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Properties

Label 20.0.32563112104...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 37^{15}$
Root discriminant $33.55$
Ramified primes $5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10256, -30696, 91296, -173528, 238937, -235037, 201934, -163476, 130181, -92370, 57953, -33492, 20202, -11695, 5989, -2458, 833, -231, 51, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 51*x^18 - 231*x^17 + 833*x^16 - 2458*x^15 + 5989*x^14 - 11695*x^13 + 20202*x^12 - 33492*x^11 + 57953*x^10 - 92370*x^9 + 130181*x^8 - 163476*x^7 + 201934*x^6 - 235037*x^5 + 238937*x^4 - 173528*x^3 + 91296*x^2 - 30696*x + 10256)
 
gp: K = bnfinit(x^20 - 9*x^19 + 51*x^18 - 231*x^17 + 833*x^16 - 2458*x^15 + 5989*x^14 - 11695*x^13 + 20202*x^12 - 33492*x^11 + 57953*x^10 - 92370*x^9 + 130181*x^8 - 163476*x^7 + 201934*x^6 - 235037*x^5 + 238937*x^4 - 173528*x^3 + 91296*x^2 - 30696*x + 10256, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 51 x^{18} - 231 x^{17} + 833 x^{16} - 2458 x^{15} + 5989 x^{14} - 11695 x^{13} + 20202 x^{12} - 33492 x^{11} + 57953 x^{10} - 92370 x^{9} + 130181 x^{8} - 163476 x^{7} + 201934 x^{6} - 235037 x^{5} + 238937 x^{4} - 173528 x^{3} + 91296 x^{2} - 30696 x + 10256 \)

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:  \(3256311210466946358286064453125=5^{10}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.55$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{3}{10} a^{10} + \frac{2}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{70} a^{13} + \frac{1}{70} a^{12} - \frac{2}{35} a^{11} + \frac{5}{14} a^{10} - \frac{9}{35} a^{9} - \frac{3}{10} a^{8} - \frac{13}{70} a^{7} + \frac{23}{70} a^{6} - \frac{11}{70} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{10} a - \frac{11}{35}$, $\frac{1}{70} a^{14} + \frac{1}{35} a^{12} - \frac{13}{70} a^{11} + \frac{13}{70} a^{10} - \frac{1}{7} a^{9} + \frac{1}{70} a^{8} + \frac{4}{35} a^{7} + \frac{1}{70} a^{6} + \frac{1}{10} a^{5} - \frac{19}{70} a^{4} + \frac{23}{70} a^{3} + \frac{17}{35} a - \frac{2}{7}$, $\frac{1}{770} a^{15} + \frac{1}{154} a^{14} - \frac{1}{154} a^{13} + \frac{1}{70} a^{12} + \frac{19}{154} a^{11} - \frac{267}{770} a^{10} - \frac{47}{110} a^{9} + \frac{19}{70} a^{8} - \frac{337}{770} a^{7} + \frac{131}{770} a^{6} - \frac{19}{770} a^{5} - \frac{289}{770} a^{4} - \frac{81}{770} a^{3} - \frac{9}{70} a^{2} + \frac{129}{770} a - \frac{1}{385}$, $\frac{1}{3850} a^{16} + \frac{1}{1925} a^{15} + \frac{1}{1925} a^{14} + \frac{3}{770} a^{13} + \frac{86}{1925} a^{12} - \frac{101}{3850} a^{11} - \frac{4}{275} a^{10} - \frac{414}{1925} a^{9} + \frac{376}{1925} a^{8} - \frac{1}{1925} a^{7} + \frac{127}{3850} a^{6} - \frac{1651}{3850} a^{5} - \frac{1051}{3850} a^{4} - \frac{73}{275} a^{3} + \frac{193}{770} a^{2} - \frac{19}{77} a - \frac{712}{1925}$, $\frac{1}{3850} a^{17} - \frac{1}{1925} a^{15} + \frac{1}{350} a^{14} - \frac{23}{3850} a^{13} + \frac{16}{385} a^{12} + \frac{18}{1925} a^{11} + \frac{1319}{3850} a^{10} + \frac{379}{1925} a^{9} + \frac{1189}{3850} a^{8} - \frac{402}{1925} a^{7} - \frac{37}{77} a^{6} - \frac{662}{1925} a^{5} + \frac{174}{385} a^{4} - \frac{73}{550} a^{3} - \frac{134}{385} a^{2} + \frac{233}{550} a + \frac{929}{1925}$, $\frac{1}{2964500} a^{18} + \frac{59}{2964500} a^{17} - \frac{173}{2964500} a^{16} - \frac{17}{423500} a^{15} + \frac{21019}{2964500} a^{14} + \frac{619}{134750} a^{13} + \frac{6997}{423500} a^{12} + \frac{707279}{2964500} a^{11} - \frac{47196}{148225} a^{10} + \frac{61561}{211750} a^{9} + \frac{151369}{592900} a^{8} + \frac{135523}{1482250} a^{7} - \frac{950811}{2964500} a^{6} + \frac{50817}{296450} a^{5} + \frac{83539}{296450} a^{4} - \frac{1151537}{2964500} a^{3} + \frac{145333}{423500} a^{2} - \frac{15069}{1482250} a - \frac{22076}{741125}$, $\frac{1}{2540334476240342826311834413000} a^{19} + \frac{32696781245062735661039}{2540334476240342826311834413000} a^{18} - \frac{49560332929389784091735553}{2540334476240342826311834413000} a^{17} - \frac{278920343494213561959294719}{2540334476240342826311834413000} a^{16} + \frac{965930976110922655080890529}{2540334476240342826311834413000} a^{15} + \frac{3732044137780862821632098759}{1270167238120171413155917206500} a^{14} - \frac{16159095308360644123791803731}{2540334476240342826311834413000} a^{13} - \frac{79159613822593038126291258871}{2540334476240342826311834413000} a^{12} + \frac{3515427509167215410362758391}{23093949784003116602834858300} a^{11} - \frac{19966939654538471148188916459}{635083619060085706577958603250} a^{10} - \frac{234287247928641142680415582331}{508066895248068565262366882600} a^{9} - \frac{599703180900347969849619905737}{1270167238120171413155917206500} a^{8} + \frac{236693814829545444596802600189}{2540334476240342826311834413000} a^{7} - \frac{21406962143840049103023452809}{127016723812017141315591720650} a^{6} + \frac{2892622924707762253035509839}{50806689524806856526236688260} a^{5} - \frac{649514523217914926418085506077}{2540334476240342826311834413000} a^{4} - \frac{140587856022110628648022539959}{2540334476240342826311834413000} a^{3} - \frac{308640344587683101392396180077}{635083619060085706577958603250} a^{2} + \frac{189417910575097446574601942}{1171740994575803886675200375} a + \frac{14524815021027644456788968733}{63508361906008570657795860325}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 5638854.77787 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.1266325.1, 5.1.1266325.1 x5, 10.2.59332423208125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1266325.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$37$37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$