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Properties

Label 20.8.20249698481...1853.2
Degree $20$
Signature $[8, 6]$
Discriminant $13^{13}\cdot 401^{8}$
Root discriminant $58.25$
Ramified primes $13, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-711, -44016, -227354, -812, 91742, -81651, 66422, 17533, 28512, -4260, -6877, 1490, 347, 1402, 29, -67, -92, -49, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 49*x^17 - 92*x^16 - 67*x^15 + 29*x^14 + 1402*x^13 + 347*x^12 + 1490*x^11 - 6877*x^10 - 4260*x^9 + 28512*x^8 + 17533*x^7 + 66422*x^6 - 81651*x^5 + 91742*x^4 - 812*x^3 - 227354*x^2 - 44016*x - 711)
 
gp: K = bnfinit(x^20 + 2*x^18 - 49*x^17 - 92*x^16 - 67*x^15 + 29*x^14 + 1402*x^13 + 347*x^12 + 1490*x^11 - 6877*x^10 - 4260*x^9 + 28512*x^8 + 17533*x^7 + 66422*x^6 - 81651*x^5 + 91742*x^4 - 812*x^3 - 227354*x^2 - 44016*x - 711, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 49 x^{17} - 92 x^{16} - 67 x^{15} + 29 x^{14} + 1402 x^{13} + 347 x^{12} + 1490 x^{11} - 6877 x^{10} - 4260 x^{9} + 28512 x^{8} + 17533 x^{7} + 66422 x^{6} - 81651 x^{5} + 91742 x^{4} - 812 x^{3} - 227354 x^{2} - 44016 x - 711 \)

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:  \(202496984818189934420549080247241853=13^{13}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.25$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{3} a^{10} - \frac{4}{9} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1053} a^{18} + \frac{40}{1053} a^{17} + \frac{149}{1053} a^{16} + \frac{10}{117} a^{15} + \frac{68}{1053} a^{14} - \frac{2}{1053} a^{13} + \frac{127}{1053} a^{12} - \frac{89}{1053} a^{11} - \frac{310}{1053} a^{10} + \frac{157}{351} a^{9} + \frac{14}{117} a^{8} - \frac{62}{351} a^{7} - \frac{5}{27} a^{6} - \frac{458}{1053} a^{5} + \frac{31}{351} a^{4} - \frac{382}{1053} a^{3} + \frac{301}{1053} a^{2} + \frac{154}{351} a - \frac{4}{117}$, $\frac{1}{23510021951939461574411614362240179193240197855817} a^{19} + \frac{9297941357342983140831127545861395866505336306}{23510021951939461574411614362240179193240197855817} a^{18} - \frac{87106527367566383061212346521388273096390101404}{7836673983979820524803871454080059731080065951939} a^{17} - \frac{85467982954792724641589377190273530753369960240}{1808463227072266274954739566326167630249245988909} a^{16} + \frac{504372565471394704898864248736825984555337812312}{23510021951939461574411614362240179193240197855817} a^{15} + \frac{966638724526276933700214634833314534210802736112}{7836673983979820524803871454080059731080065951939} a^{14} + \frac{2417673970496869823603487102038556298140372193220}{23510021951939461574411614362240179193240197855817} a^{13} + \frac{2974390960636352333988028006879934190103580277719}{23510021951939461574411614362240179193240197855817} a^{12} - \frac{279505524012976657454645838490028801467016301629}{7836673983979820524803871454080059731080065951939} a^{11} - \frac{1907907095761505222757713152161669053553161174704}{23510021951939461574411614362240179193240197855817} a^{10} - \frac{1751333905182449553220984362416789811186416943908}{7836673983979820524803871454080059731080065951939} a^{9} - \frac{3276157969290023512143721429849895263588661969579}{7836673983979820524803871454080059731080065951939} a^{8} + \frac{3084422424496133185414770148701944982882055455098}{7836673983979820524803871454080059731080065951939} a^{7} - \frac{5636610828270818282133613439151983195992472191280}{23510021951939461574411614362240179193240197855817} a^{6} - \frac{148275856466293188694319563084414312723411435100}{23510021951939461574411614362240179193240197855817} a^{5} + \frac{7951177616519462006170332961611468703845899882597}{23510021951939461574411614362240179193240197855817} a^{4} - \frac{2640709352164596771155748553649543723725979032862}{7836673983979820524803871454080059731080065951939} a^{3} - \frac{5673799532711663986679542200352999966330463738195}{23510021951939461574411614362240179193240197855817} a^{2} - \frac{3563054826350278367809778130567906716316992251559}{7836673983979820524803871454080059731080065951939} a + \frac{1158429201847490912443181211304171424259138661991}{2612224661326606841601290484693353243693355317313}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 24068080992.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.160801.1, 10.10.9600508843720093.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
401Data not computed