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Properties

Label 40.0.11302165783...0000.8
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
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![392079601, 0, -99217770, 0, -85190495, 0, -541649, 0, -71349280, 0, -87607421, 0, -27594627, 0, 21190547, 0, 43814026, 0, 54866175, 0, 52728422, 0, 37603690, 0, 19840782, 0, 7829798, 0, 2328570, 0, 521224, 0, 86696, 0, 10413, 0, 855, 0, 43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 43*x^38 + 855*x^36 + 10413*x^34 + 86696*x^32 + 521224*x^30 + 2328570*x^28 + 7829798*x^26 + 19840782*x^24 + 37603690*x^22 + 52728422*x^20 + 54866175*x^18 + 43814026*x^16 + 21190547*x^14 - 27594627*x^12 - 87607421*x^10 - 71349280*x^8 - 541649*x^6 - 85190495*x^4 - 99217770*x^2 + 392079601)
 
gp: K = bnfinit(x^40 + 43*x^38 + 855*x^36 + 10413*x^34 + 86696*x^32 + 521224*x^30 + 2328570*x^28 + 7829798*x^26 + 19840782*x^24 + 37603690*x^22 + 52728422*x^20 + 54866175*x^18 + 43814026*x^16 + 21190547*x^14 - 27594627*x^12 - 87607421*x^10 - 71349280*x^8 - 541649*x^6 - 85190495*x^4 - 99217770*x^2 + 392079601, 1)
 

Normalized defining polynomial

\( x^{40} + 43 x^{38} + 855 x^{36} + 10413 x^{34} + 86696 x^{32} + 521224 x^{30} + 2328570 x^{28} + 7829798 x^{26} + 19840782 x^{24} + 37603690 x^{22} + 52728422 x^{20} + 54866175 x^{18} + 43814026 x^{16} + 21190547 x^{14} - 27594627 x^{12} - 87607421 x^{10} - 71349280 x^{8} - 541649 x^{6} - 85190495 x^{4} - 99217770 x^{2} + 392079601 \)

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:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(389,·)$, $\chi_{660}(391,·)$, $\chi_{660}(269,·)$, $\chi_{660}(271,·)$, $\chi_{660}(659,·)$, $\chi_{660}(151,·)$, $\chi_{660}(409,·)$, $\chi_{660}(161,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(41,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(571,·)$, $\chi_{660}(191,·)$, $\chi_{660}(449,·)$, $\chi_{660}(71,·)$, $\chi_{660}(461,·)$, $\chi_{660}(589,·)$, $\chi_{660}(211,·)$, $\chi_{660}(469,·)$, $\chi_{660}(89,·)$, $\chi_{660}(349,·)$, $\chi_{660}(199,·)$, $\chi_{660}(479,·)$, $\chi_{660}(251,·)$, $\chi_{660}(101,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(109,·)$, $\chi_{660}(239,·)$, $\chi_{660}(499,·)$, $\chi_{660}(379,·)$, $\chi_{660}(281,·)$, $\chi_{660}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17711} a^{22} + \frac{22}{17711} a^{20} + \frac{209}{17711} a^{18} + \frac{1122}{17711} a^{16} + \frac{3740}{17711} a^{14} + \frac{8008}{17711} a^{12} - \frac{6700}{17711} a^{10} - \frac{8273}{17711} a^{8} + \frac{4719}{17711} a^{6} + \frac{1210}{17711} a^{4} + \frac{121}{17711} a^{2} + \frac{6767}{17711}$, $\frac{1}{17711} a^{23} + \frac{22}{17711} a^{21} + \frac{209}{17711} a^{19} + \frac{1122}{17711} a^{17} + \frac{3740}{17711} a^{15} + \frac{8008}{17711} a^{13} - \frac{6700}{17711} a^{11} - \frac{8273}{17711} a^{9} + \frac{4719}{17711} a^{7} + \frac{1210}{17711} a^{5} + \frac{121}{17711} a^{3} + \frac{6767}{17711} a$, $\frac{1}{17711} a^{24} - \frac{275}{17711} a^{20} - \frac{3476}{17711} a^{18} - \frac{3233}{17711} a^{16} - \frac{3428}{17711} a^{14} - \frac{5766}{17711} a^{12} - \frac{2561}{17711} a^{10} - \frac{8096}{17711} a^{8} + \frac{3658}{17711} a^{6} - \frac{8788}{17711} a^{4} + \frac{4105}{17711} a^{2} - \frac{7186}{17711}$, $\frac{1}{17711} a^{25} - \frac{275}{17711} a^{21} - \frac{3476}{17711} a^{19} - \frac{3233}{17711} a^{17} - \frac{3428}{17711} a^{15} - \frac{5766}{17711} a^{13} - \frac{2561}{17711} a^{11} - \frac{8096}{17711} a^{9} + \frac{3658}{17711} a^{7} - \frac{8788}{17711} a^{5} + \frac{4105}{17711} a^{3} - \frac{7186}{17711} a$, $\frac{1}{17711} a^{26} + \frac{2574}{17711} a^{20} + \frac{1109}{17711} a^{18} + \frac{4035}{17711} a^{16} - \frac{4504}{17711} a^{14} + \frac{3475}{17711} a^{12} - \frac{8652}{17711} a^{10} - \frac{4409}{17711} a^{8} - \frac{3966}{17711} a^{6} + \frac{346}{17711} a^{4} + \frac{8378}{17711} a^{2} + \frac{1270}{17711}$, $\frac{1}{17711} a^{27} + \frac{2574}{17711} a^{21} + \frac{1109}{17711} a^{19} + \frac{4035}{17711} a^{17} - \frac{4504}{17711} a^{15} + \frac{3475}{17711} a^{13} - \frac{8652}{17711} a^{11} - \frac{4409}{17711} a^{9} - \frac{3966}{17711} a^{7} + \frac{346}{17711} a^{5} + \frac{8378}{17711} a^{3} + \frac{1270}{17711} a$, $\frac{1}{17711} a^{28} - \frac{2386}{17711} a^{20} - \frac{2601}{17711} a^{18} - \frac{5639}{17711} a^{16} - \frac{6212}{17711} a^{14} - \frac{5640}{17711} a^{12} + \frac{8588}{17711} a^{10} + \frac{2114}{17711} a^{8} + \frac{3386}{17711} a^{6} - \frac{6737}{17711} a^{4} + \frac{8614}{17711} a^{2} - \frac{8345}{17711}$, $\frac{1}{17711} a^{29} - \frac{2386}{17711} a^{21} - \frac{2601}{17711} a^{19} - \frac{5639}{17711} a^{17} - \frac{6212}{17711} a^{15} - \frac{5640}{17711} a^{13} + \frac{8588}{17711} a^{11} + \frac{2114}{17711} a^{9} + \frac{3386}{17711} a^{7} - \frac{6737}{17711} a^{5} + \frac{8614}{17711} a^{3} - \frac{8345}{17711} a$, $\frac{1}{141688} a^{30} - \frac{1}{70844} a^{28} - \frac{3}{141688} a^{26} - \frac{3}{141688} a^{24} + \frac{1}{70844} a^{22} - \frac{7307}{17711} a^{20} + \frac{22635}{70844} a^{18} - \frac{22749}{70844} a^{16} - \frac{53199}{141688} a^{14} - \frac{48879}{141688} a^{12} + \frac{29721}{141688} a^{10} - \frac{60041}{141688} a^{8} - \frac{7809}{141