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Properties

Label 20.20.1719105272...6576.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{18}\cdot 7369^{4}$
Root discriminant $102.75$
Ramified primes $2, 11, 7369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6570561481, 0, -16233361694, 0, 14341314097, 0, -5709838629, 0, 1188999845, 0, -140937236, 0, 10041889, 0, -438383, 0, 11473, 0, -165, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 165*x^18 + 11473*x^16 - 438383*x^14 + 10041889*x^12 - 140937236*x^10 + 1188999845*x^8 - 5709838629*x^6 + 14341314097*x^4 - 16233361694*x^2 + 6570561481)
 
gp: K = bnfinit(x^20 - 165*x^18 + 11473*x^16 - 438383*x^14 + 10041889*x^12 - 140937236*x^10 + 1188999845*x^8 - 5709838629*x^6 + 14341314097*x^4 - 16233361694*x^2 + 6570561481, 1)
 

Normalized defining polynomial

\( x^{20} - 165 x^{18} + 11473 x^{16} - 438383 x^{14} + 10041889 x^{12} - 140937236 x^{10} + 1188999845 x^{8} - 5709838629 x^{6} + 14341314097 x^{4} - 16233361694 x^{2} + 6570561481 \)

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:  \(17191052721619032309920296885375992856576=2^{20}\cdot 11^{18}\cdot 7369^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.75$
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:  $2, 11, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{38734109831914689177721106578141} a^{18} + \frac{45488279738453771928814077666}{3521282711992244470701918779831} a^{16} - \frac{481178130927957071808123479584}{38734109831914689177721106578141} a^{14} + \frac{131377836811732871328864392360}{3521282711992244470701918779831} a^{12} - \frac{173297224677462681635604527051}{38734109831914689177721106578141} a^{10} - \frac{894019921744807353016986648009}{3521282711992244470701918779831} a^{8} - \frac{69877025421178932196979491891}{3521282711992244470701918779831} a^{6} + \frac{143735045015156517845022286114}{3521282711992244470701918779831} a^{4} - \frac{914598674561054986417378995466}{3521282711992244470701918779831} a^{2} + \frac{225706752714275549068558011}{477850822634311910802268799}$, $\frac{1}{38734109831914689177721106578141} a^{19} + \frac{45488279738453771928814077666}{3521282711992244470701918779831} a^{17} - \frac{481178130927957071808123479584}{38734109831914689177721106578141} a^{15} + \frac{131377836811732871328864392360}{3521282711992244470701918779831} a^{13} - \frac{173297224677462681635604527051}{38734109831914689177721106578141} a^{11} - \frac{894019921744807353016986648009}{3521282711992244470701918779831} a^{9} - \frac{69877025421178932196979491891}{3521282711992244470701918779831} a^{7} + \frac{143735045015156517845022286114}{3521282711992244470701918779831} a^{5} - \frac{914598674561054986417378995466}{3521282711992244470701918779831} a^{3} + \frac{225706752714275549068558011}{477850822634311910802268799} a$

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:  \( 44953700534800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
11Data not computed
7369Data not computed