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Properties

Label 40.0.11302165783...0000.3
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![1, 8, 71, 622, 5456, 47852, 419693, 3680974, 32284483, 283155444, 2483453292, -1841907254, 2050319641, -1740855678, 1625753279, -1438695542, 1302100332, -1164717018, 1048603985, -929692368, 930022655, -156288, 118286465, 14844762, 17053272, 3922868, 2779859, 747642, 541141, 80036, 153132, -44616, 13783, -3916, 1253, -338, 116, -28, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 2*x^39 + 11*x^38 - 28*x^37 + 116*x^36 - 338*x^35 + 1253*x^34 - 3916*x^33 + 13783*x^32 - 44616*x^31 + 153132*x^30 + 80036*x^29 + 541141*x^28 + 747642*x^27 + 2779859*x^26 + 3922868*x^25 + 17053272*x^24 + 14844762*x^23 + 118286465*x^22 - 156288*x^21 + 930022655*x^20 - 929692368*x^19 + 1048603985*x^18 - 1164717018*x^17 + 1302100332*x^16 - 1438695542*x^15 + 1625753279*x^14 - 1740855678*x^13 + 2050319641*x^12 - 1841907254*x^11 + 2483453292*x^10 + 283155444*x^9 + 32284483*x^8 + 3680974*x^7 + 419693*x^6 + 47852*x^5 + 5456*x^4 + 622*x^3 + 71*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^40 - 2*x^39 + 11*x^38 - 28*x^37 + 116*x^36 - 338*x^35 + 1253*x^34 - 3916*x^33 + 13783*x^32 - 44616*x^31 + 153132*x^30 + 80036*x^29 + 541141*x^28 + 747642*x^27 + 2779859*x^26 + 3922868*x^25 + 17053272*x^24 + 14844762*x^23 + 118286465*x^22 - 156288*x^21 + 930022655*x^20 - 929692368*x^19 + 1048603985*x^18 - 1164717018*x^17 + 1302100332*x^16 - 1438695542*x^15 + 1625753279*x^14 - 1740855678*x^13 + 2050319641*x^12 - 1841907254*x^11 + 2483453292*x^10 + 283155444*x^9 + 32284483*x^8 + 3680974*x^7 + 419693*x^6 + 47852*x^5 + 5456*x^4 + 622*x^3 + 71*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 2 x^{39} + 11 x^{38} - 28 x^{37} + 116 x^{36} - 338 x^{35} + 1253 x^{34} - 3916 x^{33} + 13783 x^{32} - 44616 x^{31} + 153132 x^{30} + 80036 x^{29} + 541141 x^{28} + 747642 x^{27} + 2779859 x^{26} + 3922868 x^{25} + 17053272 x^{24} + 14844762 x^{23} + 118286465 x^{22} - 156288 x^{21} + 930022655 x^{20} - 929692368 x^{19} + 1048603985 x^{18} - 1164717018 x^{17} + 1302100332 x^{16} - 1438695542 x^{15} + 1625753279 x^{14} - 1740855678 x^{13} + 2050319641 x^{12} - 1841907254 x^{11} + 2483453292 x^{10} + 283155444 x^{9} + 32284483 x^{8} + 3680974 x^{7} + 419693 x^{6} + 47852 x^{5} + 5456 x^{4} + 622 x^{3} + 71 x^{2} + 8 x + 1 \)

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:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
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$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(529,·)$, $\chi_{660}(659,·)$, $\chi_{660}(409,·)$, $\chi_{660}(541,·)$, $\chi_{660}(289,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(169,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(431,·)$, $\chi_{660}(49,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(59,·)$, $\chi_{660}(61,·)$, $\chi_{660}(191,·)$, $\chi_{660}(71,·)$, $\chi_{660}(589,·)$, $\chi_{660}(469,·)$, $\chi_{660}(599,·)$, $\chi_{660}(601,·)$, $\chi_{660}(349,·)$, $\chi_{660}(479,·)$, $\chi_{660}(481,·)$, $\chi_{660}(611,·)$, $\chi_{660}(229,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(491,·)$, $\chi_{660}(109,·)$, $\chi_{660}(239,·)$, $\chi_{660}(241,·)$, $\chi_{660}(371,·)$, $\chi_{660}(119,·)$, $\chi_{660}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{7} a^{30} + \frac{2}{7} a^{29} + \frac{3}{7} a^{28} + \frac{2}{7} a^{27} - \frac{3}{7} a^{26} - \frac{1}{7} a^{24} - \frac{3}{7} a^{23} + \frac{2}{7} a^{22} + \frac{2}{7} a^{21} + \frac{1}{7} a^{20} + \frac{3}{7} a^{19} + \frac{1}{7} a^{18} - \frac{3}{7} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{1726461333697540211407027} a^{31} + \frac{56879293781690516362613}{1726461333697540211407027} a^{30} + \frac{41639029816673223422141}{1726461333697540211407027} a^{29} + \frac{314876996838487167694017}{1726461333697540211407027} a^{28} + \frac{116753565293262359467856}{1726461333697540211407027} a^{27} + \frac{74343351191005787789985}{246637333385362887343861} a^{26} + \frac{527383651572439393938170}{1726461333697540211407027} a^{25} - \frac{245714236673063900932298}{1726461333697540211407027} a^{24} - \frac{402718532731435740832045}{1726461333697540211407027} a^{23} - \frac{668819504401181610700585}{1726461333697540211407027} a^{22} - \frac{521565597879557143005645}{1726461333697540211407027} a^{21} + \frac{239994066999188864873962}{1726461333697540