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Properties

Label 40.0.75504093671...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{20}\cdot 41^{39}$
Root discriminant $83.55$
Ramified primes $5, 41$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![370248451, -370248410, 370248410, -370245540, 370245540, -370185557, 370185557, -369594296, 369594296, -366243817, 366243817, -354060257, 354060257, -323601357, 323601357, -268775337, 268775337, -195405222, 195405222, -120747912, 120747912, -62088597, 62088597, -26151072, 26151072, -8901060, 8901060, -2413876, 2413876, -512460, 512460, -83108, 83108, -9923, 9923, -821, 821, -42, 42, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 42*x^38 - 42*x^37 + 821*x^36 - 821*x^35 + 9923*x^34 - 9923*x^33 + 83108*x^32 - 83108*x^31 + 512460*x^30 - 512460*x^29 + 2413876*x^28 - 2413876*x^27 + 8901060*x^26 - 8901060*x^25 + 26151072*x^24 - 26151072*x^23 + 62088597*x^22 - 62088597*x^21 + 120747912*x^20 - 120747912*x^19 + 195405222*x^18 - 195405222*x^17 + 268775337*x^16 - 268775337*x^15 + 323601357*x^14 - 323601357*x^13 + 354060257*x^12 - 354060257*x^11 + 366243817*x^10 - 366243817*x^9 + 369594296*x^8 - 369594296*x^7 + 370185557*x^6 - 370185557*x^5 + 370245540*x^4 - 370245540*x^3 + 370248410*x^2 - 370248410*x + 370248451)
 
gp: K = bnfinit(x^40 - x^39 + 42*x^38 - 42*x^37 + 821*x^36 - 821*x^35 + 9923*x^34 - 9923*x^33 + 83108*x^32 - 83108*x^31 + 512460*x^30 - 512460*x^29 + 2413876*x^28 - 2413876*x^27 + 8901060*x^26 - 8901060*x^25 + 26151072*x^24 - 26151072*x^23 + 62088597*x^22 - 62088597*x^21 + 120747912*x^20 - 120747912*x^19 + 195405222*x^18 - 195405222*x^17 + 268775337*x^16 - 268775337*x^15 + 323601357*x^14 - 323601357*x^13 + 354060257*x^12 - 354060257*x^11 + 366243817*x^10 - 366243817*x^9 + 369594296*x^8 - 369594296*x^7 + 370185557*x^6 - 370185557*x^5 + 370245540*x^4 - 370245540*x^3 + 370248410*x^2 - 370248410*x + 370248451, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 42 x^{38} - 42 x^{37} + 821 x^{36} - 821 x^{35} + 9923 x^{34} - 9923 x^{33} + 83108 x^{32} - 83108 x^{31} + 512460 x^{30} - 512460 x^{29} + 2413876 x^{28} - 2413876 x^{27} + 8901060 x^{26} - 8901060 x^{25} + 26151072 x^{24} - 26151072 x^{23} + 62088597 x^{22} - 62088597 x^{21} + 120747912 x^{20} - 120747912 x^{19} + 195405222 x^{18} - 195405222 x^{17} + 268775337 x^{16} - 268775337 x^{15} + 323601357 x^{14} - 323601357 x^{13} + 354060257 x^{12} - 354060257 x^{11} + 366243817 x^{10} - 366243817 x^{9} + 369594296 x^{8} - 369594296 x^{7} + 370185557 x^{6} - 370185557 x^{5} + 370245540 x^{4} - 370245540 x^{3} + 370248410 x^{2} - 370248410 x + 370248451 \)

