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Properties

Label 20.20.1506371756...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{12}\cdot 5^{15}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $40.63$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T369

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-809, 5190, -3498, -26170, 31974, 52054, -76707, -53274, 89231, 31441, -57814, -11824, 21921, 3072, -4870, -536, 608, 52, -39, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 39*x^18 + 52*x^17 + 608*x^16 - 536*x^15 - 4870*x^14 + 3072*x^13 + 21921*x^12 - 11824*x^11 - 57814*x^10 + 31441*x^9 + 89231*x^8 - 53274*x^7 - 76707*x^6 + 52054*x^5 + 31974*x^4 - 26170*x^3 - 3498*x^2 + 5190*x - 809)
 
gp: K = bnfinit(x^20 - 2*x^19 - 39*x^18 + 52*x^17 + 608*x^16 - 536*x^15 - 4870*x^14 + 3072*x^13 + 21921*x^12 - 11824*x^11 - 57814*x^10 + 31441*x^9 + 89231*x^8 - 53274*x^7 - 76707*x^6 + 52054*x^5 + 31974*x^4 - 26170*x^3 - 3498*x^2 + 5190*x - 809, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 39 x^{18} + 52 x^{17} + 608 x^{16} - 536 x^{15} - 4870 x^{14} + 3072 x^{13} + 21921 x^{12} - 11824 x^{11} - 57814 x^{10} + 31441 x^{9} + 89231 x^{8} - 53274 x^{7} - 76707 x^{6} + 52054 x^{5} + 31974 x^{4} - 26170 x^{3} - 3498 x^{2} + 5190 x - 809 \)

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:  \(150637175656330563226593017578125=3^{12}\cdot 5^{15}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.63$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1304655176239907561537557} a^{19} - \frac{627274140610401162222245}{1304655176239907561537557} a^{18} + \frac{154042480589764791431703}{1304655176239907561537557} a^{17} - \frac{25684346944139627333563}{1304655176239907561537557} a^{16} - \frac{137933690796358019612683}{1304655176239907561537557} a^{15} - \frac{116372137176306676728093}{1304655176239907561537557} a^{14} + \frac{634723715433588577987372}{1304655176239907561537557} a^{13} - \frac{356096206472951753918041}{1304655176239907561537557} a^{12} + \frac{444022069493151304412608}{1304655176239907561537557} a^{11} - \frac{476854994298893720768525}{1304655176239907561537557} a^{10} + \frac{299885736350267974273770}{1304655176239907561537557} a^{9} - \frac{49471399094773374797440}{1304655176239907561537557} a^{8} + \frac{294040972581677887745638}{1304655176239907561537557} a^{7} - \frac{623701550169118540317202}{1304655176239907561537557} a^{6} - \frac{472469083817932130484280}{1304655176239907561537557} a^{5} + \frac{565698174911163607038233}{1304655176239907561537557} a^{4} + \frac{505926348785998988842397}{1304655176239907561537557} a^{3} - \frac{272027092734606609381007}{1304655176239907561537557} a^{2} - \frac{648794233918879278266907}{1304655176239907561537557} a - \frac{542243115214876688783463}{1304655176239907561537557}$

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:  \( 4724376804.45 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T369:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n369 are not computed
Character table for t20n369 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
89Data not computed