↑

Properties

Label 20.0.81693426134...7408.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 3^{15}\cdot 11^{18}$
Root discriminant $27.90$
Ramified primes $2, 3, 11$
Class number $10$
Class group $[10]$
Galois group $C_5\times D_4$ (as 20T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![661, 95, 406, 2206, 1665, 1939, 3077, 1545, -442, 352, 1629, -836, 654, -64, 303, -300, 251, -117, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 40*x^18 - 117*x^17 + 251*x^16 - 300*x^15 + 303*x^14 - 64*x^13 + 654*x^12 - 836*x^11 + 1629*x^10 + 352*x^9 - 442*x^8 + 1545*x^7 + 3077*x^6 + 1939*x^5 + 1665*x^4 + 2206*x^3 + 406*x^2 + 95*x + 661)
 
gp: K = bnfinit(x^20 - 8*x^19 + 40*x^18 - 117*x^17 + 251*x^16 - 300*x^15 + 303*x^14 - 64*x^13 + 654*x^12 - 836*x^11 + 1629*x^10 + 352*x^9 - 442*x^8 + 1545*x^7 + 3077*x^6 + 1939*x^5 + 1665*x^4 + 2206*x^3 + 406*x^2 + 95*x + 661, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 40 x^{18} - 117 x^{17} + 251 x^{16} - 300 x^{15} + 303 x^{14} - 64 x^{13} + 654 x^{12} - 836 x^{11} + 1629 x^{10} + 352 x^{9} - 442 x^{8} + 1545 x^{7} + 3077 x^{6} + 1939 x^{5} + 1665 x^{4} + 2206 x^{3} + 406 x^{2} + 95 x + 661 \)

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:  \(81693426134005631737181457408=2^{10}\cdot 3^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.90$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{14} - \frac{2}{11} a^{13} - \frac{5}{11} a^{12} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{3}{11} a^{11} + \frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{1}{11} a^{6} + \frac{1}{11} a^{4} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{17} - \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{5}{11} a^{10} - \frac{5}{11} a^{9} - \frac{5}{11} a^{8} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11}$, $\frac{1}{253} a^{18} + \frac{10}{253} a^{17} + \frac{6}{253} a^{16} - \frac{10}{253} a^{15} + \frac{6}{253} a^{14} - \frac{53}{253} a^{13} - \frac{95}{253} a^{12} - \frac{61}{253} a^{10} - \frac{94}{253} a^{9} - \frac{50}{253} a^{8} - \frac{51}{253} a^{7} + \frac{94}{253} a^{6} + \frac{98}{253} a^{5} - \frac{95}{253} a^{4} - \frac{89}{253} a^{3} - \frac{8}{253} a^{2} - \frac{98}{253} a - \frac{83}{253}$, $\frac{1}{22431353948507038535844128809470287} a^{19} + \frac{81433707519192432396296475854}{22431353948507038535844128809470287} a^{18} + \frac{316082277912393390356473583503499}{22431353948507038535844128809470287} a^{17} - \frac{972303177232898604881573676889281}{22431353948507038535844128809470287} a^{16} + \frac{411384164888918666901496031536062}{22431353948507038535844128809470287} a^{15} + \frac{8119981471208888626613272323702124}{22431353948507038535844128809470287} a^{14} + \frac{9324138043007169363662029327487631}{22431353948507038535844128809470287} a^{13} + \frac{4857274325875180369857840623784198}{22431353948507038535844128809470287} a^{12} + \frac{93114641552759051363988633641759}{2039213995318821685076738982679117} a^{11} - \frac{6502288117437480311378536155316099}{22431353948507038535844128809470287} a^{10} + \frac{1965465695866137402612345107764369}{22431353948507038535844128809470287} a^{9} - \frac{1770897004483675254872039816182768}{22431353948507038535844128809470287} a^{8} - \frac{2206971545805324999297081613499902}{22431353948507038535844128809470287} a^{7} - \frac{507021449398969629066444553623179}{2039213995318821685076738982679117} a^{6} - \frac{138288655630054278773978109515833}{22431353948507038535844128809470287} a^{5} + \frac{9379311904305879535800583169552173}{22431353948507038535844128809470287} a^{4} + \frac{2548946808832522628008714538714569}{22431353948507038535844128809470287} a^{3} + \frac{10059138963557946959802962641616994}{22431353948507038535844128809470287} a^{2} - \frac{5772634823963015006412878791774123}{22431353948507038535844128809470287} a - \frac{1053945193214814039369760740757571}{22431353948507038535844128809470287}$

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

Class group and class number

$C_{10}$, which has order $10$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 125582.779517 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times D_4$ (as 20T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 25 conjugacy class representatives for $C_5\times D_4$
Character table for $C_5\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), 4.0.13068.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 R $20$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$