↑

Properties

Label 20.4.20249698481...1853.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{13}\cdot 401^{8}$
Root discriminant $58.25$
Ramified primes $13, 401$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T426

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29241, 98838, 17648, -277651, -131553, 197227, 184507, 57368, -72009, -73783, -2238, 5359, 2836, -2943, -1013, 311, 8, -4, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 9*x^18 - 4*x^17 + 8*x^16 + 311*x^15 - 1013*x^14 - 2943*x^13 + 2836*x^12 + 5359*x^11 - 2238*x^10 - 73783*x^9 - 72009*x^8 + 57368*x^7 + 184507*x^6 + 197227*x^5 - 131553*x^4 - 277651*x^3 + 17648*x^2 + 98838*x + 29241)
 
gp: K = bnfinit(x^20 - 2*x^19 - 9*x^18 - 4*x^17 + 8*x^16 + 311*x^15 - 1013*x^14 - 2943*x^13 + 2836*x^12 + 5359*x^11 - 2238*x^10 - 73783*x^9 - 72009*x^8 + 57368*x^7 + 184507*x^6 + 197227*x^5 - 131553*x^4 - 277651*x^3 + 17648*x^2 + 98838*x + 29241, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 9 x^{18} - 4 x^{17} + 8 x^{16} + 311 x^{15} - 1013 x^{14} - 2943 x^{13} + 2836 x^{12} + 5359 x^{11} - 2238 x^{10} - 73783 x^{9} - 72009 x^{8} + 57368 x^{7} + 184507 x^{6} + 197227 x^{5} - 131553 x^{4} - 277651 x^{3} + 17648 x^{2} + 98838 x + 29241 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{57} a^{18} - \frac{7}{57} a^{17} + \frac{7}{57} a^{16} - \frac{20}{57} a^{15} - \frac{2}{19} a^{14} - \frac{20}{57} a^{13} - \frac{1}{57} a^{12} + \frac{26}{57} a^{11} + \frac{8}{57} a^{10} - \frac{1}{57} a^{9} - \frac{10}{57} a^{8} - \frac{13}{57} a^{7} + \frac{3}{19} a^{6} - \frac{1}{3} a^{5} - \frac{2}{57} a^{4} - \frac{2}{57} a^{3} - \frac{2}{19} a^{2} + \frac{26}{57} a$, $\frac{1}{16085334451646366451883175747568407841627126225377751663} a^{19} + \frac{53902480711091884909077202854538409733917016238571734}{16085334451646366451883175747568407841627126225377751663} a^{18} + \frac{674251280153888331832161969582287736586639063167184459}{5361778150548788817294391915856135947209042075125917221} a^{17} + \frac{526143589380223196302290724068743954148288643319514719}{16085334451646366451883175747568407841627126225377751663} a^{16} + \frac{2138742807262999415911755601962062364047374580284594187}{16085334451646366451883175747568407841627126225377751663} a^{15} - \frac{4785621859029498514564168013247588412020977986099807660}{16085334451646366451883175747568407841627126225377751663} a^{14} + \frac{2400551161724620768892000347341364207446129983770795465}{16085334451646366451883175747568407841627126225377751663} a^{13} - \frac{989282642214845476370589207327710934621447468063040901}{5361778150548788817294391915856135947209042075125917221} a^{12} - \frac{1052220847549605440388877793860596206336340515125873778}{16085334451646366451883175747568407841627126225377751663} a^{11} + \frac{3828223367048535617358096508937236298233044738244205959}{16085334451646366451883175747568407841627126225377751663} a^{10} + \frac{173434874713738098367673767930554941078062730298157058}{5361778150548788817294391915856135947209042075125917221} a^{9} - \frac{161813746166226152465946545740889454438341940643856143}{16085334451646366451883175747568407841627126225377751663} a^{8} + \frac{2191878311640464562326868465579938482213452160936311859}{5361778150548788817294391915856135947209042075125917221} a^{7} - \frac{2566849053009169256129729542838109711920728226338636873}{16085334451646366451883175747568407841627126225377751663} a^{6} + \frac{5652617905012657771437461443828043010348011215995774745}{16085334451646366451883175747568407841627126225377751663} a^{5} + \frac{4250242576508047303489070884925108989538087105636374378}{16085334451646366451883175747568407841627126225377751663} a^{4} - \frac{843708360198508970035265748105400729443409064711378467}{5361778150548788817294391915856135947209042075125917221} a^{3} + \frac{7028928925795169117728295300645730222230392466020182749}{16085334451646366451883175747568407841627126225377751663} a^{2} - \frac{6688687682711430689561793051583461853005121285144884130}{16085334451646366451883175747568407841627126225377751663} a - \frac{35114789280484943327681819861945270217384842630368142}{94066283342961207320954244137826946442263896054840653}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 3377356877.65 \) (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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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