↑

Properties

Label 20.0.16314718705...2096.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 23^{4}$
Root discriminant $32.41$
Ramified primes $2, 11, 23$
Class number $28$ (GRH)
Class group $[2, 14]$ (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, 0, -97336, 0, 103684, 0, -10258, 0, 10289, 0, 232, 0, -206, 0, 64, 0, 71, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 + 71*x^16 + 64*x^14 - 206*x^12 + 232*x^10 + 10289*x^8 - 10258*x^6 + 103684*x^4 - 97336*x^2 + 279841)
 
gp: K = bnfinit(x^20 + 15*x^18 + 71*x^16 + 64*x^14 - 206*x^12 + 232*x^10 + 10289*x^8 - 10258*x^6 + 103684*x^4 - 97336*x^2 + 279841, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} + 71 x^{16} + 64 x^{14} - 206 x^{12} + 232 x^{10} + 10289 x^{8} - 10258 x^{6} + 103684 x^{4} - 97336 x^{2} + 279841 \)

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:  \(1631471870594231356489225732096=2^{20}\cdot 11^{18}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.41$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{9}{23} a^{10} + \frac{11}{23} a^{8} + \frac{7}{23} a^{6} - \frac{6}{23} a^{4} + \frac{8}{23} a^{2}$, $\frac{1}{23} a^{13} - \frac{9}{23} a^{11} + \frac{11}{23} a^{9} + \frac{7}{23} a^{7} - \frac{6}{23} a^{5} + \frac{8}{23} a^{3}$, $\frac{1}{23} a^{14} - \frac{1}{23} a^{10} - \frac{9}{23} a^{8} + \frac{11}{23} a^{6} + \frac{3}{23} a^{2}$, $\frac{1}{23} a^{15} - \frac{1}{23} a^{11} - \frac{9}{23} a^{9} + \frac{11}{23} a^{7} + \frac{3}{23} a^{3}$, $\frac{1}{529} a^{16} - \frac{8}{529} a^{14} + \frac{2}{529} a^{12} + \frac{179}{529} a^{10} - \frac{229}{529} a^{8} + \frac{25}{529} a^{6} + \frac{123}{529} a^{4} + \frac{10}{23} a^{2}$, $\frac{1}{529} a^{17} - \frac{8}{529} a^{15} + \frac{2}{529} a^{13} + \frac{179}{529} a^{11} - \frac{229}{529} a^{9} + \frac{25}{529} a^{7} + \frac{123}{529} a^{5} + \frac{10}{23} a^{3}$, $\frac{1}{11125874656120380241097} a^{18} - \frac{2926251931159083958}{11125874656120380241097} a^{16} - \frac{18122970063773427608}{11125874656120380241097} a^{14} + \frac{43251677263974864707}{11125874656120380241097} a^{12} + \frac{5156171158156640664318}{11125874656120380241097} a^{10} + \frac{2245317749117358908551}{11125874656120380241097} a^{8} - \frac{947118405962587949161}{11125874656120380241097} a^{6} + \frac{34446049326952572829}{483733680700886097439} a^{4} - \frac{9230726810299226686}{21031899160908091193} a^{2} - \frac{390344134562202272}{914430398300351791}$, $\frac{1}{11125874656120380241097} a^{19} - \frac{2926251931159083958}{11125874656120380241097} a^{17} - \frac{18122970063773427608}{11125874656120380241097} a^{15} + \frac{43251677263974864707}{11125874656120380241097} a^{13} + \frac{5156171158156640664318}{11125874656120380241097} a^{11} + \frac{2245317749117358908551}{11125874656120380241097} a^{9} - \frac{947118405962587949161}{11125874656120380241097} a^{7} + \frac{34446049326952572829}{483733680700886097439} a^{5} - \frac{9230726810299226686}{21031899160908091193} a^{3} - \frac{390344134562202272}{914430398300351791} a$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (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:  \( \frac{95071350116501926}{11125874656120380241097} a^{18} + \frac{1461930003783807594}{11125874656120380241097} a^{16} + \frac{6728997253956413954}{11125874656120380241097} a^{14} - \frac{821401577486955458}{11125874656120380241097} a^{12} - \frac{70906495176682791468}{11125874656120380241097} a^{10} - \frac{60335580420719380111}{11125874656120380241097} a^{8} + \frac{1177854349552501171394}{11125874656120380241097} a^{6} - \frac{20310679375577820757}{483733680700886097439} a^{4} + \frac{8124869876898125875}{21031899160908091193} a^{2} - \frac{405117622369293242}{914430398300351791} \) (order $22$)
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:  \( 2417625.83378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 10.0.1277290832423936.2, \(\Q(\zeta_{11})\), 10.10.116117348402176.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$