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Properties

Label 20.0.82032114573...2521.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 4903^{6}$
Root discriminant $22.17$
Ramified primes $3, 4903$
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![625, -750, -500, 420, 676, -322, -794, 852, 38, -340, 219, -126, 13, 40, -10, -2, -7, 8, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 - 7*x^16 - 2*x^15 - 10*x^14 + 40*x^13 + 13*x^12 - 126*x^11 + 219*x^10 - 340*x^9 + 38*x^8 + 852*x^7 - 794*x^6 - 322*x^5 + 676*x^4 + 420*x^3 - 500*x^2 - 750*x + 625)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 + 8*x^17 - 7*x^16 - 2*x^15 - 10*x^14 + 40*x^13 + 13*x^12 - 126*x^11 + 219*x^10 - 340*x^9 + 38*x^8 + 852*x^7 - 794*x^6 - 322*x^5 + 676*x^4 + 420*x^3 - 500*x^2 - 750*x + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} + 8 x^{17} - 7 x^{16} - 2 x^{15} - 10 x^{14} + 40 x^{13} + 13 x^{12} - 126 x^{11} + 219 x^{10} - 340 x^{9} + 38 x^{8} + 852 x^{7} - 794 x^{6} - 322 x^{5} + 676 x^{4} + 420 x^{3} - 500 x^{2} - 750 x + 625 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(820321145738470682226262521=3^{10}\cdot 4903^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.17$
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:  $3, 4903$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{11} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{30} a^{16} + \frac{1}{30} a^{15} + \frac{1}{30} a^{14} + \frac{1}{5} a^{13} + \frac{1}{30} a^{12} + \frac{1}{30} a^{11} - \frac{1}{15} a^{10} - \frac{1}{30} a^{9} - \frac{1}{5} a^{7} - \frac{7}{15} a^{6} - \frac{1}{15} a^{5} - \frac{4}{15} a^{4} + \frac{13}{30} a^{3} + \frac{1}{3} a^{2} - \frac{1}{15} a - \frac{1}{6}$, $\frac{1}{90} a^{17} - \frac{1}{90} a^{16} - \frac{1}{15} a^{15} + \frac{2}{45} a^{14} + \frac{2}{45} a^{13} - \frac{1}{90} a^{12} + \frac{7}{30} a^{11} - \frac{2}{15} a^{10} - \frac{14}{45} a^{9} + \frac{22}{45} a^{8} + \frac{19}{45} a^{7} - \frac{17}{45} a^{6} + \frac{11}{90} a^{5} + \frac{1}{10} a^{4} - \frac{8}{45} a^{3} + \frac{4}{45} a^{2} - \frac{7}{30} a - \frac{2}{9}$, $\frac{1}{67050} a^{18} - \frac{22}{11175} a^{17} - \frac{487}{67050} a^{16} - \frac{2476}{33525} a^{15} - \frac{3367}{67050} a^{14} - \frac{1802}{11175} a^{13} - \frac{1522}{6705} a^{12} - \frac{159}{1490} a^{11} - \frac{7747}{67050} a^{10} - \frac{21821}{67050} a^{9} + \frac{3949}{67050} a^{8} - \frac{3254}{6705} a^{7} - \frac{3107}{67050} a^{6} - \frac{10423}{67050} a^{5} + \frac{9928}{33525} a^{4} + \frac{17113}{67050} a^{3} - \frac{3707}{33525} a^{2} - \frac{347}{6705} a - \frac{1097}{2682}$, $\frac{1}{21946270775230608750} a^{19} + \frac{15180720040439}{10973135387615304375} a^{18} - \frac{99959035109690737}{21946270775230608750} a^{17} - \frac{9109807883551603}{2438474530581178750} a^{16} - \frac{84660182612113863}{2438474530581178750} a^{15} + \frac{510285021577394269}{10973135387615304375} a^{14} + \frac{396865498387843568}{2194627077523060875} a^{13} - \frac{369987699695582806}{2194627077523060875} a^{12} + \frac{5429472665782786463}{21946270775230608750} a^{11} + \frac{895698420725749813}{7315423591743536250} a^{10} + \frac{7166341928449637789}{21946270775230608750} a^{9} + \frac{1052803538531940721}{4389254155046121750} a^{8} + \frac{8968954207070901563}{21946270775230608750} a^{7} + \frac{2260651046897000389}{7315423591743536250} a^{6} + \frac{2821576149928015958}{10973135387615304375} a^{5} - \frac{1517348057436334582}{3657711795871768125} a^{4} - \frac{1377871749875423089}{3657711795871768125} a^{3} + \frac{6866862434470897}{19507796244649430} a^{2} + \frac{1670801894768108}{87785083100922435} a - \frac{23559722344817198}{87785083100922435}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{104306246168966}{24548401314575625} a^{19} + \frac{94109647648702}{24548401314575625} a^{18} + \frac{348449556661742}{24548401314575625} a^{17} - \frac{174657291942781}{8182800438191875} a^{16} + \frac{28872089820574}{8182800438191875} a^{15} + \frac{613065108873842}{24548401314575625} a^{14} + \frac{273834772771084}{4909680262915125} a^{13} - \frac{516886068793583}{4909680262915125} a^{12} - \frac{4801743744279508}{24548401314575625} a^{11} + \frac{3150612988577617}{8182800438191875} a^{10} - \frac{12070122905405524}{24548401314575625} a^{9} + \frac{3427151464030519}{4909680262915125} a^{8} + \frac{24870833921347892}{24548401314575625} a^{7} - \frac{26008988392491174}{8182800438191875} a^{6} + \frac{7223865474269344}{24548401314575625} a^{5} + \frac{21049039864053124}{8182800438191875} a^{4} - \frac{9068755743524952}{8182800438191875} a^{3} - \frac{1118406615244307}{327312017527675} a^{2} - \frac{126127559158766}{196387210516605} a + \frac{752132128883066}{196387210516605} \) (order $6$)
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:  \( 545995.081874 \) (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

\(\Q(\sqrt{-3}) \), 5.3.4903.1, 10.0.5841576387.2, 10.6.28641249025461.1, 10.0.117865222327.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
4903Data not computed