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Properties

Label 20.0.83086102921...3888.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}$
Root discriminant $88.30$
Ramified primes $2, 3, 7, 11, 13$
Class number $1450$ (GRH)
Class group $[5, 290]$ (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17487613, 4594876, 45176274, 3387604, 24408428, -3181752, 3541782, -313688, -413105, 250450, -138456, 20752, -11369, -1574, 684, -298, 177, -18, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^18 - 18*x^17 + 177*x^16 - 298*x^15 + 684*x^14 - 1574*x^13 - 11369*x^12 + 20752*x^11 - 138456*x^10 + 250450*x^9 - 413105*x^8 - 313688*x^7 + 3541782*x^6 - 3181752*x^5 + 24408428*x^4 + 3387604*x^3 + 45176274*x^2 + 4594876*x + 17487613)
 
gp: K = bnfinit(x^20 - 2*x^19 + 10*x^18 - 18*x^17 + 177*x^16 - 298*x^15 + 684*x^14 - 1574*x^13 - 11369*x^12 + 20752*x^11 - 138456*x^10 + 250450*x^9 - 413105*x^8 - 313688*x^7 + 3541782*x^6 - 3181752*x^5 + 24408428*x^4 + 3387604*x^3 + 45176274*x^2 + 4594876*x + 17487613, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 10 x^{18} - 18 x^{17} + 177 x^{16} - 298 x^{15} + 684 x^{14} - 1574 x^{13} - 11369 x^{12} + 20752 x^{11} - 138456 x^{10} + 250450 x^{9} - 413105 x^{8} - 313688 x^{7} + 3541782 x^{6} - 3181752 x^{5} + 24408428 x^{4} + 3387604 x^{3} + 45176274 x^{2} + 4594876 x + 17487613 \)

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:  \(830861029216363101082988400989215653888=2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.30$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{16} - \frac{2}{13} a^{14} + \frac{5}{13} a^{13} + \frac{1}{13} a^{12} - \frac{1}{13} a^{11} - \frac{2}{13} a^{10} - \frac{4}{13} a^{9} - \frac{3}{13} a^{8} - \frac{6}{13} a^{7} + \frac{1}{13} a^{6} + \frac{1}{13} a^{5} + \frac{4}{13} a^{4} - \frac{3}{13} a^{3} + \frac{2}{13} a^{2}$, $\frac{1}{299} a^{18} - \frac{3}{299} a^{17} - \frac{80}{299} a^{16} - \frac{119}{299} a^{15} - \frac{107}{299} a^{14} + \frac{112}{299} a^{13} + \frac{81}{299} a^{12} - \frac{19}{299} a^{11} + \frac{4}{13} a^{10} - \frac{136}{299} a^{9} - \frac{44}{299} a^{8} + \frac{68}{299} a^{7} + \frac{135}{299} a^{6} - \frac{148}{299} a^{5} + \frac{9}{23} a^{4} - \frac{62}{299} a^{3} + \frac{60}{299} a^{2} + \frac{11}{23} a$, $\frac{1}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{19} + \frac{6669763952918214033872120220997722172636148102807029196013317753816}{21101796282140126786212621099279299208177546951277032844465424368187871} a^{18} - \frac{77403407682751759782166994994041065340913094289407763218755690774623}{2828075790389913899183134786501349378415547529552592030701551719447859} a^{17} - \frac{129423333839445137995055916097748055777664557219135525496206958617930883}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{16} - \frac{20084698114730680879030111350888751354970885902771577140736321178845252}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{15} - \frac{39187367142429840491865456678768697859505877869817290236363610004455095}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{14} + \frac{77226131836053912063789212797239945983134585924385339386554000511817861}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{13} + \frac{123881079294492304787498393940920947950536927758043614806962364508145227}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{12} - \frac{65689300231663841221358077201753139277247780368136879682679098869353306}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{11} - \frac{101369354837939649525140261033874563534439496976427308544190309427995184}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{10} - \frac{26931694394844371938631675504236377658373908979388913085703048323552384}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{9} + \frac{111352520400929696103495212034074213999428085781487021729359985558595653}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{8} - \frac{112002646194057066697424842449372117185848174594036557129375742665649586}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{7} - \frac{111596420706140072176979994795661821402427724682419477078610735851885753}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{6} + \frac{24363081822616424692116830028136724732307761191962462140648314125604192}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{5} - \frac{44732083085441349488833134997768327886546855833568585025625673148142121}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{4} - \frac{44943268362829890419546960215851261350085852395769991972681071906274882}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{3} - \frac{67183128602438401802975577825035129921441152598460083785260098075937777}{274323351667821648220764074290630889706308110366601426978050516786442323} a^{2} + \frac{8237760803961315219353335923451300196985705319583541681004861518419819}{21101796282140126786212621099279299208177546951277032844465424368187871} a + \frac{233604053025801202763227866895785886059254544525798057605943363689897}{917469403571309860270113960838230400355545519620740558455018450790777}$

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

Class group and class number

$C_{5}\times C_{290}$, which has order $1450$ (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:  \( -1 \) (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:  \( 161877573.325 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.0.7488.1, 10.10.249828821987576832.1

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 R R $20$ R R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$