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Properties

Label 20.0.17598488843...1792.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{15}\cdot 11^{13}$
Root discriminant $57.85$
Ramified primes $2, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2524841438603, -941629298346, 1451235311405, -475906101396, 375875514806, -106965671572, 57369907328, -14042016534, 5709357517, -1190535180, 388231353, -68079464, 18371119, -2650522, 600600, -68434, 13039, -1076, 170, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 170*x^18 - 1076*x^17 + 13039*x^16 - 68434*x^15 + 600600*x^14 - 2650522*x^13 + 18371119*x^12 - 68079464*x^11 + 388231353*x^10 - 1190535180*x^9 + 5709357517*x^8 - 14042016534*x^7 + 57369907328*x^6 - 106965671572*x^5 + 375875514806*x^4 - 475906101396*x^3 + 1451235311405*x^2 - 941629298346*x + 2524841438603)
 
gp: K = bnfinit(x^20 - 8*x^19 + 170*x^18 - 1076*x^17 + 13039*x^16 - 68434*x^15 + 600600*x^14 - 2650522*x^13 + 18371119*x^12 - 68079464*x^11 + 388231353*x^10 - 1190535180*x^9 + 5709357517*x^8 - 14042016534*x^7 + 57369907328*x^6 - 106965671572*x^5 + 375875514806*x^4 - 475906101396*x^3 + 1451235311405*x^2 - 941629298346*x + 2524841438603, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 170 x^{18} - 1076 x^{17} + 13039 x^{16} - 68434 x^{15} + 600600 x^{14} - 2650522 x^{13} + 18371119 x^{12} - 68079464 x^{11} + 388231353 x^{10} - 1190535180 x^{9} + 5709357517 x^{8} - 14042016534 x^{7} + 57369907328 x^{6} - 106965671572 x^{5} + 375875514806 x^{4} - 475906101396 x^{3} + 1451235311405 x^{2} - 941629298346 x + 2524841438603 \)

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:  \(175984888437863295253314247476641792=2^{30}\cdot 7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.85$
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:  $2, 7, 11$
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}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{19} - \frac{140892362831888230969328323363691759732640655471125310371868066367536234554322070419999311828666084}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{18} - \frac{261401454663539620481417055221714262375516004160055930564802954412732292779665049056490390891089940}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{17} - \frac{197284810004051020906462471951493920001920733244455275354739910749610197254250386564076992318198913}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{16} - \frac{218091524962168126702799133776502686111736764461820965458638170599504272982126292760565041558458052}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{15} + \frac{105537671820441275206347581203271600380823948556345664889193252124882106438489172237283161702466135}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{14} - \frac{123558924055108110794325705903116675260138219857352911203742780301033784161137310270835857297210209}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{13} + \frac{124663913464298853252606141574382192223210167662597461289071663750712139219311804396293811501801981}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{12} - \frac{116794008432007157753192234110026272339411081395878245446516552678850321622432511564008044570829393}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{11} - \frac{270647814625610190834080180765853684557060283441360740779207530580798936824833348888506232335745337}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{10} + \frac{164331281929119596757616011151409563738462022078484057563397370514911656556194202860921277773653116}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{9} + \frac{196173115832693611028158500049725647955320048111292878464532503592717451658551059351893373714244356}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{8} - \frac{243923134299324452513330270116426813794539586416433639320153741958699044763758989622180514258094707}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{7} + \frac{310295810079886806318270009938680716210064925676701683913657879574543946229593005481098669260056863}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{6} - \frac{272667129992085354778706847367980102251323229256126744202671199833515413704148831867731215845033031}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{5} - \frac{150272478711384961293713713182211797430737792071135354385112083119107134955961656318002245512549533}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{4} + \frac{336496685219285833113112709399167993619228915640878565002236573812982330136876189912152050851014770}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{3} + \frac{245327665407486929309653978193267669367532014656210273603106398756753252531347385273722777170230304}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a^{2} - \frac{215696224019247445005952499096469171527005733252485390853510691384065154632242842341247165373264437}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209} a + \frac{199903355981522340050475817847704124345181285625133302643183567991025408663299428473204962196921362}{675908593370681482232221125953624194825386949652985534558355828981488873594910330030267708181474209}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.241472.2, 10.0.246071287.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
7Data not computed
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$