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Properties

Label 20.16.5109347819...0625.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{2}\cdot 5^{22}\cdot 47^{8}$
Root discriminant $30.58$
Ramified primes $3, 5, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 130, -835, -1005, 6090, 9461, 140, -6465, -6420, 1335, 5922, -2235, -2440, 2075, 355, -706, 40, 100, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 15*x^18 + 100*x^17 + 40*x^16 - 706*x^15 + 355*x^14 + 2075*x^13 - 2440*x^12 - 2235*x^11 + 5922*x^10 + 1335*x^9 - 6420*x^8 - 6465*x^7 + 140*x^6 + 9461*x^5 + 6090*x^4 - 1005*x^3 - 835*x^2 + 130*x - 4)
 
gp: K = bnfinit(x^20 - 5*x^19 - 15*x^18 + 100*x^17 + 40*x^16 - 706*x^15 + 355*x^14 + 2075*x^13 - 2440*x^12 - 2235*x^11 + 5922*x^10 + 1335*x^9 - 6420*x^8 - 6465*x^7 + 140*x^6 + 9461*x^5 + 6090*x^4 - 1005*x^3 - 835*x^2 + 130*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 15 x^{18} + 100 x^{17} + 40 x^{16} - 706 x^{15} + 355 x^{14} + 2075 x^{13} - 2440 x^{12} - 2235 x^{11} + 5922 x^{10} + 1335 x^{9} - 6420 x^{8} - 6465 x^{7} + 140 x^{6} + 9461 x^{5} + 6090 x^{4} - 1005 x^{3} - 835 x^{2} + 130 x - 4 \)

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:  \(510934781922934055328369140625=3^{2}\cdot 5^{22}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.58$
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:  $3, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{124} a^{18} + \frac{7}{62} a^{17} + \frac{1}{62} a^{16} + \frac{7}{124} a^{14} - \frac{15}{124} a^{13} + \frac{1}{124} a^{12} + \frac{1}{124} a^{11} + \frac{27}{124} a^{10} - \frac{9}{62} a^{9} - \frac{27}{124} a^{8} + \frac{17}{62} a^{7} + \frac{19}{124} a^{6} + \frac{1}{4} a^{5} - \frac{3}{124} a^{4} - \frac{5}{31} a^{3} - \frac{53}{124} a^{2} - \frac{2}{31} a - \frac{1}{31}$, $\frac{1}{81380344584391195440454292} a^{19} + \frac{57516191042844937467178}{20345086146097798860113573} a^{18} - \frac{2477254504225358555951590}{20345086146097798860113573} a^{17} + \frac{2536606893927417223895442}{20345086146097798860113573} a^{16} - \frac{3034762251520734973867677}{81380344584391195440454292} a^{15} + \frac{8872671975998797429985841}{81380344584391195440454292} a^{14} - \frac{8538679826446975778516833}{81380344584391195440454292} a^{13} + \frac{6362149599491646147876713}{81380344584391195440454292} a^{12} - \frac{6982793052273889712765509}{81380344584391195440454292} a^{11} - \frac{9555008825417627158663351}{20345086146097798860113573} a^{10} - \frac{17764326566000063584321269}{81380344584391195440454292} a^{9} + \frac{6587972504843034337358591}{40690172292195597720227146} a^{8} - \frac{3539219198727095907876843}{81380344584391195440454292} a^{7} + \frac{5312030009204381841578269}{81380344584391195440454292} a^{6} + \frac{36020450738059101023014287}{81380344584391195440454292} a^{5} + \frac{7491114785887191747494084}{20345086146097798860113573} a^{4} - \frac{17838353401945129608905399}{81380344584391195440454292} a^{3} + \frac{2577560611272702922539267}{20345086146097798860113573} a^{2} - \frac{14844127552419962126731419}{40690172292195597720227146} a + \frac{1059731662474948552973310}{20345086146097798860113573}$

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:  \( 154283370.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.6903125.1, 10.8.714797021484375.1, 10.8.142959404296875.1, 10.10.238265673828125.1

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
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$