↑

Properties

Label 20.4.17287511078...4801.1
Degree $20$
Signature $[4, 8]$
Discriminant $401^{12}$
Root discriminant $36.47$
Ramified prime $401$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^4:D_5$ (as 20T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 567, 2745, 4197, 7697, 5475, 10393, 1900, 5801, -45, 4868, 2570, 1743, -843, -775, -97, 85, 30, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 14*x^18 + 30*x^17 + 85*x^16 - 97*x^15 - 775*x^14 - 843*x^13 + 1743*x^12 + 2570*x^11 + 4868*x^10 - 45*x^9 + 5801*x^8 + 1900*x^7 + 10393*x^6 + 5475*x^5 + 7697*x^4 + 4197*x^3 + 2745*x^2 + 567*x + 81)
 
gp: K = bnfinit(x^20 - 2*x^19 - 14*x^18 + 30*x^17 + 85*x^16 - 97*x^15 - 775*x^14 - 843*x^13 + 1743*x^12 + 2570*x^11 + 4868*x^10 - 45*x^9 + 5801*x^8 + 1900*x^7 + 10393*x^6 + 5475*x^5 + 7697*x^4 + 4197*x^3 + 2745*x^2 + 567*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 14 x^{18} + 30 x^{17} + 85 x^{16} - 97 x^{15} - 775 x^{14} - 843 x^{13} + 1743 x^{12} + 2570 x^{11} + 4868 x^{10} - 45 x^{9} + 5801 x^{8} + 1900 x^{7} + 10393 x^{6} + 5475 x^{5} + 7697 x^{4} + 4197 x^{3} + 2745 x^{2} + 567 x + 81 \)

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:  \(17287511078984605626766090564801=401^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.47$
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:  $401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{15} + \frac{1}{9} a^{14} - \frac{1}{6} a^{13} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{9} a^{10} - \frac{1}{6} a^{9} + \frac{4}{9} a^{8} - \frac{5}{18} a^{7} + \frac{2}{9} a^{6} - \frac{1}{2} a^{5} + \frac{1}{18} a^{4} + \frac{5}{18} a^{3} - \frac{2}{9} a^{2} - \frac{1}{6} a$, $\frac{1}{18} a^{17} + \frac{1}{18} a^{15} - \frac{1}{18} a^{14} + \frac{1}{9} a^{13} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{6} a^{8} - \frac{1}{18} a^{7} - \frac{5}{18} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{18} a^{3} + \frac{5}{18} a^{2} + \frac{1}{6} a$, $\frac{1}{486} a^{18} - \frac{1}{162} a^{17} - \frac{4}{243} a^{16} + \frac{29}{486} a^{15} - \frac{49}{486} a^{14} + \frac{13}{162} a^{13} - \frac{35}{243} a^{12} + \frac{73}{486} a^{11} + \frac{1}{243} a^{10} + \frac{11}{162} a^{9} + \frac{31}{243} a^{8} + \frac{235}{486} a^{7} + \frac{1}{486} a^{6} - \frac{23}{162} a^{5} - \frac{73}{243} a^{4} - \frac{175}{486} a^{3} + \frac{8}{27} a^{2} - \frac{2}{27} a + \frac{4}{9}$, $\frac{1}{543645992555640451547997242921957826} a^{19} + \frac{46859782603855986581573906970611}{60405110283960050171999693657995314} a^{18} - \frac{7447726140990545020806343328247796}{271822996277820225773998621460978913} a^{17} + \frac{5631682705377930699852545938082221}{271822996277820225773998621460978913} a^{16} - \frac{665298728096918908276434823965871}{11818391142513922859739070498303431} a^{15} - \frac{8331978086661158193170871780842369}{60405110283960050171999693657995314} a^{14} + \frac{24705411550256572435397018231314265}{543645992555640451547997242921957826} a^{13} + \frac{34509064594782643166739417015704318}{271822996277820225773998621460978913} a^{12} + \frac{4229513454780571142216050161814571}{543645992555640451547997242921957826} a^{11} - \frac{5941054042026991195883261153158361}{181215330851880150515999080973985942} a^{10} - \frac{26210009966031326305293987729826585}{543645992555640451547997242921957826} a^{9} - \frac{560112616544259495621020886690799}{23636782285027845719478140996606862} a^{8} - \frac{27826359374631761736447476475184984}{271822996277820225773998621460978913} a^{7} + \frac{33177341551023640977163117292220041}{181215330851880150515999080973985942} a^{6} + \frac{44586927036033561901421747597683105}{543645992555640451547997242921957826} a^{5} - \frac{64548787775704144437240961485276371}{271822996277820225773998621460978913} a^{4} - \frac{65584828688800042693428434896341607}{181215330851880150515999080973985942} a^{3} + \frac{14908105079965531603682663181747130}{30202555141980025085999846828997657} a^{2} + \frac{5318934194291314408209058382870053}{20135036761320016723999897885998438} a + \frac{1634116753899063775485981200344411}{3355839460220002787333316314333073}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 35976171.7676 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.10368641602001.1, 10.6.10368641602001.2, 10.2.4157825282402401.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
401Data not computed