211407027} a^{20} - \frac{270451707092918815428600}{1726461333697540211407027} a^{19} - \frac{533660214200302911532065}{1726461333697540211407027} a^{18} - \frac{110784721074298034345756}{1726461333697540211407027} a^{17} - \frac{738275779903443066114624}{1726461333697540211407027} a^{16} + \frac{155151173172375837795090}{1726461333697540211407027} a^{15} - \frac{651466358970313867405450}{1726461333697540211407027} a^{14} + \frac{46492079389092527619348}{1726461333697540211407027} a^{13} + \frac{779159161107148064212552}{1726461333697540211407027} a^{12} + \frac{306089523364625633499750}{1726461333697540211407027} a^{11} + \frac{685954074011218636627517}{1726461333697540211407027} a^{10} + \frac{611665220329121637917446}{1726461333697540211407027} a^{9} - \frac{854363587102895698800129}{1726461333697540211407027} a^{8} - \frac{764495346236412029201569}{1726461333697540211407027} a^{7} - \frac{244186728625691366485714}{1726461333697540211407027} a^{6} - \frac{224438547673007219908844}{1726461333697540211407027} a^{5} + \frac{383816146967935185233375}{1726461333697540211407027} a^{4} - \frac{74777943020959389698747}{1726461333697540211407027} a^{3} - \frac{441514071638441528160946}{1726461333697540211407027} a^{2} + \frac{201627598151575607392453}{1726461333697540211407027} a + \frac{100498658883538345640886}{246637333385362887343861}$, $\frac{1}{1726461333697540211407027} a^{32} - \frac{51709123218587637979413}{1726461333697540211407027} a^{30} + \frac{86423379742215211224266}{246637333385362887343861} a^{29} + \frac{154570154776413788411411}{1726461333697540211407027} a^{28} + \frac{59009644671936246516854}{1726461333697540211407027} a^{27} + \frac{96572311683006122594771}{246637333385362887343861} a^{26} - \frac{307347270842728161987390}{1726461333697540211407027} a^{25} + \frac{484860815659211880620899}{1726461333697540211407027} a^{24} + \frac{501425948661345512229767}{1726461333697540211407027} a^{23} - \frac{743610536231076014175758}{1726461333697540211407027} a^{22} + \frac{499917933156899797742053}{1726461333697540211407027} a^{21} + \frac{671936005968515013042321}{1726461333697540211407027} a^{20} + \frac{105183616086990574049638}{1726461333697540211407027} a^{19} - \frac{98094886758068791441989}{1726461333697540211407027} a^{18} + \frac{628661199623671273606878}{1726461333697540211407027} a^{17} - \frac{6856050018410613792267}{246637333385362887343861} a^{16} + \frac{153747720076826179940280}{1726461333697540211407027} a^{15} - \frac{695441408400954089625711}{1726461333697540211407027} a^{14} - \frac{271944477012514418547475}{1726461333697540211407027} a^{13} + \frac{406695873356914368597136}{1726461333697540211407027} a^{12} + \frac{198024595401812493062570}{1726461333697540211407027} a^{11} - \frac{195742988580677539885034}{1726461333697540211407027} a^{10} - \frac{482654065705792274652947}{1726461333697540211407027} a^{9} - \frac{807856178322845711738505}{1726461333697540211407027} a^{8} + \frac{86920962535851251234023}{246637333385362887343861} a^{7} - \frac{178608258642256888342237}{1726461333697540211407027} a^{6} + \frac{362609654361791879303535}{1726461333697540211407027} a^{5} + \frac{247030293716053381027010}{1726461333697540211407027} a^{4} + \frac{6894186211422527694668}{1726461333697540211407027} a^{3} - \frac{384474823966500099650830}{1726461333697540211407027} a^{2} + \frac{704380313082052850669854}{1726461333697540211407027} a - \frac{839038530559199135551130}{1726461333697540211407027}$, $\frac{1}{1726461333697540211407027} a^{33} - \frac{7315986794770423733897}{246637333385362887343861} a^{22} + \frac{560010830121248336221270}{1726461333697540211407027} a^{11} - \frac{73123807354264478075876}{1726461333697540211407027}$, $\frac{1}{1726461333697540211407027} a^{34} - \frac{7315986794770423733897}{246637333385362887343861} a^{23} + \frac{560010830121248336221270}{1726461333697540211407027} a^{12} - \frac{73123807354264478075876}{1726461333697540211407027} a$, $\frac{1}{1726461333697540211407027} a^{35} - \frac{7315986794770423733897}{246637333385362887343861} a^{24} + \frac{560010830121248336221270}{1726461333697540211407027} a^{13} - \frac{73123807354264478075876}{1726461333697540211407027} a^{2}$, $\frac{1}{1726461333697540211407027} a^{36} - \frac{7315986794770423733897}{246637333385362887343861} a^{25} + \frac{560010830121248336221270}{1726461333697540211407027} a^{14} - \frac{73123807354264478075876