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Properties

Label 20.0.11687418886...1241.2
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 71^{18}$
Root discriminant $401.22$
Ramified primes $11, 71$
Class number $1015260925$ (GRH)
Class group $[5, 203052185]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![671391534375, 808179039000, -115389084975, -750849440890, 76901734366, 272079548399, 19864173718, -34794231463, 12569556082, -2329750576, 446289007, -135265045, 31434424, -3467410, 211942, -27301, 7927, -988, -17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 17*x^18 - 988*x^17 + 7927*x^16 - 27301*x^15 + 211942*x^14 - 3467410*x^13 + 31434424*x^12 - 135265045*x^11 + 446289007*x^10 - 2329750576*x^9 + 12569556082*x^8 - 34794231463*x^7 + 19864173718*x^6 + 272079548399*x^5 + 76901734366*x^4 - 750849440890*x^3 - 115389084975*x^2 + 808179039000*x + 671391534375)
 
gp: K = bnfinit(x^20 - x^19 - 17*x^18 - 988*x^17 + 7927*x^16 - 27301*x^15 + 211942*x^14 - 3467410*x^13 + 31434424*x^12 - 135265045*x^11 + 446289007*x^10 - 2329750576*x^9 + 12569556082*x^8 - 34794231463*x^7 + 19864173718*x^6 + 272079548399*x^5 + 76901734366*x^4 - 750849440890*x^3 - 115389084975*x^2 + 808179039000*x + 671391534375, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 17 x^{18} - 988 x^{17} + 7927 x^{16} - 27301 x^{15} + 211942 x^{14} - 3467410 x^{13} + 31434424 x^{12} - 135265045 x^{11} + 446289007 x^{10} - 2329750576 x^{9} + 12569556082 x^{8} - 34794231463 x^{7} + 19864173718 x^{6} + 272079548399 x^{5} + 76901734366 x^{4} - 750849440890 x^{3} - 115389084975 x^{2} + 808179039000 x + 671391534375 \)

