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Properties

Label 20.20.2463593732...0161.2
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, -1, -2091, -5475, 8971, 31918, -14084, -73500, 12862, 85401, -13592, -52015, 11851, 15259, -4437, -2064, 707, 115, -46, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 46*x^18 + 115*x^17 + 707*x^16 - 2064*x^15 - 4437*x^14 + 15259*x^13 + 11851*x^12 - 52015*x^11 - 13592*x^10 + 85401*x^9 + 12862*x^8 - 73500*x^7 - 14084*x^6 + 31918*x^5 + 8971*x^4 - 5475*x^3 - 2091*x^2 - x + 23)
 
gp: K = bnfinit(x^20 - 2*x^19 - 46*x^18 + 115*x^17 + 707*x^16 - 2064*x^15 - 4437*x^14 + 15259*x^13 + 11851*x^12 - 52015*x^11 - 13592*x^10 + 85401*x^9 + 12862*x^8 - 73500*x^7 - 14084*x^6 + 31918*x^5 + 8971*x^4 - 5475*x^3 - 2091*x^2 - x + 23, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 46 x^{18} + 115 x^{17} + 707 x^{16} - 2064 x^{15} - 4437 x^{14} + 15259 x^{13} + 11851 x^{12} - 52015 x^{11} - 13592 x^{10} + 85401 x^{9} + 12862 x^{8} - 73500 x^{7} - 14084 x^{6} + 31918 x^{5} + 8971 x^{4} - 5475 x^{3} - 2091 x^{2} - x + 23 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{50636555126688253840362622807} a^{19} - \frac{19585119431895071452884346526}{50636555126688253840362622807} a^{18} - \frac{14352311052536609212534128296}{50636555126688253840362622807} a^{17} + \frac{7211836129512402135355597032}{50636555126688253840362622807} a^{16} + \frac{2163177907086457315519931289}{50636555126688253840362622807} a^{15} + \frac{13563563896772499502207505566}{50636555126688253840362622807} a^{14} + \frac{2767184848836920901518551143}{50636555126688253840362622807} a^{13} + \frac{18862606474343406567651200529}{50636555126688253840362622807} a^{12} + \frac{13702132231069360362026067576}{50636555126688253840362622807} a^{11} + \frac{2643880547454236031277228467}{50636555126688253840362622807} a^{10} - \frac{9014318199073930308364029880}{50636555126688253840362622807} a^{9} - \frac{2429299886090836392237053353}{50636555126688253840362622807} a^{8} - \frac{9552057637543303686393859009}{50636555126688253840362622807} a^{7} - \frac{7900123424177464710744301751}{50636555126688253840362622807} a^{6} - \frac{2902005792528143973700139960}{50636555126688253840362622807} a^{5} - \frac{11574857139318279185936144347}{50636555126688253840362622807} a^{4} - \frac{14348658106601331756361756275}{50636555126688253840362622807} a^{3} - \frac{14462901708316465955062015571}{50636555126688253840362622807} a^{2} + \frac{22549220994875267463314480569}{50636555126688253840362622807} a - \frac{5128697325611880303766983659}{50636555126688253840362622807}$

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:  \( 5374624052.84 \) (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