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:  \(75504093671268370020376748996904037084465397328249745498031917667388916015625=5^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.55$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(131,·)$, $\chi_{205}(134,·)$, $\chi_{205}(129,·)$, $\chi_{205}(141,·)$, $\chi_{205}(14,·)$, $\chi_{205}(16,·)$, $\chi_{205}(146,·)$, $\chi_{205}(19,·)$, $\chi_{205}(21,·)$, $\chi_{205}(24,·)$, $\chi_{205}(156,·)$, $\chi_{205}(29,·)$, $\chi_{205}(31,·)$, $\chi_{205}(34,·)$, $\chi_{205}(36,·)$, $\chi_{205}(166,·)$, $\chi_{205}(44,·)$, $\chi_{205}(46,·)$, $\chi_{205}(179,·)$, $\chi_{205}(54,·)$, $\chi_{205}(61,·)$, $\chi_{205}(51,·)$, $\chi_{205}(196,·)$, $\chi_{205}(69,·)$, $\chi_{205}(199,·)$, $\chi_{205}(194,·)$, $\chi_{205}(201,·)$, $\chi_{205}(79,·)$, $\chi_{205}(81,·)$, $\chi_{205}(66,·)$, $\chi_{205}(86,·)$, $\chi_{205}(89,·)$, $\chi_{205}(91,·)$, $\chi_{205}(94,·)$, $\chi_{205}(99,·)$, $\chi_{205}(104,·)$, $\chi_{205}(109,·)$, $\chi_{205}(121,·)$, $\chi_{205}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{165580141} a^{21} - \frac{63245986}{165580141} a^{20} + \frac{21}{165580141} a^{19} + \frac{59721408}{165580141} a^{18} + \frac{189}{165580141} a^{17} + \frac{10891545}{165580141} a^{16} + \frac{952}{165580141} a^{15} + \frac{70734346}{165580141} a^{14} + \frac{2940}{165580141} a^{13} + \frac{4524379}{165580141} a^{12} + \frac{5733}{165580141} a^{11} - \frac{64892355}{165580141} a^{10} + \frac{7007}{165580141} a^{9} + \frac{60571159}{165580141} a^{8} + \frac{5148}{165580141} a^{7} - \frac{64620912}{165580141} a^{6} + \frac{2079}{165580141} a^{5} - \frac{20194035}{165580141} a^{4} + \frac{385}{165580141} a^{3} - \frac{32553242}{165580141} a^{2} + \frac{21}{165580141} a + \frac{39088169}{165580141}$, $\frac{1}{165580141} a^{22} + \frac{22}{165580141} a^{20} + \frac{63245986}{165580141} a^{19} + \frac{209}{165580141} a^{18} + \frac{42612747}{165580141} a^{17} + \frac{1122}{165580141} a^{16} + \frac{9741694}{165580141} a^{15} + \frac{3740}{165580141} a^{14} + \frac{1224876}{165580141} a^{13} + \frac{8008}{165580141} a^{12} + \frac{69416734}{165580141} a^{11} + \frac{11011}{165580141} a^{10} - \frac{32842396}{165580141} a^{9} + \frac{9438}{165580141} a^{8} - \frac{4842190}{165580141} a^{7} + \frac{4719}{165580141} a^{6} - \frac{2421095}{165580141} a^{5} + \frac{1210}{165580141} a^{4} - \frac{23129359}{165580141} a^{3} + \frac{121}{165580141} a^{2} + \frac{42612747}{165580141} a + \frac{2}{165580141}$, $\frac{1}{165580141} a^{23} - \frac{35563591}{165580141} a^{20} - \frac{253}{165580141} a^{19} + \frac{53382899}{165580141} a^{18} - \frac{3036}{165580141} a^{17} - \frac{64292155}{165580141} a^{16} - \frac{17204}{165580141} a^{15} - \frac{64709467}{165580141} a^{14} - \frac{56672}{165580141} a^{13} - \frac{30119604}{165580141} a^{12} - \frac{115115}{165580141} a^{11} + \frac{70148286}{165580141} a^{10} - \frac{144716}{165580141} a^{9} - \frac{12766560}{165580141} a^{8} - \frac{108537}{165580141} a^{7} - \frac{70982300}{165580141} a^{6} - \frac{44528}{165580141} a^{5} - \frac{75601012}{165580141} a^{4} - \frac{8349}{165580141} a^{3} - \frac{69116634}{165580141} a^{2} - \frac{460}{165580141} a - \frac{32039013}{165580141}$, $\frac{1}{165580141} a^{24} - \frac{276}{165580141} a^{20} - \frac{27682395}{165580141} a^{19} - \frac{3496}{165580141} a^{18} + \frac{34020904}{165580141} a^{17} - \frac{21114}{165580141} a^{16} + \frac{13480401}{165580141} a^{15} - \frac{75072}{165580141} a^{14} + \frac{45768965}{165580141} a^{13} - \frac{167440}{165580141} a^{12} - \frac{38518223}{165580141} a^{11} - \frac{236808}{165580141} a^{10} - \frac{16796628}{165580141} a^{9} - \frac{207207}{165580141} a^{8} + \frac{44328363}{165580141} a^{7} - \frac{105248}{165580141} a^{6} + \frac{12361791}{165580141} a^{5} - \frac{27324}{165580141} a^{4} + \frac{45294339}{165580141} a^{3} - \frac{2760}{165580141} a^{2} + \frac{52475834}{165580141} a - \frac{46}{165580141}$, $\frac{1}{165580141} a^{25} + \frac{67920415}{165580141} a^{20} + \frac{2300}{165580141} a^{19} - \frac{40884588}{165580141} a^{18} + \frac{31050}{165580141} a^{17} + \frac{39104283}{165580141} a^{16} + \frac{187680}{165580141} a^{15} + \frac{29991823}{165580141} a^{14} + \frac{644000}{165580141} a^{13} + \frac{51149394}{165580141} a^{12} + \frac{1345500}{165580141} a^{11} - \frac{44431380}{165580141} a^{10} + \frac{1726725}{165580141} a^{9} + \frac{38374006}{165580141} a^{8} + \frac{1315600}{165580141} a^{7} + \frac{59645307}{165580141} a^{6} + \frac{546480}{165580141} a^{5} - \frac{64114668}{165580141} a^{4} + \frac{103500}{165580141} a^{3} + \frac{9108656}{165580141} a^{2} + \frac{5750}{165580141} a + \frac{25625479}{165580141}$, $\frac{1}{165580141} a^{26} + \frac{2600}{165580141} a^{20} + \frac{23007966}{165580141} a^{19} + \frac{37050}{165580141} a^{18} - \frac{48183295}{165580141} a^{17} + \frac{238680}{165580141} a^{16} - \frac{53988267}{165580141} a^{15} + \frac{884000}{165580141} a^{14} + \frac{54779340}{165580141} a^{13} + \frac{2028000}{165580141} a^{12} + \frac{12321057}{165580141} a^{11} + \frac{2927925}{165580141} a^{10} - \frac{2648665}{165580141} a^{9} + \frac{2602600}{165580141} a^{8} - \frac{54973462}{165580141} a^{7} + \frac{1338480}{165580141} a^{6} - \frac{30797180}{165580141} a^{5} + \frac{351000}{165580141} a^{4} + \frac{21411159}{165580141} a^{3} + \frac{35750}{165580141} a^{2} - \frac{76062108}{165580141} a + \frac{600}{165580141}$, $\frac{1}{165580141} a^{27} + \frac{41491553}{165580141} a^{20} - \frac{17550}{165580141} a^{19} - \frac{9671837}{165580141} a^{18} - \frac{252720}{165580141} a^{17} - \frac{57801156}{165580141} a^{16} - \frac{1591200}{165580141} a^{15} - \frac{60563750}{165580141} a^{14} - \frac{5616000}{165580141} a^{13} + \frac{5125668}{165580141} a^{12} - \frac{11977875}{165580141} a^{11} - \frac{8689344}{165580141} a^{10} - \frac{15615600}{165580141} a^{9} - \frac{73272771}{165580141} a^{8} - \frac{12046320}{165580141} a^{7} - \frac{80269095}{165580141} a^{6} - \frac{5054400}{165580141} a^{5} + \frac{36997462}{165580141} a^{4} - \frac{965250}{165580141} a^{3} - \frac{49084959}{165580141} a^{2} - \frac{54000}{165580141} a + \frac{36967174}{165580141}$, $\frac{1}{165580141} a^{28} - \frac{20475}{165580141} a^{20} - \frac{53093745}{165580141} a^{19} - \frac{311220}{165580141} a^{18} + \frac{48142095}{165580141} a^{17} - \frac{2088450}{165580141} a^{16} + \frac{13131493}{165580141} a^{15} - \frac{7956000}{165580141} a^{14} + \frac{52523765}{165580141} a^{13} - \frac{18632250}{165580141} a^{12} + \frac{58899924}{165580141} a^{11} - \frac{27327300}{165580141} a^{10} - \frac{45857046}{165580141} a^{9} - \frac{24594570}{165580141} a^{8} - \frac{80402049}{165580141} a^{7} - \frac{12776400}{165580141} a^{6} + \frac{43312236}{165580141} a^{5} - \frac{3378375}{165580141} a^{4} + \frac{37940813}{165580141} a^{3} - \frac{346500}{165580141} a^{2} - \frac{6454734}{165580141} a - \frac{5850}{165580141}$, $\frac{1}{165580141} a^{29} - \frac{12374334}{165580141} a^{20} + \frac{118755}{165580141} a^{19} + \frac{34629610}{165580141} a^{18} + \frac{1781325}{165580141} a^{17} - \frac{18934559}{165580141} a^{16} + \frac{11536200}{165580141} a^{15} + \frac{8764788}{165580141} a^{14} + \frac{41564250}{165580141} a^{13} - \frac{29319011}{165580141} a^{12} - \frac{75524266}{165580141} a^{11} + \frac{63805854}{165580141} a^{10} - \frac{46706386}{165580141} a^{9} - \frac{81177614}{165580141} a^{8} - \frac{72951241}{165580141} a^{7} + \frac{81045767}{165580141} a^{6} + \frac{39189150}{165580141} a^{5} + \frac{18686265}{165580141} a^{4} + \frac{7536375}{165580141} a^{3} - \frac{74017159}{165580141} a^{2} + \frac{424125}{165580141} a + \frac{81438822}{165580141}$, $\frac{1}{165580141} a^{30} + \frac{142506}{165580141} a^{20} - \frac{36669658}{165580141} a^{19} + \frac{2256345}{165580141} a^{18} + \frac{1692593}{165580141} a^{17} + \frac{15573870}{165580141} a^{16} + \frac{32940745}{165580141} a^{15} + \frac{60565050}{165580141} a^{14} - \frac{76408071}{165580141} a^{13} - \frac{21490741}{165580141} a^{12} - \frac{28017813}{165580141} a^{11} + \frac{48392618}{165580141} a^{10} + \frac{27366981}{165580141} a^{9} + \frac{28940549}{165580141} a^{8} + \frac{35762914}{165580141} a^{7} - \frac{63688351}{165580141} a^{6} + \frac{80004796}{165580141} a^{5} + \frac{27130950}{165580141} a^{4} + \frac{53857483}{165580141} a^{3} + \frac{2799225}{165580141} a^{2} + \frac{10139554}{165580141} a + \frac{47502}{165580141}$, $\frac{1}{165580141} a^{31} + \frac{37576346}{165580141} a^{20} - \frac{736281}{165580141} a^{19} - \frac{3608596}{165580141} a^{18} - \frac{11359764}{165580141} a^{17} + \frac{70670709}{165580141} a^{16} - \frac{75100662}{165580141} a^{15} + \frac{42704651}{165580141} a^{14} + \frac{56282042}{165580141} a^{13} - \frac{10102533}{165580141} a^{12} + \frac{59306425}{165580141} a^{11} - \frac{73566239}{165580141} a^{10} + \frac{23881853}{165580141} a^{9} - \frac{25071210}{165580141} a^{8} + \frac{30591466}{165580141} a^{7} + \frac{42568412}{165580141} a^{6} + \frac{62021258}{165580141} a^{5} + \frac{42158613}{165580141} a^{4} - \frac{52065585}{165580141} a^{3} - \frac{16366391}{165580141} a^{2} - \frac{2945124}{165580141} a - \frac{17088133}{165580141}$, $\frac{1}{165580141} a^{32} - \frac{906192}{165580141} a^{20} + \frac{35188843}{165580141} a^{19} - \frac{14757984}{165580141} a^{18} - \frac{76892763}{165580141} a^{17} + \frac{61594609}{165580141} a^{16} + \frac{35333715}{165580141} a^{15} - \frac{79646758}{165580141} a^{14} - \frac{42605726}{165580141} a^{13} + \frac{3919182}{165580141} a^{12} - \frac{78994416}{165580141} a^{11} + \frac{5878773}{165580141} a^{10} - \frac{50103442}{165580141} a^{9} - \frac{36006160}{165580141} a^{8} - \frac{2856108}{165580141} a^{7} - \frac{55383500}{165580141} a^{6} + \frac{74761831}{165580141} a^{5} - \frac{26662019}{165580141} a^{4} - \frac{77787334}{165580141} a^{3} - \frac{19936224}{165580141} a^{2} + \frac{21709306}{165580141} a - \frac{339822}{165580141}$, $\frac{1}{165580141} a^{33} - \frac{54831575}{165580141} a^{20} + \frac{4272048}{165580141} a^{19} - \frac{55919572}{165580141} a^{18} + \frac{67284756}{165580141} a^{17} - \frac{34764373}{165580141} a^{16} - \frac{44852679}{165580141} a^{15} - \frac{31578791}{165580141} a^{14} + \frac{18841406}{165580141} a^{13} - \frac{52810949}{165580141} a^{12} + \frac{68093138}{165580141} a^{11} - \frac{23890157}{165580141} a^{10} + \frac{21635826}{165580141} a^{9} - \frac{57560516}{165580141} a^{8} - \frac{26551032}{165580141} a^{7} + \frac{28360646}{165580141} a^{6} + \frac{35929598}{165580141} a^{5} + \frac{851125}{165580141} a^{4} - \frac{2212586}{165580141} a^{3} - \frac{39004880}{165580141} a^{2} + \frac{18690210}{165580141} a - \frac{14460695}{165580141}$, $\frac{1}{165580141} a^{34} + \frac{5379616}{165580141} a^{20} - \frac{63517484}{165580141} a^{19} - \frac{76144025}{165580141} a^{18} + \frac{62434560}{165580141} a^{17} - \frac{22146260}{165580141} a^{16} + \frac{10336194}{165580141} a^{15} + \frac{76995101}{165580141} a^{14} + \frac{42542358}{165580141} a^{13} - \frac{61471918}{165580141} a^{12} + \frac{54421700}{165580141} a^{11} - \frac{14325709}{165580141} a^{10} + \frac{1358389}{165580141} a^{9} - \frac{76908401}{165580141} a^{8} - \frac{12831659}{165580141} a^{7} - \frac{20533420}{165580141} a^{6} + \frac{76558542}{165580141} a^{5} - \frac{81949832}{165580141} a^{4} + \frac{42473588}{165580141} a^{3} - \frac{36133131}{165580141} a^{2} - \frac{22058607}{165580141} a + \frac{2215136}{165580141}$, $\frac{1}{165580141} a^{35} + \frac{13572862}{165580141} a^{20} - \frac{23535820}{165580141} a^{19} - \frac{51559930}{165580141} a^{18} - \frac{45412838}{165580141} a^{17} + \frac{34863875}{165580141} a^{16} - \frac{76995101}{165580141} a^{15} - \frac{46573435}{165580141} a^{14} + \frac{18150578}{165580141} a^{13} - \frac{79990610}{165580141} a^{12} - \frac{57758011}{165580141} a^{11} + \frac{29720949}{165580141} a^{10} - \frac{19605565}{165580141} a^{9} + \frac{41211104}{165580141} a^{8} - \frac{62913041}{165580141} a^{7} - \frac{68501448}{165580141} a^{6} - \frac{6721908}{165580141} a^{5} + \frac{59234894}{165580141} a^{4} + \frac{45256542}{165580141} a^{3} + \frac{70289507}{165580141} a^{2} + \frac{54823341}{165580141} a - \frac{11399449}{165580141}$, $\frac{1}{165580141} a^{36} - \frac{30260340}{165580141} a^{20} - \frac{5429750}{165580141} a^{19} - \frac{14323097}{165580141} a^{18} - \frac{46704928}{165580141} a^{17} - \frac{61102514}{165580141} a^{16} - \frac{52687061}{165580141} a^{15} - \frac{62900910}{165580141} a^{14} - \frac{79390909}{165580141} a^{13} + \frac{42912102}{165580141} a^{12} + \frac{39169373}{165580141} a^{11} + \frac{44979338}{165580141} a^{10} - \frac{20831996}{165580141} a^{9} + \frac{65420834}{165580141} a^{8} - \frac{66775522}{165580141} a^{7} - \frac{42379621}{165580141} a^{6} - \frac{10121234}{165580141} a^{5} - \frac{38327805}{165580141} a^{4} - \frac{22277992}{165580141} a^{3} + \frac{44691905}{165580141} a^{2} + \frac{34730731}{165580141} a - \frac{13449040}{165580141}$, $\frac{1}{165580141} a^{37} - \frac{66507629}{165580141} a^{20} - \frac{41176521}{165580141} a^{19} + \frac{56378620}{165580141} a^{18} + \frac{28376952}{165580141} a^{17} - \frac{4377608}{165580141} a^{16} - \frac{66001764}{165580141} a^{15} + \frac{70027627}{165580141} a^{14} - \frac{73804156}{165580141} a^{13} + \frac{8732947}{165580141} a^{12} - \frac{479210}{165580141} a^{11} - \frac{14791255}{165580141} a^{10} - \frac{8537407}{165580141} a^{9} + \frac{22486771}{165580141} a^{8} - \frac{73061982}{165580141} a^{7} + \frac{19335540}{165580141} a^{6} - \frac{47534525}{165580141} a^{5} - \frac{75165303}{165580141} a^{4} - \frac{61267206}{165580141} a^{3} + \frac{53408048}{165580141} a^{2} - \frac{40302464}{165580141} a + \frac{43424383}{165580141}$, $\frac{1}{165580141} a^{38} - \frac{2568501}{165580141} a^{20} - \frac{37182440}{165580141} a^{19} - \frac{27363353}{165580141} a^{18} - \frac{18526443}{165580141} a^{17} + \frac{39736137}{165580141} a^{16} - \frac{31903568}{165580141} a^{15} + \frac{14705618}{165580141} a^{14} - \frac{8984314}{165580141} a^{13} + \frac{75291560}{165580141} a^{12} - \frac{57618921}{165580141} a^{11} - \frac{73877021}{165580141} a^{10} - \frac{66653741}{165580141} a^{9} - \frac{24797182}{165580141} a^{8} - \frac{19121956}{165580141} a^{7} + \frac{45851560}{165580141} a^{6} - \frac{65222347}{165580141} a^{5} - \frac{1037778}{165580141} a^{4} - \frac{6076642}{165580141} a^{3} + \frac{12156293}{165580141} a^{2} - \frac{50136677}{165580141} a + \frac{77216040}{165580141}$, $\frac{1}{165580141} a^{39} - \frac{50737146}{165580141} a^{20} + \frac{26575168}{165580141} a^{19} + \frac{41539719}{165580141} a^{18} + \frac{28442403}{165580141} a^{17} - \frac{18081614}{165580141} a^{16} - \frac{23783545}{165580141} a^{15} + \frac{75540192}{165580141} a^{14} + \frac{9998014}{165580141} a^{13} + \frac{68911296}{165580141} a^{12} + \frac{80286804}{165580141} a^{11} - \frac{29410317}{165580141} a^{10} - \frac{75546044}{165580141} a^{9} - \frac{50181205}{165580141} a^{8} + \frac{22083428}{165580141} a^{7} + \frac{81244410}{165580141} a^{6} + \frac{40311289}{165580141} a^{5} + \frac{70740496}{165580141} a^{4} + \frac{7548332}{165580141} a^{3} - \frac{80966149}{165580141} a^{2} - \frac{34425580}{165580141} a - \frac{27109412}{165580141}$

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

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.121721421175625.1, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ R $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$ $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
5Data not computed
41Data not computed