}{1726461333697540211407027} a^{3}$, $\frac{1}{1726461333697540211407027} a^{37} - \frac{7315986794770423733897}{246637333385362887343861} a^{26} + \frac{560010830121248336221270}{1726461333697540211407027} a^{15} - \frac{73123807354264478075876}{1726461333697540211407027} a^{4}$, $\frac{1}{1726461333697540211407027} a^{38} - \frac{7315986794770423733897}{246637333385362887343861} a^{27} + \frac{560010830121248336221270}{1726461333697540211407027} a^{16} - \frac{73123807354264478075876}{1726461333697540211407027} a^{5}$, $\frac{1}{1726461333697540211407027} a^{39} - \frac{7315986794770423733897}{246637333385362887343861} a^{28} + \frac{560010830121248336221270}{1726461333697540211407027} a^{17} - \frac{73123807354264478075876}{1726461333697540211407027} a^{6}$

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{83361312013381621096}{246637333385362887343861} a^{39} - \frac{458487216073598916028}{246637333385362887343861} a^{38} + \frac{1167058368187342695344}{246637333385362887343861} a^{37} - \frac{4834956096776134023568}{246637333385362887343861} a^{36} + \frac{14088061730261493965224}{246637333385362887343861} a^{35} - \frac{52225861976383585616644}{246637333385362887343861} a^{34} + \frac{163221448922201214105968}{246637333385362887343861} a^{33} - \frac{574484481740219441783084}{246637333385362887343861} a^{32} + \frac{1859624148394517203409568}{246637333385362887343861} a^{31} - \frac{6382642215616577200836336}{246637333385362887343861} a^{30} + \frac{21023556167134659594917164}{246637333385362887343861} a^{29} - \frac{22555111872116671910755268}{246637333385362887343861} a^{28} - \frac{31162209018154330979727816}{246637333385362887343861} a^{27} - \frac{115866346726103509919152732}{246637333385362887343861} a^{26} - \frac{163507711667655166592811664}{246637333385362887343861} a^{25} - \frac{710791564020532212175513056}{246637333385362887343861} a^{24} - \frac{618739418423195490172149576}{246637333385362887343861} a^{23} - \frac{4930257457912472327707632820}{246637333385362887343861} a^{22} + \frac{6514186365973693398925824}{246637333385362887343861} a^{21} - \frac{38763954361484285389952964940}{246637333385362887343861} a^{20} + \frac{38750187782653803502209497664}{246637333385362887343861} a^{19} - \frac{343865685156301933197305891609}{246637333385362887343861} a^{18} + \frac{48546169372396708909469505864}{246637333385362887343861} a^{17} - \frac{54272396024289898635899901936}{246637333385362887343861} a^{16} + \frac{59965773984461591307774177016}{246637333385362887343861} a^{15} - \frac{67762463173748631187578786892}{246637333385362887343861} a^{14} + \frac{72560006672012503532908091544}{246637333385362887343861} a^{13} - \frac{85458667660282796280774373268}{246637333385362887343861} a^{12} + \frac{76771902650202476483500915192}{246637333385362887343861} a^{11} - \frac{103511962372535867481578924016}{246637333385362887343861} a^{10} - \frac{11802104657785803431438823312}{246637333385362887343861} a^{9} - \frac{1345638430276857359393126684}{246637333385362887343861} a^{8} - \frac{984790941589312332626021935438}{246637333385362887343861} a^{7} - \frac{17493079561416086351321764}{246637333385362887343861} a^{6} - \frac{1994502751232168666342896}{246637333385362887343861} a^{5} - \frac{227409659172505062349888}{246637333385362887343861} a^{4} - \frac{25925368036161684160856}{246637333385362887343861} a^{3} - \frac{2959326576475047548908}{246637333385362887343861} a^{2} - \frac{333445248053526484384}{246637333385362887343861} a - \frac{41680656006690810548}{246637333385362887343861} \) (order $22$)
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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), \(\Q(\sqrt{-11}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{-55})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{15}, \sqrt{-33})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.6, 10.0.1833540124521600000.1, 10.10.669871503125.1, 10.0.586732839846912.1, 10.10.166685465865600000.1, \(\Q(\zeta_{11})\), 10.10.53339349076992.1, 10.0.7368586534375.1, 20.0.3361869388230684433628866560000000000.6, 20.0.3361869388230684433628866560000000000.10, 20.0.3361869388230684433628866560000000000.4, 20.20.27784044530832102757263360000000000.1, 20.0.54296067514572573056640625.1, 20.0.3361869388230684433628866560000000000.8, 20.0.344255425354822086003595935744.3

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$