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Properties

Label 40.0.13030509980...5936.6
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 3^{20}\cdot 11^{36}$
Root discriminant $59.96$
Ramified primes $2, 3, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 60, 0, 3025, 0, 30356, 0, 202622, 0, 753104, 0, 1959963, 0, 3548680, 0, 4817692, 0, 4954644, 0, 3994121, 0, 2544812, 0, 1304886, 0, 540148, 0, 181674, 0, 49280, 0, 10709, 0, 1812, 0, 231, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 20*x^38 + 231*x^36 + 1812*x^34 + 10709*x^32 + 49280*x^30 + 181674*x^28 + 540148*x^26 + 1304886*x^24 + 2544812*x^22 + 3994121*x^20 + 4954644*x^18 + 4817692*x^16 + 3548680*x^14 + 1959963*x^12 + 753104*x^10 + 202622*x^8 + 30356*x^6 + 3025*x^4 + 60*x^2 + 1)
 
gp: K = bnfinit(x^40 + 20*x^38 + 231*x^36 + 1812*x^34 + 10709*x^32 + 49280*x^30 + 181674*x^28 + 540148*x^26 + 1304886*x^24 + 2544812*x^22 + 3994121*x^20 + 4954644*x^18 + 4817692*x^16 + 3548680*x^14 + 1959963*x^12 + 753104*x^10 + 202622*x^8 + 30356*x^6 + 3025*x^4 + 60*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 20 x^{38} + 231 x^{36} + 1812 x^{34} + 10709 x^{32} + 49280 x^{30} + 181674 x^{28} + 540148 x^{26} + 1304886 x^{24} + 2544812 x^{22} + 3994121 x^{20} + 4954644 x^{18} + 4817692 x^{16} + 3548680 x^{14} + 1959963 x^{12} + 753104 x^{10} + 202622 x^{8} + 30356 x^{6} + 3025 x^{4} + 60 x^{2} + 1 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.96$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(137,·)$, $\chi_{264}(139,·)$, $\chi_{264}(19,·)$, $\chi_{264}(151,·)$, $\chi_{264}(25,·)$, $\chi_{264}(157,·)$, $\chi_{264}(133,·)$, $\chi_{264}(35,·)$, $\chi_{264}(37,·)$, $\chi_{264}(167,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(257,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(185,·)$, $\chi_{264}(181,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(215,·)$, $\chi_{264}(89,·)$, $\chi_{264}(221,·)$, $\chi_{264}(95,·)$, $\chi_{264}(263,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(229,·)$, $\chi_{264}(259,·)$, $\chi_{264}(107,·)$, $\chi_{264}(239,·)$, $\chi_{264}(113,·)$, $\chi_{264}(211,·)$, $\chi_{264}(245,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{43} a^{32} + \frac{7}{43} a^{30} + \frac{11}{43} a^{28} + \frac{14}{43} a^{26} + \frac{2}{43} a^{24} - \frac{16}{43} a^{22} + \frac{6}{43} a^{20} - \frac{4}{43} a^{18} - \frac{21}{43} a^{16} - \frac{14}{43} a^{14} - \frac{16}{43} a^{12} - \frac{10}{43} a^{10} - \frac{20}{43} a^{8} + \frac{8}{43} a^{6} - \frac{18}{43} a^{4} - \frac{11}{43} a^{2} - \frac{20}{43}$, $\frac{1}{43} a^{33} + \frac{7}{43} a^{31} + \frac{11}{43} a^{29} + \frac{14}{43} a^{27} + \frac{2}{43} a^{25} - \frac{16}{43} a^{23} + \frac{6}{43} a^{21} - \frac{4}{43} a^{19} - \frac{21}{43} a^{17} - \frac{14}{43} a^{15} - \frac{16}{43} a^{13} - \frac{10}{43} a^{11} - \frac{20}{43} a^{9} + \frac{8}{43} a^{7} - \frac{18}{43} a^{5} - \frac{11}{43} a^{3} - \frac{20}{43} a$, $\frac{1}{43} a^{34} + \frac{5}{43} a^{30} - \frac{20}{43} a^{28} - \frac{10}{43} a^{26} + \frac{13}{43} a^{24} - \frac{11}{43} a^{22} - \frac{3}{43} a^{20} + \frac{7}{43} a^{18} + \frac{4}{43} a^{16} - \frac{4}{43} a^{14} + \frac{16}{43} a^{12} + \frac{7}{43} a^{10} + \frac{19}{43} a^{8} + \frac{12}{43} a^{6} - \frac{14}{43} a^{4} + \frac{14}{43} a^{2} + \frac{11}{43}$, $\frac{1}{43} a^{35} + \frac{5}{43} a^{31} - \frac{20}{43} a^{29} - \frac{10}{43} a^{27} + \frac{13}{43} a^{25} - \frac{11}{43} a^{23} - \frac{3}{43} a^{21} + \frac{7}{43} a^{19} + \frac{4}{43} a^{17} - \frac{4}{43} a^{15} + \frac{16}{43} a^{13} + \frac{7}{43} a^{11} + \frac{19}{43} a^{9} + \frac{12}{43} a^{7} - \frac