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:  \(11687418886731512831139064930618445074696899279531241=11^{18}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $401.22$
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:  $11, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(781=11\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{781}(1,·)$, $\chi_{781}(259,·)$, $\chi_{781}(196,·)$, $\chi_{781}(585,·)$, $\chi_{781}(522,·)$, $\chi_{781}(780,·)$, $\chi_{781}(14,·)$, $\chi_{781}(401,·)$, $\chi_{781}(147,·)$, $\chi_{781}(85,·)$, $\chi_{781}(279,·)$, $\chi_{781}(409,·)$, $\chi_{781}(285,·)$, $\chi_{781}(496,·)$, $\chi_{781}(372,·)$, $\chi_{781}(502,·)$, $\chi_{781}(696,·)$, $\chi_{781}(634,·)$, $\chi_{781}(380,·)$, $\chi_{781}(767,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{5} - \frac{1}{15} a$, $\frac{1}{45} a^{6} - \frac{1}{9} a^{4} + \frac{4}{45} a^{2}$, $\frac{1}{45} a^{7} + \frac{1}{45} a^{5} + \frac{4}{45} a^{3} - \frac{2}{15} a$, $\frac{1}{135} a^{8} + \frac{1}{45} a^{5} - \frac{7}{45} a^{4} - \frac{1}{9} a^{3} + \frac{4}{27} a^{2} + \frac{4}{45} a$, $\frac{1}{405} a^{9} - \frac{1}{135} a^{7} + \frac{1}{135} a^{5} - \frac{37}{405} a^{3} + \frac{4}{45} a$, $\frac{1}{2025} a^{10} + \frac{1}{675} a^{6} - \frac{1}{45} a^{5} - \frac{4}{81} a^{4} + \frac{1}{9} a^{3} + \frac{32}{675} a^{2} - \frac{4}{45} a$, $\frac{1}{6075} a^{11} + \frac{1}{1215} a^{9} - \frac{1}{405} a^{8} - \frac{19}{2025} a^{7} + \frac{1}{135} a^{6} - \frac{8}{1215} a^{5} + \frac{2}{135} a^{4} + \frac{631}{6075} a^{3} - \frac{8}{405} a^{2} - \frac{4}{45} a$, $\frac{1}{6075} a^{12} - \frac{1}{6075} a^{10} - \frac{4}{2025} a^{8} - \frac{58}{6075} a^{6} + \frac{1}{45} a^{5} + \frac{286}{6075} a^{4} - \frac{1}{9} a^{3} - \frac{8}{225} a^{2} + \frac{4}{45} a$, $\frac{1}{18225} a^{13} + \frac{1}{18225} a^{11} + \frac{1}{6075} a^{10} + \frac{13}{18225} a^{9} + \frac{4}{1215} a^{8} + \frac{53}{18225} a^{7} + \frac{11}{2025} a^{6} - \frac{559}{18225} a^{5} - \frac{134}{1215} a^{4} - \frac{1129}{18225} a^{3} + \frac{2641}{6075} a^{2} + \frac{4}{45} a - \frac{1}{3}$, $\frac{1}{273375} a^{14} + \frac{1}{273375} a^{13} + \frac{1}{273375} a^{12} - \frac{14}{273375} a^{11} - \frac{47}{273375} a^{10} + \frac{163}{273375} a^{9} + \frac{113}{273375} a^{8} + \frac{1043}{273375} a^{7} - \frac{2269}{273375} a^{6} + \frac{2336}{273375} a^{5} - \frac{25189}{273375} a^{4} - \frac{19729}{273375} a^{3} + \frac{7091}{18225} a^{2} - \frac{28}{135} a + \frac{4}{9}$, $\frac{1}{12301875} a^{15} - \frac{1}{4100625} a^{14} - \frac{26}{4100625} a^{13} + \frac{23}{1366875} a^{12} + \frac{173}{4100625} a^{11} + \frac{304}{1366875} a^{10} + \frac{8161}{12301875} a^{9} + \frac{9722}{4100625} a^{8} - \frac{21337}{4100625} a^{7} - \frac{512}{1366875} a^{6} - \frac{120086}{4100625} a^{5} - \frac{19072}{1366875} a^{4} + \frac{298216}{12301875} a^{3} + \frac{23468}{164025} a^{2} + \frac{2731}{6075} a + \frac{25}{81}$, $\frac{1}{12301875} a^{16} + \frac{1}{4100625} a^{14} + \frac{7}{1366875} a^{13} - \frac{53}{820125} a^{12} + \frac{112}{1366875} a^{11} + \frac{2014}{12301875} a^{10} - \frac{284}{1366875} a^{9} - \frac{931}{4100625} a^{8} - \frac{8719}{1366875} a^{7} - \frac{13889}{4100625} a^{6} + \frac{24827}{1366875} a^{5} + \frac{1250362}{12301875} a^{4} + \frac{83957}{1366875} a^{3} + \frac{17153}{54675} a^{2} + \frac{311}{2025} a - \frac{5}{27}$, $\frac{1}{184528125} a^{17} - \frac{7}{184528125} a^{15} - \frac{44}{61509375} a^{14} + \frac{209}{12301875} a^{13} - \frac{368}{20503125} a^{12} - \frac{8801}{184528125} a^{11} + \frac{4601}{20503125} a^{10} + \frac{74672}{184528125} a^{9} - \frac{117002}{61509375} a^{8} - \frac{165394}{61509375} a^{7} + \frac{31547}{20503125} a^{6} - \frac{2723033}{184528125} a^{5} + \frac{1945927}{20503125} a^{4} - \frac{5047247}{36905625} a^{3} - \frac{867029}{2460375} a^{2} - \frac{4292}{18225} a - \frac{32}{243}$, $\frac{1}{52590515625} a^{18} - \frac{23}{52590515625} a^{17} + \frac{1148}{52590515625} a^{16} - \frac{49}{2767921875} a^{15} + \frac{7997}{17530171875} a^{14} - \frac{377099}{17530171875} a^{13} + \frac{770456}{10518103125} a^{12} - \frac{313943}{52590515625} a^{11} + \frac{504782}{2103620625} a^{10} + \frac{62826473}{52590515625} a^{9} - \frac{3840266}{5843390625} a^{8} + \frac{172006613}{17530171875} a^{7} - \frac{63014492}{52590515625} a^{6} + \frac{1006725547}{52590515625} a^{5} + \frac{8085149291}{52590515625} a^{4} - \frac{123250928}{10518103125} a^{3} - \frac{269712608}{701206875} a^{2} - \frac{745763}{5194125} a + \frac{16177}{69255}$, $\frac{1}{217641114055040977361485380441609365562600816618307429998046875} a^{19} + \frac{411821448094595813397283994603243299762092675845126}{72547038018346992453828460147203121854200272206102476666015625} a^{18} - \frac{75198824764868243089421871569137507047285841653560476}{43528222811008195472297076088321873112520163323661485999609375} a^{17} - \frac{1312890956364601607788298038949106906510563499274693711}{72547038018346992453828460147203121854200272206102476666015625} a^{16} + \frac{128168409182412338528075669684480519854403463521710079}{43528222811008195472297076088321873112520163323661485999609375} a^{15} + \frac{90220100810799442258454103932305949816268504826421019968}{72547038018346992453828460147203121854200272206102476666015625} a^{14} + \frac{3505172628026812985534370774082114019697605214871577802433}{217641114055040977361485380441609365562600816618307429998046875} a^{13} - \frac{1807443051133565019468302662723515742273512063589769030101}{72547038018346992453828460147203121854200272206102476666015625} a^{12} + \frac{15161941173404192823680625737550331560533955997012218048337}{217641114055040977361485380441609365562600816618307429998046875} a^{11} + \frac{16723920660096508709564082831202797453651908543651963486351}{72547038018346992453828460147203121854200272206102476666015625} a^{10} - \frac{2879347827015991741806969585456305396407634776513505437406}{217641114055040977361485380441609365562600816618307429998046875} a^{9} + \frac{10206788023201522041287688731844549695014945734656608158584}{2901881520733879698153138405888124874168010888244099066640625} a^{8} + \frac{870506023874837614752383596479390755037841901135004770609407}{217641114055040977361485380441609365562600816618307429998046875} a^{7} + \frac{136767611026551017624836761029640280048910135862934691516086}{14509407603669398490765692029440624370840054441220495333203125} a^{6} + \frac{2012889031815434782561283185878584929659941399158987746372628}{217641114055040977361485380441609365562600816618307429998046875} a^{5} + \frac{2209538708126058039748406179172997693482603250353643352686887}{72547038018346992453828460147203121854200272206102476666015625} a^{4} + \frac{3274324661568070816505843946214759332935978115206705259876977}{43528222811008195472297076088321873112520163323661485999609375} a^{3} + \frac{1204211306553264930420176968311962657674200766148792789614532}{2901881520733879698153138405888124874168010888244099066640625} a^{2} - \frac{3005942497240162826619022919977363210757392168545805645438}{21495418672102812578912136339912036104948228801808141234375} a - \frac{3678629546768942998976999671683266196495542173646393}{550106683866994563759747571079002843376794083219658125}$

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

Class group and class number

$C_{5}\times C_{203052185}$, which has order $1015260925$ (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:  \( 70496491595348.39 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{781}) \), \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{-71})\), 5.5.372052421521.1, 10.10.108108366404878733747637421.2, 10.0.9828033309534430340694311.2, 10.0.1522653047956038503487851.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
$71$71.10.9.9$x^{10} + 9088$$10$$1$$9$$C_{10}$$[\ ]_{10}$
71.10.9.9$x^{10} + 9088$$10$$1$$9$$C_{10}$$[\ ]_{10}$