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Properties

Label 20.0.27884995438...8601.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{10}\cdot 401^{10}$
Root discriminant $66.42$
Ramified primes $11, 401$
Class number $20886$ (GRH)
Class group $[59, 354]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16791547, -25350279, 41501494, -42150471, 40784529, -31190787, 22178110, -13413591, 7448650, -3594273, 1564827, -585015, 188347, -47673, 8221, -90, -435, 120, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + x^18 + 120*x^17 - 435*x^16 - 90*x^15 + 8221*x^14 - 47673*x^13 + 188347*x^12 - 585015*x^11 + 1564827*x^10 - 3594273*x^9 + 7448650*x^8 - 13413591*x^7 + 22178110*x^6 - 31190787*x^5 + 40784529*x^4 - 42150471*x^3 + 41501494*x^2 - 25350279*x + 16791547)
 
gp: K = bnfinit(x^20 - 6*x^19 + x^18 + 120*x^17 - 435*x^16 - 90*x^15 + 8221*x^14 - 47673*x^13 + 188347*x^12 - 585015*x^11 + 1564827*x^10 - 3594273*x^9 + 7448650*x^8 - 13413591*x^7 + 22178110*x^6 - 31190787*x^5 + 40784529*x^4 - 42150471*x^3 + 41501494*x^2 - 25350279*x + 16791547, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + x^{18} + 120 x^{17} - 435 x^{16} - 90 x^{15} + 8221 x^{14} - 47673 x^{13} + 188347 x^{12} - 585015 x^{11} + 1564827 x^{10} - 3594273 x^{9} + 7448650 x^{8} - 13413591 x^{7} + 22178110 x^{6} - 31190787 x^{5} + 40784529 x^{4} - 42150471 x^{3} + 41501494 x^{2} - 25350279 x + 16791547 \)

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:  \(2788499543846837793819477624443028601=11^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.42$
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:  $11, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{4}{27} a^{11} - \frac{2}{27} a^{10} + \frac{1}{9} a^{9} + \frac{4}{27} a^{8} + \frac{1}{27} a^{7} - \frac{1}{27} a^{6} - \frac{1}{27} a^{4} - \frac{4}{27} a^{3} + \frac{11}{27} a^{2} - \frac{2}{27} a + \frac{5}{27}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{12} + \frac{2}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{9} a^{8} - \frac{2}{27} a^{7} - \frac{2}{27} a^{6} - \frac{1}{27} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{10}{27} a^{2} + \frac{7}{27} a - \frac{8}{27}$, $\frac{1}{81} a^{16} + \frac{1}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{81} a^{12} - \frac{4}{81} a^{11} - \frac{13}{81} a^{10} + \frac{13}{81} a^{9} + \frac{1}{9} a^{8} + \frac{4}{81} a^{7} + \frac{13}{81} a^{6} - \frac{4}{81} a^{5} - \frac{32}{81} a^{4} - \frac{1}{3} a^{3} - \frac{11}{81} a^{2} + \frac{19}{81} a + \frac{38}{81}$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{1}{27} a^{11} - \frac{4}{81} a^{10} - \frac{1}{81} a^{9} + \frac{13}{81} a^{8} + \frac{1}{27} a^{7} + \frac{4}{81} a^{6} - \frac{31}{81} a^{5} - \frac{4}{81} a^{4} - \frac{20}{81} a^{3} - \frac{1}{3} a^{2} + \frac{40}{81} a + \frac{19}{81}$, $\frac{1}{161109} a^{18} - \frac{19}{3159} a^{17} + \frac{302}{53703} a^{16} + \frac{10}{1377} a^{15} + \frac{35}{17901} a^{14} + \frac{176}{17901} a^{13} - \frac{5801}{161109} a^{12} + \frac{61}{1053} a^{11} + \frac{1322}{53703} a^{10} - \frac{2402}{53703} a^{9} + \frac{443}{17901} a^{8} - \frac{631}{5967} a^{7} - \frac{19262}{161109} a^{6} + \frac{8897}{17901} a^{5} + \frac{23293}{53703} a^{4} - \frac{7507}{17901} a^{3} - \frac{7283}{17901} a^{2} + \frac{898}{4131} a - \frac{77057}{161109}$, $\frac{1}{140291403645290794825489071610217846865365378709} a^{19} + \frac{25861359253125228803801470086432400244039}{15587933738365643869498785734468649651707264301} a^{18} + \frac{216129137641802563896491570699939044344723947}{46763801215096931608496357203405948955121792903} a^{17} - \frac{65565513434506315723752423942131526432706694}{15587933738365643869498785734468649651707264301} a^{16} - \frac{167702248290225814236359193246764797238667937}{15587933738365643869498785734468649651707264301} a^{15} + \frac{229796782427789320288808618733669045728148199}{15587933738365643869498785734468649651707264301} a^{14} + \frac{971084611788867134481758973266100784362314764}{140291403645290794825489071610217846865365378709} a^{13} + \frac{593312906707913082246064977865656492162689258}{46763801215096931608496357203405948955121792903} a^{12} + \frac{6011536676705619744524970435062039879040747074}{46763801215096931608496357203405948955121792903} a^{11} + \frac{3297298310654903198181977348866000528028002738}{46763801215096931608496357203405948955121792903} a^{10} + \frac{36668601884631456671076707330476477691232758}{1199071826028126451499906594959126896285174177} a^{9} + \frac{1989964601390464577267581414728370287201778783}{15587933738365643869498785734468649651707264301} a^{8} + \frac{3001827932329503456996879159128286297770252}{545880948036150952628362146343260104534495637} a^{7} + \frac{7457756470623861661412272725658432589981983775}{46763801215096931608496357203405948955121792903} a^{6} - \frac{10260391258840955371184684720594342574344295701}{46763801215096931608496357203405948955121792903} a^{5} - \frac{6648785675548495305268126816007628141020963584}{15587933738365643869498785734468649651707264301} a^{4} + \frac{2028726376090536953746612196365277912718253103}{15587933738365643869498785734468649651707264301} a^{3} - \frac{427814155243074088726778125364415393421068383}{46763801215096931608496357203405948955121792903} a^{2} - \frac{4943263660178360835355203590178811967186288369}{140291403645290794825489071610217846865365378709} a - \frac{21940820249708242771660955100420470102403781303}{46763801215096931608496357203405948955121792903}$

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

Class group and class number

$C_{59}\times C_{354}$, which has order $20886$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-4411}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.1669880098643863051.1 x5, 10.0.4164289522802651.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed