11} - \frac{7217129}{21236797} a^{10} + \frac{8179522}{21236797} a^{9} - \frac{22991}{21236797} a^{8} + \frac{1087760}{21236797} a^{7} - \frac{8401845}{21236797} a^{6} + \frac{378124}{21236797} a^{5} - \frac{1685846}{21236797} a^{4} + \frac{1765398}{21236797} a^{3} - \frac{8444670}{21236797} a^{2} + \frac{8922947}{21236797} a - \frac{8396126}{21236797}$

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:  \( 766699.913561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.4.28804474634375.1, 10.10.669871503125.1, 10.4.9217431883.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$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}$
43.2.0.1$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}$