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Properties

Label 20.20.4059140893...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 29^{5}\cdot 331^{4}\cdot 641^{4}$
Root discriminant $60.31$
Ramified primes $5, 29, 331, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T299

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 68, 850, 3036, -3291, -25774, 14376, 71947, -57464, -67087, 77412, 13604, -39332, 7619, 6719, -2579, -320, 254, -13, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 13*x^18 + 254*x^17 - 320*x^16 - 2579*x^15 + 6719*x^14 + 7619*x^13 - 39332*x^12 + 13604*x^11 + 77412*x^10 - 67087*x^9 - 57464*x^8 + 71947*x^7 + 14376*x^6 - 25774*x^5 - 3291*x^4 + 3036*x^3 + 850*x^2 + 68*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 - 13*x^18 + 254*x^17 - 320*x^16 - 2579*x^15 + 6719*x^14 + 7619*x^13 - 39332*x^12 + 13604*x^11 + 77412*x^10 - 67087*x^9 - 57464*x^8 + 71947*x^7 + 14376*x^6 - 25774*x^5 - 3291*x^4 + 3036*x^3 + 850*x^2 + 68*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 13 x^{18} + 254 x^{17} - 320 x^{16} - 2579 x^{15} + 6719 x^{14} + 7619 x^{13} - 39332 x^{12} + 13604 x^{11} + 77412 x^{10} - 67087 x^{9} - 57464 x^{8} + 71947 x^{7} + 14376 x^{6} - 25774 x^{5} - 3291 x^{4} + 3036 x^{3} + 850 x^{2} + 68 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(405914089364717749725605850283203125=5^{10}\cdot 29^{5}\cdot 331^{4}\cdot 641^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.31$
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:  $5, 29, 331, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{317310706106319558925560181} a^{19} - \frac{97493047667646710737940975}{317310706106319558925560181} a^{18} + \frac{2024543212471461007590654}{317310706106319558925560181} a^{17} - \frac{5405596056288378095667726}{16700563479279976785555799} a^{16} + \frac{17646926341027076613033671}{317310706106319558925560181} a^{15} - \frac{26725668775585101586785252}{317310706106319558925560181} a^{14} - \frac{79636697749364969055046563}{317310706106319558925560181} a^{13} + \frac{56769544129179460690631241}{317310706106319558925560181} a^{12} + \frac{95078033540048373256689465}{317310706106319558925560181} a^{11} - \frac{3830588155907212131938556}{317310706106319558925560181} a^{10} + \frac{12149275822216367211048238}{317310706106319558925560181} a^{9} - \frac{85065813898227933743857975}{317310706106319558925560181} a^{8} + \frac{105532634994116732830007962}{317310706106319558925560181} a^{7} - \frac{151425442210972415014506522}{317310706106319558925560181} a^{6} - \frac{37974674658141546748731761}{317310706106319558925560181} a^{5} - \frac{72694625785440031555335366}{317310706106319558925560181} a^{4} - \frac{125990284334034442337516437}{317310706106319558925560181} a^{3} + \frac{146247213589083859092992466}{317310706106319558925560181} a^{2} - \frac{86479198422460663671914103}{317310706106319558925560181} a - \frac{6083078722815076251583673}{317310706106319558925560181}$

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:  \( 253215720131 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T299:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5000
The 230 conjugacy class representatives for t20n299 are not computed
Character table for t20n299 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$29$29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
331Data not computed
641Data not computed