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Properties

Label 20.16.4916306985...0625.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 11^{16}\cdot 331^{2}$
Root discriminant $27.20$
Ramified primes $5, 11, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -39, -18, 271, 210, -702, -1038, 583, 1990, 117, -1724, -121, 834, -135, -232, 98, 30, -19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 19*x^18 + 30*x^17 + 98*x^16 - 232*x^15 - 135*x^14 + 834*x^13 - 121*x^12 - 1724*x^11 + 117*x^10 + 1990*x^9 + 583*x^8 - 1038*x^7 - 702*x^6 + 210*x^5 + 271*x^4 - 18*x^3 - 39*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 19*x^18 + 30*x^17 + 98*x^16 - 232*x^15 - 135*x^14 + 834*x^13 - 121*x^12 - 1724*x^11 + 117*x^10 + 1990*x^9 + 583*x^8 - 1038*x^7 - 702*x^6 + 210*x^5 + 271*x^4 - 18*x^3 - 39*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 19 x^{18} + 30 x^{17} + 98 x^{16} - 232 x^{15} - 135 x^{14} + 834 x^{13} - 121 x^{12} - 1724 x^{11} + 117 x^{10} + 1990 x^{9} + 583 x^{8} - 1038 x^{7} - 702 x^{6} + 210 x^{5} + 271 x^{4} - 18 x^{3} - 39 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49163069859207319641806640625=5^{10}\cdot 11^{16}\cdot 331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.20$
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:  $5, 11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{11}{43} a^{17} - \frac{1}{43} a^{16} + \frac{11}{43} a^{15} + \frac{19}{43} a^{13} + \frac{7}{43} a^{12} - \frac{1}{43} a^{11} + \frac{15}{43} a^{10} - \frac{11}{43} a^{9} - \frac{5}{43} a^{8} + \frac{2}{43} a^{7} + \frac{16}{43} a^{6} + \frac{1}{43} a^{5} - \frac{20}{43} a^{4} - \frac{15}{43} a^{3} + \frac{6}{43} a^{2} + \frac{1}{43} a + \frac{20}{43}$, $\frac{1}{5634799755403460423} a^{19} + \frac{11830643011209281}{5634799755403460423} a^{18} - \frac{1923584990212288711}{5634799755403460423} a^{17} + \frac{505554031024594108}{5634799755403460423} a^{16} + \frac{255296273924977206}{5634799755403460423} a^{15} + \frac{1834741160715526991}{5634799755403460423} a^{14} - \frac{2731122007654423406}{5634799755403460423} a^{13} - \frac{2513266794476705411}{5634799755403460423} a^{12} - \frac{2118508688688003685}{5634799755403460423} a^{11} + \frac{2556908515572493730}{5634799755403460423} a^{10} - \frac{1565118565800262336}{5634799755403460423} a^{9} - \frac{2355635272533354988}{5634799755403460423} a^{8} + \frac{1358513324787135643}{5634799755403460423} a^{7} - \frac{2520869517211447694}{5634799755403460423} a^{6} + \frac{1268872356396592468}{5634799755403460423} a^{5} + \frac{44962616766237735}{131041854776824661} a^{4} + \frac{2704707846111401286}{5634799755403460423} a^{3} - \frac{2320413048376626545}{5634799755403460423} a^{2} - \frac{1021238027319735204}{5634799755403460423} a + \frac{1035458189793682762}{5634799755403460423}$

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:  $17$
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:  \( 24362077.3195 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.8.221727467534375.1, 10.8.70952789611.1, 10.10.669871503125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$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}$
331Data not computed