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Properties

Label 20.20.2463593732...0161.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 43^{4}\cdot 199^{4}$
Root discriminant $41.65$
Ramified primes $11, 43, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T314

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 242, 2043, -271, -19062, -11943, 58922, 36187, -85521, -29820, 65619, 6515, -27237, 2333, 5856, -1310, -552, 201, 9, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 9*x^18 + 201*x^17 - 552*x^16 - 1310*x^15 + 5856*x^14 + 2333*x^13 - 27237*x^12 + 6515*x^11 + 65619*x^10 - 29820*x^9 - 85521*x^8 + 36187*x^7 + 58922*x^6 - 11943*x^5 - 19062*x^4 - 271*x^3 + 2043*x^2 + 242*x - 23)
 
gp: K = bnfinit(x^20 - 10*x^19 + 9*x^18 + 201*x^17 - 552*x^16 - 1310*x^15 + 5856*x^14 + 2333*x^13 - 27237*x^12 + 6515*x^11 + 65619*x^10 - 29820*x^9 - 85521*x^8 + 36187*x^7 + 58922*x^6 - 11943*x^5 - 19062*x^4 - 271*x^3 + 2043*x^2 + 242*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 9 x^{18} + 201 x^{17} - 552 x^{16} - 1310 x^{15} + 5856 x^{14} + 2333 x^{13} - 27237 x^{12} + 6515 x^{11} + 65619 x^{10} - 29820 x^{9} - 85521 x^{8} + 36187 x^{7} + 58922 x^{6} - 11943 x^{5} - 19062 x^{4} - 271 x^{3} + 2043 x^{2} + 242 x - 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(246359373213147467442410887070161=11^{16}\cdot 43^{4}\cdot 199^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.65$
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:  $11, 43, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{11}{23} a^{16} - \frac{8}{23} a^{15} + \frac{5}{23} a^{14} + \frac{3}{23} a^{13} - \frac{6}{23} a^{12} - \frac{6}{23} a^{11} - \frac{4}{23} a^{10} + \frac{11}{23} a^{9} + \frac{2}{23} a^{8} - \frac{8}{23} a^{7} - \frac{2}{23} a^{6} - \frac{2}{23} a^{5} - \frac{1}{23} a^{4} - \frac{7}{23} a^{3} + \frac{8}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{23} a^{18} + \frac{9}{23} a^{16} + \frac{1}{23} a^{15} - \frac{6}{23} a^{14} + \frac{7}{23} a^{13} - \frac{9}{23} a^{12} - \frac{7}{23} a^{11} + \frac{9}{23} a^{10} - \frac{4}{23} a^{9} - \frac{7}{23} a^{8} - \frac{6}{23} a^{7} - \frac{3}{23} a^{6} - \frac{2}{23} a^{5} + \frac{4}{23} a^{4} - \frac{7}{23} a^{3} - \frac{5}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{1474556187240368443531} a^{19} + \frac{28310129605725710226}{1474556187240368443531} a^{18} - \frac{21592429283028884572}{1474556187240368443531} a^{17} + \frac{670289997081452608532}{1474556187240368443531} a^{16} - \frac{313692098268042259589}{1474556187240368443531} a^{15} - \frac{642813352625343266390}{1474556187240368443531} a^{14} - \frac{711652086693019695282}{1474556187240368443531} a^{13} - \frac{342688385941601305006}{1474556187240368443531} a^{12} + \frac{281934784009851368697}{1474556187240368443531} a^{11} + \frac{256507758230304772911}{1474556187240368443531} a^{10} + \frac{414879421788235705722}{1474556187240368443531} a^{9} - \frac{11022561587525570003}{64111138575668193197} a^{8} + \frac{281551096651316618671}{1474556187240368443531} a^{7} + \frac{540470468680028101461}{1474556187240368443531} a^{6} + \frac{42862716793398300453}{1474556187240368443531} a^{5} - \frac{11644654090077279118}{64111138575668193197} a^{4} - \frac{129427291085941971482}{1474556187240368443531} a^{3} + \frac{157764807024783118785}{1474556187240368443531} a^{2} + \frac{512594985779209067951}{1474556187240368443531} a + \frac{9177479123942858551}{64111138575668193197}$

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:  $19$
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:  \( 5171060698.93 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T314:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n314
Character table for t20n314 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1834268944717.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
43Data not computed
199Data not computed