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Properties

Label 20.10.9331576415...1527.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 727^{5}$
Root discriminant $35.36$
Ramified primes $11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-109, 661, 12300, 30453, 3738, -48171, -20987, 30897, 10725, -10864, 328, 2199, -1413, -31, 237, -121, 49, 28, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 17*x^18 + 28*x^17 + 49*x^16 - 121*x^15 + 237*x^14 - 31*x^13 - 1413*x^12 + 2199*x^11 + 328*x^10 - 10864*x^9 + 10725*x^8 + 30897*x^7 - 20987*x^6 - 48171*x^5 + 3738*x^4 + 30453*x^3 + 12300*x^2 + 661*x - 109)
 
gp: K = bnfinit(x^20 - 2*x^19 - 17*x^18 + 28*x^17 + 49*x^16 - 121*x^15 + 237*x^14 - 31*x^13 - 1413*x^12 + 2199*x^11 + 328*x^10 - 10864*x^9 + 10725*x^8 + 30897*x^7 - 20987*x^6 - 48171*x^5 + 3738*x^4 + 30453*x^3 + 12300*x^2 + 661*x - 109, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 17 x^{18} + 28 x^{17} + 49 x^{16} - 121 x^{15} + 237 x^{14} - 31 x^{13} - 1413 x^{12} + 2199 x^{11} + 328 x^{10} - 10864 x^{9} + 10725 x^{8} + 30897 x^{7} - 20987 x^{6} - 48171 x^{5} + 3738 x^{4} + 30453 x^{3} + 12300 x^{2} + 661 x - 109 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9331576415771428587203963681527=-\,11^{16}\cdot 727^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.36$
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:  $11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{23739840943936329594003295850241637373} a^{19} - \frac{8775394915786397095570312386569072133}{23739840943936329594003295850241637373} a^{18} + \frac{907396262082373815485140683659063381}{23739840943936329594003295850241637373} a^{17} + \frac{11063523446434946468200678662793977331}{23739840943936329594003295850241637373} a^{16} - \frac{8881656593012885243075963433222621104}{23739840943936329594003295850241637373} a^{15} + \frac{10479502964071603209906930578416363917}{23739840943936329594003295850241637373} a^{14} + \frac{7491995654869461205953733754199299542}{23739840943936329594003295850241637373} a^{13} - \frac{5633969529567928337657222660286927214}{23739840943936329594003295850241637373} a^{12} - \frac{3043499627373120362956298230196148566}{23739840943936329594003295850241637373} a^{11} + \frac{5234930372249119241337726639869060878}{23739840943936329594003295850241637373} a^{10} - \frac{4869899868363738383460639861335805248}{23739840943936329594003295850241637373} a^{9} + \frac{1530603125799148280692643552566546530}{23739840943936329594003295850241637373} a^{8} - \frac{11461538684044677934903483670663813591}{23739840943936329594003295850241637373} a^{7} + \frac{10610020314170242459215491187626582386}{23739840943936329594003295850241637373} a^{6} + \frac{8920191991049729211673050162558539730}{23739840943936329594003295850241637373} a^{5} - \frac{3459948128178400560880153312001089070}{23739840943936329594003295850241637373} a^{4} + \frac{11426167133785506367599460385846753300}{23739840943936329594003295850241637373} a^{3} + \frac{11591168399745460966889205986220313644}{23739840943936329594003295850241637373} a^{2} - \frac{10403881302965646035216317696917469571}{23739840943936329594003295850241637373} a + \frac{5768352714907081725322326700179050118}{23739840943936329594003295850241637373}$

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:  $14$
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:  \( 86307674.8427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
11Data not computed
727Data not computed