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Properties

Label 20.8.15418170861...0000.4
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -40, -66, 350, 529, -142, -740, -786, -997, -682, -186, -156, -61, -34, -56, 44, -20, -4, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 - 4*x^17 - 20*x^16 + 44*x^15 - 56*x^14 - 34*x^13 - 61*x^12 - 156*x^11 - 186*x^10 - 682*x^9 - 997*x^8 - 786*x^7 - 740*x^6 - 142*x^5 + 529*x^4 + 350*x^3 - 66*x^2 - 40*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 - 4*x^17 - 20*x^16 + 44*x^15 - 56*x^14 - 34*x^13 - 61*x^12 - 156*x^11 - 186*x^10 - 682*x^9 - 997*x^8 - 786*x^7 - 740*x^6 - 142*x^5 + 529*x^4 + 350*x^3 - 66*x^2 - 40*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} - 4 x^{17} - 20 x^{16} + 44 x^{15} - 56 x^{14} - 34 x^{13} - 61 x^{12} - 156 x^{11} - 186 x^{10} - 682 x^{9} - 997 x^{8} - 786 x^{7} - 740 x^{6} - 142 x^{5} + 529 x^{4} + 350 x^{3} - 66 x^{2} - 40 x - 1 \)

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:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{14} - \frac{4}{11} a^{13} + \frac{4}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{9} - \frac{4}{11} a^{8} - \frac{3}{11} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{5}{11} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{14} - \frac{4}{11} a^{13} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{9} - \frac{5}{11} a^{7} - \frac{5}{11} a^{5} + \frac{3}{11} a^{4} - \frac{3}{11} a^{3} + \frac{5}{11} a^{2} - \frac{5}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{17} + \frac{2}{11} a^{14} + \frac{2}{11} a^{13} - \frac{4}{11} a^{12} - \frac{5}{11} a^{11} - \frac{1}{11} a^{10} + \frac{1}{11} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{2}{11} a^{5} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{14} + \frac{4}{11} a^{13} - \frac{2}{11} a^{12} + \frac{2}{11} a^{11} + \frac{1}{11} a^{10} + \frac{4}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} - \frac{4}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{4633568785643964182055721} a^{19} - \frac{46590311982252212883547}{4633568785643964182055721} a^{18} + \frac{17204639942858640537358}{4633568785643964182055721} a^{17} + \frac{115504454699605049515894}{4633568785643964182055721} a^{16} + \frac{202007723286231277128451}{4633568785643964182055721} a^{15} - \frac{1383059200527286326844314}{4633568785643964182055721} a^{14} - \frac{2151424845426094295488978}{4633568785643964182055721} a^{13} - \frac{245324102157992675559482}{4633568785643964182055721} a^{12} - \frac{1422985601461871151609841}{4633568785643964182055721} a^{11} - \frac{458906132496058859084952}{4633568785643964182055721} a^{10} + \frac{611783005850656795920635}{4633568785643964182055721} a^{9} + \frac{1665330388322879768790918}{4633568785643964182055721} a^{8} - \frac{270895900996295153036470}{4633568785643964182055721} a^{7} - \frac{1254800877236386096796824}{4633568785643964182055721} a^{6} - \frac{1076099951474872534542679}{4633568785643964182055721} a^{5} + \frac{66571308415433674162799}{4633568785643964182055721} a^{4} + \frac{1743262099925146602348907}{4633568785643964182055721} a^{3} - \frac{976283844224958319016576}{4633568785643964182055721} a^{2} + \frac{243105080488714758233428}{4633568785643964182055721} a + \frac{697394856853708491974300}{4633568785643964182055721}$

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:  \( 9358132.20404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 R $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$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}$