688} a^{6} + \frac{50001}{141688} a^{4} + \frac{66527}{141688} a^{2} - \frac{11531}{141688}$, $\frac{1}{2805564088} a^{31} + \frac{7579}{1402782044} a^{29} + \frac{48869}{2805564088} a^{27} + \frac{55965}{2805564088} a^{25} - \frac{37151}{1402782044} a^{23} + \frac{107841800}{350695511} a^{21} - \frac{393191741}{1402782044} a^{19} - \frac{695854705}{1402782044} a^{17} + \frac{1110975129}{2805564088} a^{15} - \frac{1158632959}{2805564088} a^{13} + \frac{1087140377}{2805564088} a^{11} + \frac{35920319}{2805564088} a^{9} + \frac{1010725239}{2805564088} a^{7} + \frac{1188618217}{2805564088} a^{5} - \frac{629805481}{2805564088} a^{3} + \frac{196778421}{2805564088} a$, $\frac{1}{27777890035288} a^{32} + \frac{10885907}{27777890035288} a^{30} + \frac{555387715}{27777890035288} a^{28} + \frac{218007291}{13888945017644} a^{26} - \frac{700376269}{27777890035288} a^{24} + \frac{128669313}{13888945017644} a^{22} - \frac{5903096082593}{13888945017644} a^{20} + \frac{1930687521265}{6944472508822} a^{18} + \frac{6011553364295}{27777890035288} a^{16} - \frac{5379285522689}{13888945017644} a^{14} - \frac{257474578073}{13888945017644} a^{12} - \frac{623678022123}{6944472508822} a^{10} + \frac{5594606213197}{13888945017644} a^{8} + \frac{1326921755663}{6944472508822} a^{6} - \frac{2723838241309}{6944472508822} a^{4} + \frac{930020627052}{3472236254411} a^{2} - \frac{389717935}{1402852888}$, $\frac{1}{27777890035288} a^{33} + \frac{1177}{6944472508822} a^{31} + \frac{299922113}{27777890035288} a^{29} + \frac{369489763}{27777890035288} a^{27} + \frac{54998901}{3472236254411} a^{25} - \frac{135244751}{6944472508822} a^{23} + \frac{901193454363}{13888945017644} a^{21} - \frac{4261107307667}{13888945017644} a^{19} + \frac{13415944511253}{27777890035288} a^{17} + \frac{3549837609551}{27777890035288} a^{15} - \frac{879387649297}{27777890035288} a^{13} + \frac{8740088423525}{27777890035288} a^{11} - \frac{8764418428207}{27777890035288} a^{9} + \frac{8009599061195}{27777890035288} a^{7} + \frac{11457783417581}{27777890035288} a^{5} + \frac{1132774817943}{27777890035288} a^{3} + \frac{6815661016011}{13888945017644} a$, $\frac{1}{27777890035288} a^{34} + \frac{21994217}{27777890035288} a^{30} + \frac{122940079}{27777890035288} a^{28} - \frac{17097695}{6944472508822} a^{26} - \frac{181368843}{6944472508822} a^{24} - \frac{224620349}{13888945017644} a^{22} + \frac{197576230489}{13888945017644} a^{20} - \frac{10644401974171}{27777890035288} a^{18} - \frac{10497975510805}{27777890035288} a^{16} + \frac{13469449472963}{27777890035288} a^{14} + \frac{3157602152665}{27777890035288} a^{12} + \frac{4701676189205}{27777890035288} a^{10} + \frac{10739167542679}{27777890035288} a^{8} - \frac{2762034426879}{27777890035288} a^{6} - \frac{7787798546949}{27777890035288} a^{4} + \frac{4160182283849}{13888945017644} a^{2} + \frac{18701160}{175356611}$, $\frac{1}{27777890035288} a^{35} + \frac{512}{3472236254411} a^{31} - \frac{703021143}{27777890035288} a^{29} - \frac{351252449}{27777890035288} a^{27} - \frac{210474857}{27777890035288} a^{25} + \frac{189704979}{6944472508822} a^{23} - \frac{6258709196087}{13888945017644} a^{21} - \frac{12264026069041}{27777890035288} a^{19} - \frac{9042824728923}{27777890035288} a^{17} - \frac{4578748106683}{13888945017644} a^{15} - \frac{89616439243}{3472236254411} a^{13} + \frac{58901462963}{6944472508822} a^{11} + \frac{540935533702}{3472236254411} a^{9} - \frac{4558192524275}{13888945017644} a^{7} - \frac{1769994560463}{13888945017644} a^{5} - \frac{4069492287493}{27777890035288} a^{3} + \frac{8645767620787}{27777890035288} a$, $\frac{1}{27777890035288} a^{36} - \frac{1053821}{6944472508822} a^{30} - \frac{650702087}{27777890035288} a^{28} - \frac{350787861}{13888945017644} a^{26} - \frac{79455797}{27777890035288} a^