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Properties

Label 20.16.5165883938...8125.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $34.33$
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![-499, -590, 9133, 7912, -42210, -10085, 86117, -18979, -80598, 42497, 32442, -27558, -3445, 7153, -558, -765, 33, 77, 0, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 77*x^17 + 33*x^16 - 765*x^15 - 558*x^14 + 7153*x^13 - 3445*x^12 - 27558*x^11 + 32442*x^10 + 42497*x^9 - 80598*x^8 - 18979*x^7 + 86117*x^6 - 10085*x^5 - 42210*x^4 + 7912*x^3 + 9133*x^2 - 590*x - 499)
 
gp: K = bnfinit(x^20 - 7*x^19 + 77*x^17 + 33*x^16 - 765*x^15 - 558*x^14 + 7153*x^13 - 3445*x^12 - 27558*x^11 + 32442*x^10 + 42497*x^9 - 80598*x^8 - 18979*x^7 + 86117*x^6 - 10085*x^5 - 42210*x^4 + 7912*x^3 + 9133*x^2 - 590*x - 499, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 77 x^{17} + 33 x^{16} - 765 x^{15} - 558 x^{14} + 7153 x^{13} - 3445 x^{12} - 27558 x^{11} + 32442 x^{10} + 42497 x^{9} - 80598 x^{8} - 18979 x^{7} + 86117 x^{6} - 10085 x^{5} - 42210 x^{4} + 7912 x^{3} + 9133 x^{2} - 590 x - 499 \)

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:  \(5165883938831637970733642578125=3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{85829508300224759510929664883} a^{19} + \frac{12563863417944778522228008400}{85829508300224759510929664883} a^{18} + \frac{7606731959548180208877915866}{85829508300224759510929664883} a^{17} + \frac{8801912273615349775462959785}{85829508300224759510929664883} a^{16} - \frac{2818513517752458830849002756}{85829508300224759510929664883} a^{15} - \frac{1828734561830456301528755001}{28609836100074919836976554961} a^{14} - \frac{20461238838610732186687282348}{85829508300224759510929664883} a^{13} - \frac{37170734150422056882541244014}{85829508300224759510929664883} a^{12} + \frac{22266214305611646906949320314}{85829508300224759510929664883} a^{11} + \frac{11294664260967017519234743091}{85829508300224759510929664883} a^{10} - \frac{11616071374229958570778159967}{85829508300224759510929664883} a^{9} - \frac{15479107667838249238454688316}{85829508300224759510929664883} a^{8} + \frac{29965083197681935025134789639}{85829508300224759510929664883} a^{7} - \frac{9901887273830526241386290618}{85829508300224759510929664883} a^{6} + \frac{21427180548093699681073012252}{85829508300224759510929664883} a^{5} - \frac{24215293625951768656948917596}{85829508300224759510929664883} a^{4} + \frac{42081677915486910932474111338}{85829508300224759510929664883} a^{3} - \frac{7813769476692908340983766961}{28609836100074919836976554961} a^{2} - \frac{14072550629541147371876882383}{28609836100074919836976554961} a + \frac{30640273019261907276448375906}{85829508300224759510929664883}$

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:  \( 383994609.129 \) (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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]_{4}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$23$23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$