{14}{43} a^{5} + \frac{14}{43} a^{3} + \frac{11}{43} a$, $\frac{1}{43} a^{36} - \frac{12}{43} a^{30} + \frac{21}{43} a^{28} - \frac{14}{43} a^{26} - \frac{21}{43} a^{24} - \frac{9}{43} a^{22} + \frac{20}{43} a^{20} - \frac{19}{43} a^{18} + \frac{15}{43} a^{16} + \frac{1}{43} a^{12} - \frac{17}{43} a^{10} - \frac{17}{43} a^{8} - \frac{11}{43} a^{6} + \frac{18}{43} a^{4} - \frac{20}{43} a^{2} + \frac{14}{43}$, $\frac{1}{43} a^{37} - \frac{12}{43} a^{31} + \frac{21}{43} a^{29} - \frac{14}{43} a^{27} - \frac{21}{43} a^{25} - \frac{9}{43} a^{23} + \frac{20}{43} a^{21} - \frac{19}{43} a^{19} + \frac{15}{43} a^{17} + \frac{1}{43} a^{13} - \frac{17}{43} a^{11} - \frac{17}{43} a^{9} - \frac{11}{43} a^{7} + \frac{18}{43} a^{5} - \frac{20}{43} a^{3} + \frac{14}{43} a$, $\frac{1}{224658185921597823206597305721673242429} a^{38} - \frac{2374344175392738427827280366860414928}{224658185921597823206597305721673242429} a^{36} - \frac{285817055617074802527238122251899232}{224658185921597823206597305721673242429} a^{34} - \frac{478984279929454491756230919551141021}{224658185921597823206597305721673242429} a^{32} - \frac{107673681794240679588774801985928040591}{224658185921597823206597305721673242429} a^{30} + \frac{2400223195729263538350937155486925638}{9767747213982514052460752422681445323} a^{28} - \frac{30437857782170667456149719282473579991}{224658185921597823206597305721673242429} a^{26} + \frac{110532439597435151544133323215691817415}{224658185921597823206597305721673242429} a^{24} - \frac{91870023879315432381228634584653724999}{224658185921597823206597305721673242429} a^{22} - \frac{77956631493468250661210837243011076131}{224658185921597823206597305721673242429} a^{20} + \frac{85673898183265813244371749103575000352}{224658185921597823206597305721673242429} a^{18} - \frac{29425089331420342408526207219836986433}{224658185921597823206597305721673242429} a^{16} + \frac{104242849965171210434514982108138137859}{224658185921597823206597305721673242429} a^{14} - \frac{23463589075018459856113909201501780962}{224658185921597823206597305721673242429} a^{12} - \frac{78898761176656569827479473790585015419}{224658185921597823206597305721673242429} a^{10} - \frac{78093783310368026344393388290776871130}{224658185921597823206597305721673242429} a^{8} + \frac{11461521967243433960178548894521116743}{224658185921597823206597305721673242429} a^{6} - \frac{31734736323324652260049632891251703692}{224658185921597823206597305721673242429} a^{4} + \frac{31873816148452625663929970040650155025}{224658185921597823206597305721673242429} a^{2} - \frac{4701273404819260251598919387234994898}{224658185921597823206597305721673242429}$, $\frac{1}{224658185921597823206597305721673242429} a^{39} - \frac{2374344175392738427827280366860414928}{224658185921597823206597305721673242429} a^{37} - \frac{285817055617074802527238122251899232}{224658185921597823206597305721673242429} a^{35} - \frac{478984279929454491756230919551141021}{224658185921597823206597305721673242429} a^{33} - \frac{107673681794240679588774801985928040591}{224658185921597823206597305721673242429} a^{31} + \frac{2400223195729263538350937155486925638}{9767747213982514052460752422681445323} a^{29} - \frac{30437857782170667456149719282473579991}{224658185921597823206597305721673242429} a^{27} + \frac{110532439597435151544133323215691817415}{224658185921597823206597305721673242429} a^{25} - \frac{91870023879315432381228634584653724999}{224658185921597823206597305721673242429} a^{23} - \frac{77956631493468250661210837243011076131}{224658185921597823206597305721673242429} a^{21} + \frac{85673898183265813244371749103575000352}{224658185921597823206597305721673242429} a^{19} - \frac{29425089331420342408526207219836986433}{224658185921597823206597305721673242429} a^{17} + \frac{104242849965171210434514982108138137859}{224658185921597823206597305721673242429} a^{15} - \frac{23463589075018459856113909201501780962}{224658185921597823206597305721673242429} a^{13} - \frac{78898761176656569827479473790585015419}{224658185921597823206597305721673242429} a^{11} - \frac{78093783310368026344393388290776871130}{224658185921597823206597305721673242429} a^{9} + \frac{11461521967243433960178548894521116743}{224658185921597823206597305721673242429} a^{7} - \frac{31734736323324652260049632891251703692}{224658185921597823206597305721673242429} a^{5} + \frac{31873816148452625663929970040650155025}{224658185921597823206597305721673242429} a^{3} - \frac{4701273404819260251598919387234994898}{224658185921597823206597305721673242429} a$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{5080791259918300299740496912046655776}{224658185921597823206597305721673242429} a^{38} - \frac{101451924046779396192754697460220609903}{224658185921597823206597305721673242429} a^{36} - \frac{1170410703975544732317290899165648530944}{224658185921597823206597305721673242429} a^{34} - \frac{9169045072830155735773051351456043006888}{224658185921597823206597305721673242429} a^{32} - \frac{54119072247195440536245166968075232034124}{224658185921597823206597305721673242429} a^{30} - \frac{10811815539627020670491264589715384316209}{9767747213982514052460752422681445323} a^{28} - \frac{915237434801245657191272986145370561091636}{224658185921597823206597305721673242429} a^{26} - \frac{2715817121311291661864357525611831054438697}{224658185921597823206597305721673242429} a^{24} - \frac{6545746692282391260349052949038788511323256}{224658185921597823206597305721673242429} a^{22} - \frac{12728755983245911553739024764537008612303686}{224658185921597823206597305721673242429} a^{20} - \frac{19907024450807145541181119779646685184616180}{224658185921597823206597305721673242429} a^{18} - \frac{24577788814115491124269186355572593226390486}{224658185921597823206597305721673242429} a^{16} - \frac{23755874145055474657407286129028929607548560}{224658185921597823206597305721673242429} a^{14} - \frac{17348934078521056249858340015679985489178873}{224658185921597823206597305721673242429} a^{12} - \frac{9478004883568827701515967247945978499664712}{224658185921597823206597305721673242429} a^{10} - \frac{3575881750535688428642484243944196077931123}{224658185921597823206597305721673242429} a^{8} - \frac{942779388565629517685965251399746338891980}{224658185921597823206597305721673242429} a^{6} - \frac{133670352136312976626545402469902327786412}{224658185921597823206597305721673242429} a^{4} - \frac{13728457126047526199356492463843048242376}{224658185921597823206597305721673242429} a^{2} - \frac{47473013766506234347649214060219294152}{224658185921597823206597305721673242429} \) (order $6$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-33})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.7, 10.10.77265229938688.1, 10.0.1706859170463744.1, 10.0.586732839846912.1, 10.0.52089208083.1, 10.0.18775450875101184.1, 10.10.7024111812608.1, \(\Q(\zeta_{44})^+\), 20.0.360977976896857923653306611918700544.4, 20.0.352517555563337816067682238201856.5, \(\Q(\zeta_{88})^+\), 20.0.2913368227796180298080018497536.4, 20.0.360977976896857923653306611918700544.3, 20.0.344255425354822086003595935744.2, 20.0.360977976896857923653306611918700544.7

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{8}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2Data not computed
3Data not computed
11Data not computed