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Properties

Label 20.8.33966664043...5184.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $26.70$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -26, 119, 118, -1201, 1364, 1929, -5428, 3775, -22, 364, -2972, 2758, -300, -1036, 734, -153, -52, 40, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 40*x^18 - 52*x^17 - 153*x^16 + 734*x^15 - 1036*x^14 - 300*x^13 + 2758*x^12 - 2972*x^11 + 364*x^10 - 22*x^9 + 3775*x^8 - 5428*x^7 + 1929*x^6 + 1364*x^5 - 1201*x^4 + 118*x^3 + 119*x^2 - 26*x - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 40*x^18 - 52*x^17 - 153*x^16 + 734*x^15 - 1036*x^14 - 300*x^13 + 2758*x^12 - 2972*x^11 + 364*x^10 - 22*x^9 + 3775*x^8 - 5428*x^7 + 1929*x^6 + 1364*x^5 - 1201*x^4 + 118*x^3 + 119*x^2 - 26*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 40 x^{18} - 52 x^{17} - 153 x^{16} + 734 x^{15} - 1036 x^{14} - 300 x^{13} + 2758 x^{12} - 2972 x^{11} + 364 x^{10} - 22 x^{9} + 3775 x^{8} - 5428 x^{7} + 1929 x^{6} + 1364 x^{5} - 1201 x^{4} + 118 x^{3} + 119 x^{2} - 26 x - 1 \)

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:  \(33966664043402470640067805184=2^{20}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{14} - \frac{5}{11} a^{13} + \frac{3}{11} a^{12} + \frac{5}{11} a^{11} + \frac{2}{11} a^{9} + \frac{4}{11} a^{8} - \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{14} + \frac{5}{11} a^{13} - \frac{5}{11} a^{12} - \frac{2}{11} a^{11} + \frac{2}{11} a^{10} + \frac{1}{11} a^{9} + \frac{4}{11} a^{7} - \frac{1}{11} a^{6} - \frac{5}{11} a^{5} - \frac{4}{11} a^{4} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} - \frac{2}{11} a^{14} - \frac{5}{11} a^{12} - \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{7} + \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{14} - \frac{4}{11} a^{13} + \frac{3}{11} a^{12} - \frac{2}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{5}{11} a^{7} - \frac{2}{11} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{3}{11} a - \frac{2}{11}$, $\frac{1}{2450681298553852111} a^{19} + \frac{86322823070879091}{2450681298553852111} a^{18} - \frac{88180311351762849}{2450681298553852111} a^{17} - \frac{29061100561727928}{2450681298553852111} a^{16} - \frac{100079206347638839}{2450681298553852111} a^{15} - \frac{204337052905773477}{2450681298553852111} a^{14} - \frac{56687555065718504}{2450681298553852111} a^{13} + \frac{33602176333614127}{2450681298553852111} a^{12} + \frac{581267848060994517}{2450681298553852111} a^{11} - \frac{263390246674784091}{2450681298553852111} a^{10} - \frac{599945918234801112}{2450681298553852111} a^{9} + \frac{85132162313488168}{222789208959441101} a^{8} - \frac{679399912145080699}{2450681298553852111} a^{7} + \frac{566596504236298746}{2450681298553852111} a^{6} + \frac{776654981163167486}{2450681298553852111} a^{5} - \frac{866841438835797417}{2450681298553852111} a^{4} + \frac{614698332073315966}{2450681298553852111} a^{3} - \frac{265246885261888364}{2450681298553852111} a^{2} - \frac{906545392356374400}{2450681298553852111} a + \frac{88512859478434470}{2450681298553852111}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 4440787.52556 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.19535810978816.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
89Data not computed