{24} - \frac{65625206}{3472236254411} a^{22} - \frac{6666789034209}{27777890035288} a^{20} + \frac{13146905031503}{27777890035288} a^{18} - \frac{2235376950559}{6944472508822} a^{16} - \frac{12240758341317}{27777890035288} a^{14} + \frac{1481169115063}{27777890035288} a^{12} - \frac{4114083811485}{27777890035288} a^{10} - \frac{4028578423097}{27777890035288} a^{8} - \frac{11755691944017}{27777890035288} a^{6} + \frac{4467653239039}{13888945017644} a^{4} + \frac{1126629232659}{3472236254411} a^{2} - \frac{617737121}{1402852888}$, $\frac{1}{27777890035288} a^{37} + \frac{1271}{13888945017644} a^{31} + \frac{547200101}{27777890035288} a^{29} - \frac{5090447}{3472236254411} a^{27} + \frac{711535093}{27777890035288} a^{25} - \frac{59596875}{6944472508822} a^{23} - \frac{8719054373601}{27777890035288} a^{21} - \frac{12807194308709}{27777890035288} a^{19} + \frac{1046341624392}{3472236254411} a^{17} - \frac{3856285303387}{27777890035288} a^{15} + \frac{11772851347329}{27777890035288} a^{13} - \frac{12384111784475}{27777890035288} a^{11} + \frac{8095149958045}{27777890035288} a^{9} + \frac{1167989553909}{27777890035288} a^{7} - \frac{17511683022}{39013890499} a^{5} - \frac{1788641097189}{13888945017644} a^{3} + \frac{4678597226977}{27777890035288} a$, $\frac{1}{27777890035288} a^{38} - \frac{69916755}{27777890035288} a^{30} + \frac{252875449}{13888945017644} a^{28} + \frac{43278299}{27777890035288} a^{26} + \frac{94725655}{6944472508822} a^{24} + \frac{319143319}{27777890035288} a^{22} + \frac{11279185459095}{27777890035288} a^{20} + \frac{3428528193155}{6944472508822} a^{18} - \frac{12823836638617}{27777890035288} a^{16} - \frac{5149279078217}{27777890035288} a^{14} + \frac{8212905013435}{27777890035288} a^{12} + \frac{10778688571111}{27777890035288} a^{10} + \frac{300737029783}{27777890035288} a^{8} - \frac{3069323213281}{13888945017644} a^{6} + \frac{330287531253}{3472236254411} a^{4} - \frac{66697872177}{27777890035288} a^{2} + \frac{167069501}{350713222}$, $\frac{1}{27777890035288} a^{39} + \frac{4107}{27777890035288} a^{31} + \frac{64697043}{13888945017644} a^{29} - \frac{532504455}{27777890035288} a^{27} + \frac{173956245}{13888945017644} a^{25} - \frac{407867309}{27777890035288} a^{23} + \frac{7566516399895}{27777890035288} a^{21} - \frac{1221836715758}{3472236254411} a^{19} + \frac{13065705330107}{27777890035288} a^{17} - \frac{4724144177835}{27777890035288} a^{15} + \frac{10533068343497}{27777890035288} a^{13} + \frac{11277494673877}{27777890035288} a^{11} - \frac{10596994085759}{27777890035288} a^{9} - \frac{796636245992}{3472236254411} a^{7} - \frac{2079984612753}{13888945017644} a^{5} + \frac{13817316744497}{27777890035288} a^{3} - \frac{2509532420475}{13888945017644} a$

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:  \( -1 \) (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

$C_2^2\times C_{10}$ (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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-5}, \sqrt{33})\), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\sqrt{3}, \sqrt{-55})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.4, 10.0.1833540124521600000.1, 10.0.685948419200000.1, \(\Q(\zeta_{33})^+\), 10.0.162778775259375.1, \(\Q(\zeta_{44})^+\), 10.10.53339349076992.1, 10.0.7368586534375.1, 20.0.3361869388230684433628866560000000000.7, 20.0.3361869388230684433628866560000000000.5, 20.0.3361869388230684433628866560000000000.4, 20.0.27784044530832102757263360000000000.1, 20.0.56933553290160450365440000000000.1, 20.0.3206128490667995866421572265625.1, \(\Q(\zeta_{132})^+\)

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ 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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
5Data not computed
11Data not computed