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Properties

Label 20.20.7328106503...5625.2
Degree $20$
Signature $[20, 0]$
Discriminant $5^{16}\cdot 6029^{6}$
Root discriminant $49.35$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1181, 34267, -11881, -324785, -187919, 976031, 1479197, 67624, -1254600, -819879, 127754, 303701, 62822, -39300, -16187, 1718, 1471, 17, -60, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 60*x^18 + 17*x^17 + 1471*x^16 + 1718*x^15 - 16187*x^14 - 39300*x^13 + 62822*x^12 + 303701*x^11 + 127754*x^10 - 819879*x^9 - 1254600*x^8 + 67624*x^7 + 1479197*x^6 + 976031*x^5 - 187919*x^4 - 324785*x^3 - 11881*x^2 + 34267*x - 1181)
 
gp: K = bnfinit(x^20 - 2*x^19 - 60*x^18 + 17*x^17 + 1471*x^16 + 1718*x^15 - 16187*x^14 - 39300*x^13 + 62822*x^12 + 303701*x^11 + 127754*x^10 - 819879*x^9 - 1254600*x^8 + 67624*x^7 + 1479197*x^6 + 976031*x^5 - 187919*x^4 - 324785*x^3 - 11881*x^2 + 34267*x - 1181, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 60 x^{18} + 17 x^{17} + 1471 x^{16} + 1718 x^{15} - 16187 x^{14} - 39300 x^{13} + 62822 x^{12} + 303701 x^{11} + 127754 x^{10} - 819879 x^{9} - 1254600 x^{8} + 67624 x^{7} + 1479197 x^{6} + 976031 x^{5} - 187919 x^{4} - 324785 x^{3} - 11881 x^{2} + 34267 x - 1181 \)

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:  \(7328106503776662626239166259765625=5^{16}\cdot 6029^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.35$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{15} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{17} + \frac{2}{25} a^{16} + \frac{7}{25} a^{15} - \frac{1}{5} a^{14} + \frac{6}{25} a^{13} - \frac{9}{25} a^{12} - \frac{3}{25} a^{11} + \frac{12}{25} a^{10} - \frac{2}{5} a^{9} + \frac{2}{25} a^{8} + \frac{12}{25} a^{7} + \frac{4}{25} a^{6} - \frac{6}{25} a^{5} + \frac{2}{5} a^{4} + \frac{12}{25} a^{3} - \frac{8}{25} a^{2} - \frac{11}{25} a - \frac{1}{25}$, $\frac{1}{3340391922807967031989323275} a^{19} + \frac{26650764486795135182300276}{3340391922807967031989323275} a^{18} + \frac{98072319319910422193841932}{3340391922807967031989323275} a^{17} - \frac{56941190257028420028793268}{3340391922807967031989323275} a^{16} + \frac{208455886477260415894664506}{668078384561593406397864655} a^{15} + \frac{929362071534680729919225831}{3340391922807967031989323275} a^{14} - \frac{1205575804594158638189391184}{3340391922807967031989323275} a^{13} - \frac{1658576737128685339094345273}{3340391922807967031989323275} a^{12} - \frac{447231589304220520128681813}{3340391922807967031989323275} a^{11} - \frac{589998530862989791438154}{133615676912318681279572931} a^{10} - \frac{1162307994505826785752759873}{3340391922807967031989323275} a^{9} + \frac{536390371846541467630594412}{3340391922807967031989323275} a^{8} - \frac{380737201982520256741081286}{3340391922807967031989323275} a^{7} - \frac{1091916733150441345896146181}{3340391922807967031989323275} a^{6} + \frac{253630342394699230654477591}{668078384561593406397864655} a^{5} + \frac{1402647998882495462374279412}{3340391922807967031989323275} a^{4} - \frac{1179531543066507050742884558}{3340391922807967031989323275} a^{3} + \frac{1010571064118850914053227774}{3340391922807967031989323275} a^{2} - \frac{87079469708575221899108626}{3340391922807967031989323275} a - \frac{148973261309897586941491}{565688725285007118033755}$

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:  \( 27849473193.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1, 10.10.85604360308203125.2, 10.10.17120872061640625.3

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed