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Properties

Label 40.40.1358711128...0361.1
Degree $40$
Signature $[40, 0]$
Discriminant $23^{20}\cdot 41^{39}$
Root discriminant $179.20$
Ramified primes $23, 41$
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![15037528415907031, 134864967626674985, -134864967626674985, -1613997486203448535, 1613997486203448535, 4477873394638148393, -4477873394638148393, -5530200195315903703, 5530200195315903703, 3921869306307367721, -3921869306307367721, -1806657664373402839, 1806657664373402839, 580228573410251561, -580228573410251561, -135837297924844759, 135837297924844759, 23873472397443881, -23873472397443881, -3211979879435479, 3211979879435479, 334924585393961, -334924585393961, -27242866548439, 27242866548439, 1730529606953, -1730529606953, -85466733271, 85466733271, 3245731625, -3245731625, -92909527, 92909527, 1938233, -1938233, -27799, 27799, 245, -245, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 245*x^38 + 245*x^37 + 27799*x^36 - 27799*x^35 - 1938233*x^34 + 1938233*x^33 + 92909527*x^32 - 92909527*x^31 - 3245731625*x^30 + 3245731625*x^29 + 85466733271*x^28 - 85466733271*x^27 - 1730529606953*x^26 + 1730529606953*x^25 + 27242866548439*x^24 - 27242866548439*x^23 - 334924585393961*x^22 + 334924585393961*x^21 + 3211979879435479*x^20 - 3211979879435479*x^19 - 23873472397443881*x^18 + 23873472397443881*x^17 + 135837297924844759*x^16 - 135837297924844759*x^15 - 580228573410251561*x^14 + 580228573410251561*x^13 + 1806657664373402839*x^12 - 1806657664373402839*x^11 - 3921869306307367721*x^10 + 3921869306307367721*x^9 + 5530200195315903703*x^8 - 5530200195315903703*x^7 - 4477873394638148393*x^6 + 4477873394638148393*x^5 + 1613997486203448535*x^4 - 1613997486203448535*x^3 - 134864967626674985*x^2 + 134864967626674985*x + 15037528415907031)
 
gp: K = bnfinit(x^40 - x^39 - 245*x^38 + 245*x^37 + 27799*x^36 - 27799*x^35 - 1938233*x^34 + 1938233*x^33 + 92909527*x^32 - 92909527*x^31 - 3245731625*x^30 + 3245731625*x^29 + 85466733271*x^28 - 85466733271*x^27 - 1730529606953*x^26 + 1730529606953*x^25 + 27242866548439*x^24 - 27242866548439*x^23 - 334924585393961*x^22 + 334924585393961*x^21 + 3211979879435479*x^20 - 3211979879435479*x^19 - 23873472397443881*x^18 + 23873472397443881*x^17 + 135837297924844759*x^16 - 135837297924844759*x^15 - 580228573410251561*x^14 + 580228573410251561*x^13 + 1806657664373402839*x^12 - 1806657664373402839*x^11 - 3921869306307367721*x^10 + 3921869306307367721*x^9 + 5530200195315903703*x^8 - 5530200195315903703*x^7 - 4477873394638148393*x^6 + 4477873394638148393*x^5 + 1613997486203448535*x^4 - 1613997486203448535*x^3 - 134864967626674985*x^2 + 134864967626674985*x + 15037528415907031, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 245 x^{38} + 245 x^{37} + 27799 x^{36} - 27799 x^{35} - 1938233 x^{34} + 1938233 x^{33} + 92909527 x^{32} - 92909527 x^{31} - 3245731625 x^{30} + 3245731625 x^{29} + 85466733271 x^{28} - 85466733271 x^{27} - 1730529606953 x^{26} + 1730529606953 x^{25} + 27242866548439 x^{24} - 27242866548439 x^{23} - 334924585393961 x^{22} + 334924585393961 x^{21} + 3211979879435479 x^{20} - 3211979879435479 x^{19} - 23873472397443881 x^{18} + 23873472397443881 x^{17} + 135837297924844759 x^{16} - 135837297924844759 x^{15} - 580228573410251561 x^{14} + 580228573410251561 x^{13} + 1806657664373402839 x^{12} - 1806657664373402839 x^{11} - 3921869306307367721 x^{10} + 3921869306307367721 x^{9} + 5530200195315903703 x^{8} - 5530200195315903703 x^{7} - 4477873394638148393 x^{6} + 4477873394638148393 x^{5} + 1613997486203448535 x^{4} - 1613997486203448535 x^{3} - 134864967626674985 x^{2} + 134864967626674985 x + 15037528415907031 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1358711128258825934115575439892361036364148090821174339024587245535856279071374753274170361=23^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.20$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(943=23\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{943}(896,·)$, $\chi_{943}(1,·)$, $\chi_{943}(898,·)$, $\chi_{943}(643,·)$, $\chi_{943}(392,·)$, $\chi_{943}(137,·)$, $\chi_{943}(139,·)$, $\chi_{943}(783,·)$, $\chi_{943}(275,·)$, $\chi_{943}(277,·)$, $\chi_{943}(22,·)$, $\chi_{943}(919,·)$, $\chi_{943}(413,·)$, $\chi_{943}(415,·)$, $\chi_{943}(162,·)$, $\chi_{943}(553,·)$, $\chi_{943}(298,·)$, $\chi_{943}(436,·)$, $\chi_{943}(183,·)$, $\chi_{943}(185,·)$, $\chi_{943}(827,·)$, $\chi_{943}(829,·)$, $\chi_{943}(576,·)$, $\chi_{943}(321,·)$, $\chi_{943}(323,·)$, $\chi_{943}(68,·)$, $\chi_{943}(712,·)$, $\chi_{943}(461,·)$, $\chi_{943}(850,·)$, $\chi_{943}(852,·)$, $\chi_{943}(599,·)$, $\chi_{943}(346,·)$, $\chi_{943}(735,·)$, $\chi_{943}(737,·)$, $\chi_{943}(484,·)$, $\chi_{943}(229,·)$, $\chi_{943}(873,·)$, $\chi_{943}(505,·)$, $\chi_{943}(252,·)$, $\chi_{943}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2029056410309741} a^{21} - \frac{632136812045358}{2029056410309741} a^{20} - \frac{126}{2029056410309741} a^{19} + \frac{781330263982543}{2029056410309741} a^{18} + \frac{6804}{2029056410309741} a^{17} + \frac{733284743085127}{2029056410309741} a^{16} - \frac{205632}{2029056410309741} a^{15} - \frac{1010727587044335}{2029056410309741} a^{14} + \frac{3810240}{2029056410309741} a^{13} - \frac{381888110621909}{2029056410309741} a^{12} - \frac{44579808}{2029056410309741} a^{11} - \frac{918159192988409}{2029056410309741} a^{10} + \frac{326918592}{2029056410309741} a^{9} + \frac{974743529887544}{2029056410309741} a^{8} - \frac{1441110528}{2029056410309741} a^{7} - \frac{789590311463598}{2029056410309741} a^{6} + \frac{3491921664}{2029056410309741} a^{5} + \frac{973217731416811}{2029056410309741} a^{4} - \frac{3879912960}{2029056410309741} a^{3} - \frac{154415692764115}{2029056410309741} a^{2} + \frac{1269789696}{2029056410309741} a + \frac{424341165193642}{2029056410309741}$, $\frac{1}{2029056410309741} a^{22} - \frac{132}{2029056410309741} a^{20} + \frac{265291948347334}{2029056410309741} a^{19} + \frac{7524}{2029056410309741} a^{18} + \frac{192564043050039}{2029056410309741} a^{17} - \frac{242352}{2029056410309741} a^{16} + \frac{902207985146833}{2029056410309741} a^{15} + \frac{4847040}{2029056410309741} a^{14} - \frac{856074994119780}{2029056410309741} a^{13} - \frac{62270208}{2029056410309741} a^{12} - \frac{848624963899619}{2029056410309741} a^{11} + \frac{513729216}{2029056410309741} a^{10} + \frac{464825849903084}{2029056410309741} a^{9} - \frac{2642035968}{2029056410309741} a^{8} + \frac{547205154859444}{2029056410309741} a^{7} + \frac{7926107904}{2029056410309741} a^{6} + \frac{387440945731409}{2029056410309741} a^{5} - \frac{12194012160}{2029056410309741} a^{4} + \frac{947348826955163}{2029056410309741} a^{3} + \frac{7316407296}{2029056410309741} a^{2} + \frac{26871751279883}{2029056410309741} a - \frac{725594112}{2029056410309741}$, $\frac{1}{2029056410309741} a^{23} + \frac{14545581059459}{2029056410309741} a^{20} - \frac{9108}{2029056410309741} a^{19} - \frac{153718037051076}{2029056410309741} a^{18} + \frac{655776}{2029056410309741} a^{17} + \frac{301086377516029}{2029056410309741} a^{16} - \frac{22296384}{2029056410309741} a^{15} - \frac{354393403529094}{2029056410309741} a^{14} + \frac{440681472}{2029056410309741} a^{13} - \frac{531445308248082}{2029056410309741} a^{12} - \frac{5370805440}{2029056410309741} a^{11} + \frac{1011196994017556}{2029056410309741} a^{10} + \frac{40511218176}{2029056410309741} a^{9} - \frac{646259159808172}{2029056410309741} a^{8} - \frac{182300481792}{2029056410309741} a^{7} - \frac{356603241666736}{2029056410309741} a^{6} + \frac{448739647488}{2029056410309741} a^{5} - \frac{447520885849209}{2029056410309741} a^{4} - \frac{504832103424}{2029056410309741} a^{3} - \frac{65435590485887}{2029056410309741} a^{2} + \frac{166886645760}{2029056410309741} a - \frac{800545683112004}{2029056410309741}$, $\frac{1}{2029056410309741} a^{24} - \frac{9936}{2029056410309741} a^{20} - \frac{350031233868983}{2029056410309741} a^{19} + \frac{755136}{2029056410309741} a^{18} + \frac{756716954134302}{2029056410309741} a^{17} - \frac{27363744}{2029056410309741} a^{16} - \frac{146617781414240}{2029056410309741} a^{15} + \frac{583759872}{2029056410309741} a^{14} + \frac{989626309267173}{2029056410309741} a^{13} - \frac{7812080640}{2029056410309741} a^{12} + \frac{461638556721871}{2029056410309741} a^{11} + \frac{66291084288}{2029056410309741} a^{10} + \frac{2075363177283}{2029056410309741} a^{9} - \frac{348028192512}{2029056410309741} a^{8} - \frac{522962227273371}{2029056410309741} a^{7} + \frac{1060657348608}{2029056410309741} a^{6} - \frac{122460898045045}{2029056410309741} a^{5} - \frac{1652177793024}{2029056410309741} a^{4} - \frac{201162642456098}{2029056410309741} a^{3} + \frac{1001319874560}{2029056410309741} a^{2} - \frac{866801724847739}{2029056410309741} a - \frac{100131987456}{2029056410309741}$, $\frac{1}{2029056410309741} a^{25} + \frac{697250602412065}{2029056410309741} a^{20} - \frac{496800}{2029056410309741} a^{19} + \frac{884394039612484}{2029056410309741} a^{18} + \frac{40240800}{2029056410309741} a^{17} - \frac{570979909872299}{2029056410309741} a^{16} - \frac{1459399680}{2029056410309741} a^{15} + \frac{200496059662822}{2029056410309741} a^{14} + \frac{30046464000}{2029056410309741} a^{13} + \frac{356858696649717}{2029056410309741} a^{12} - \frac{376653888000}{2029056410309741} a^{11} - \frac{190045417059005}{2029056410309741} a^{10} + \frac{2900234937600}{2029056410309741} a^{9} - \frac{157495673029980}{2029056410309741} a^{8} - \frac{13258216857600}{2029056410309741} a^{7} + \frac{869343067413674}{2029056410309741} a^{6} + \frac{33043555860480}{2029056410309741} a^{5} - \frac{792634821247608}{2029056410309741} a^{4} - \frac{37549495296000}{2029056410309741} a^{3} + \frac{854577575379558}{2029056410309741} a^{2} + \frac{12516498432000}{2029056410309741} a - \frac{125403259614886}{2029056410309741}$, $\frac{1}{2029056410309741} a^{26} - \frac{561600}{2029056410309741} a^{20} - \frac{540512110095930}{2029056410309741} a^{19} + \frac{48016800}{2029056410309741} a^{18} - \frac{730191417388101}{2029056410309741} a^{17} - \frac{1855975680}{2029056410309741} a^{16} + \frac{52305950494360}{2029056410309741} a^{15} + \frac{41243904000}{2029056410309741} a^{14} + \frac{476896541429201}{2029056410309741} a^{13} - \frac{567710208000}{2029056410309741} a^{12} + \frac{28756927268825}{2029056410309741} a^{11} + \frac{4917789676800}{2029056410309741} a^{10} - \frac{409341924560906}{2029056410309741} a^{9} - \frac{26228211609600}{2029056410309741} a^{8} - \frac{505036407849682}{2029056410309741} a^{7} + \frac{80932767252480}{2029056410309741} a^{6} - \frac{584032877816770}{2029056410309741} a^{5} - \frac{127341766656000}{2029056410309741} a^{4} - \frac{504455923049007}{2029056410309741} a^{3} + \frac{77819968512000}{2029056410309741} a^{2} + \frac{688290687199870}{2029056410309741} a - \frac{7836416409600}{2029056410309741}$, $\frac{1}{2029056410309741} a^{27} - \frac{806496170243888}{2029056410309741} a^{20} - \frac{22744800}{2029056410309741} a^{19} + \frac{752049645720744}{2029056410309741} a^{18} + \frac{1965150720}{2029056410309741} a^{17} - \frac{466901086596318}{2029056410309741} a^{16} - \frac{74239027200}{2029056410309741} a^{15} + \frac{336683772318469}{2029056410309741} a^{14} + \frac{1572120576000}{2029056410309741} a^{13} + \frac{899344992488384}{2029056410309741} a^{12} - \frac{20118230496000}{2029056410309741} a^{11} - \frac{593741431504199}{2029056410309741} a^{10} + \frac{157369269657600}{2029056410309741} a^{9} + \frac{390523792455810}{2029056410309741} a^{8} - \frac{728394905272320}{2029056410309741} a^{7} - \frac{456928923035948}{2029056410309741} a^{6} - \frac{195334970463341}{2029056410309741} a^{5} - \frac{235511735685613}{2029056410309741} a^{4} - \frac{72082739514259}{2029056410309741} a^{3} + \frac{677154588236469}{2029056410309741} a^{2} + \frac{705277476864000}{2029056410309741} a - \frac{647961719423509}{2029056410309741}$, $\frac{1}{2029056410309741} a^{28} - \frac{26535600}{2029056410309741} a^{20} + \frac{586352710477906}{2029056410309741} a^{19} + \frac{2420046720}{2029056410309741} a^{18} + \frac{364507775277970}{2029056410309741} a^{17} - \frac{97438723200}{2029056410309741} a^{16} - \frac{216211972797594}{2029056410309741} a^{15} + \frac{2227170816000}{2029056410309741} a^{14} - \frac{194666729150766}{2029056410309741} a^{13} - \frac{31295025216000}{2029056410309741} a^{12} + \frac{31857249815410}{2029056410309741} a^{11} + \frac{275396221900800}{2029056410309741} a^{10} - \frac{139087049935174}{2029056410309741} a^{9} + \frac{541916812045421}{2029056410309741} a^{8} + \frac{304128186447366}{2029056410309741} a^{7} + \frac{577127485658918}{2029056410309741} a^{6} - \frac{206636506840036}{2029056410309741} a^{5} + \frac{762238616854964}{2029056410309741} a^{4} - \frac{707633272909555}{2029056410309741} a^{3} + \frac{467417655924518}{2029056410309741} a^{2} + \frac{284836732115674}{2029056410309741} a - \frac{458430359961600}{2029056410309741}$, $\frac{1}{2029056410309741} a^{29} - \frac{821583874809534}{2029056410309741} a^{20} - \frac{923438880}{2029056410309741} a^{19} + \frac{905215954062267}{2029056410309741} a^{18} + \frac{83109499200}{2029056410309741} a^{17} + \frac{614460653539633}{2029056410309741} a^{16} - \frac{3229397683200}{2029056410309741} a^{15} - \frac{732550838703630}{2029056410309741} a^{14} + \frac{69811979328000}{2029056410309741} a^{13} - \frac{935777182487553}{2029056410309741} a^{12} - \frac{907555731264000}{2029056410309741} a^{11} - \frac{228474009189369}{2029056410309741} a^{10} - \frac{928384249628084}{2029056410309741} a^{9} + \frac{9163034377302}{2029056410309741} a^{8} + \frac{888466754747197}{2029056410309741} a^{7} + \frac{726384330779451}{2029056410309741} a^{6} + \frac{85880249845278}{2029056410309741} a^{5} + \frac{840169323307236}{2029056410309741} a^{4} + \frac{993476240345309}{2029056410309741} a^{3} + \frac{265920158672412}{2029056410309741} a^{2} + \frac{771298532260144}{2029056410309741} a + \frac{326919014462750}{2029056410309741}$, $\frac{1}{2029056410309741} a^{30} - \frac{1108126656}{2029056410309741} a^{20} + \frac{867524653857774}{2029056410309741} a^{19} + \frac{105272032320}{2029056410309741} a^{18} + \frac{620734454272514}{2029056410309741} a^{17} - \frac{4359686872320}{2029056410309741} a^{16} + \frac{655995947165765}{2029056410309741} a^{15} + \frac{101726027020800}{2029056410309741} a^{14} + \frac{548494817616066}{2029056410309741} a^{13} + \frac{576967240287341}{2029056410309741} a^{12} - \frac{753288204830861}{2029056410309741} a^{11} + \frac{763776043041138}{2029056410309741} a^{10} - \frac{876664488307819}{2029056410309741} a^{9} + \frac{445440697752295}{2029056410309741} a^{8} - \frac{503498321841401}{2029056410309741} a^{7} + \frac{629099931659671}{2029056410309741} a^{6} + \frac{542563088003565}{2029056410309741} a^{5} + \frac{735835490809475}{2029056410309741} a^{4} - \frac{79758383315645}{2029056410309741} a^{3} + \frac{220834928173572}{2029056410309741} a^{2} + \frac{403158238925937}{2029056410309741} a - \frac{15106623922001}{2029056410309741}$, $\frac{1}{2029056410309741} a^{31} - \frac{302705094949076}{2029056410309741} a^{20} - \frac{34351926336}{2029056410309741} a^{19} - \frac{985644410587356}{2029056410309741} a^{18} + \frac{3180006895104}{2029056410309741} a^{17} - \frac{609994767680246}{2029056410309741} a^{16} - \frac{126140273505792}{2029056410309741} a^{15} + \frac{487237172508006}{2029056410309741} a^{14} + \frac{741082929425299}{2029056410309741} a^{13} + \frac{799184467402848}{2029056410309741} a^{12} + \frac{61056326312874}{2029056410309741} a^{11} - \frac{145539804081572}{2029056410309741} a^{10} - \frac{488450610502992}{2029056410309741} a^{9} - \frac{565742354016438}{2029056410309741} a^{8} + \frac{563504526391470}{2029056410309741} a^{7} + \frac{304143419809146}{2029056410309741} a^{6} + \frac{816737572408972}{2029056410309741} a^{5} + \frac{113646443260173}{2029056410309741} a^{4} + \frac{356394438652991}{2029056410309741} a^{3} + \frac{262500183234433}{2029056410309741} a^{2} + \frac{936610683164062}{2029056410309741} a - \frac{551256828203747}{2029056410309741}$, $\frac{1}{2029056410309741} a^{32} - \frac{42279293952}{2029056410309741} a^{20} - \frac{574414578285853}{2029056410309741} a^{19} + \frac{4131291009024}{2029056410309741} a^{18} - \frac{496785198554257}{2029056410309741} a^{17} - \frac{174655763315712}{2029056410309741} a^{16} - \frac{3348323963369}{2029056410309741} a^{15} + \frac{81875643160358}{2029056410309741} a^{14} - \frac{733261951955024}{2029056410309741} a^{13} - \frac{992373839294375}{2029056410309741} a^{12} + \frac{740009455803706}{2029056410309741} a^{11} + \frac{815138912410411}{2029056410309741} a^{10} - \frac{972601028912587}{2029056410309741} a^{9} + \frac{589377007517092}{2029056410309741} a^{8} - \frac{511143928597905}{2029056410309741} a^{7} - \frac{527441687322396}{2029056410309741} a^{6} - \frac{954434212602738}{2029056410309741} a^{5} + \frac{847674140339565}{2029056410309741} a^{4} + \frac{96552193986729}{2029056410309741} a^{3} - \frac{527441687322396}{2029056410309741} a^{2} + \frac{536027595344097}{2029056410309741} a - \frac{960585305315080}{2029056410309741}$, $\frac{1}{2029056410309741} a^{33} - \frac{115695565107580}{2029056410309741} a^{20} - \frac{1195900028928}{2029056410309741} a^{19} - \frac{758543701584010}{2029056410309741} a^{18} + \frac{113012552733696}{2029056410309741} a^{17} - \frac{857059541614153}{2029056410309741} a^{16} - \frac{495874489538342}{2029056410309741} a^{15} - \frac{261013997187518}{2029056410309741} a^{14} - \frac{193573266095434}{2029056410309741} a^{13} + \frac{37608214955946}{2029056410309741} a^{12} + \frac{1005737334438584}{2029056410309741} a^{11} - \frac{859847002675309}{2029056410309741} a^{10} + \frac{544359519516984}{2029056410309741} a^{9} - \frac{76530434231983}{2029056410309741} a^{8} + \frac{871872869963437}{2029056410309741} a^{7} + \frac{250301760936576}{2029056410309741} a^{6} + \frac{656693206250792}{2029056410309741} a^{5} - \frac{640535448667179}{2029056410309741} a^{4} + \frac{586562199580570}{2029056410309741} a^{3} - \frac{772505401189189}{2029056410309741} a^{2} + \frac{76725124276134}{2029056410309741} a - \frac{898085373777071}{2029056410309741}$, $\frac{1}{2029056410309741} a^{34} - \frac{1505948184576}{2029056410309741} a^{20} + \frac{896266377338838}{2029056410309741} a^{19} + \frac{150218331411456}{2029056410309741} a^{18} - \frac{938321749819341}{2029056410309741} a^{17} - \frac{364312791794361}{2029056410309741} a^{16} - \frac{285047317364853}{2029056410309741} a^{15} + \frac{627281361825700}{2029056410309741} a^{14} - \frac{829987353135032}{2029056410309741} a^{13} - \frac{64860138741166}{2029056410309741} a^{12} - \frac{42638343435675}{2029056410309741} a^{11} - \frac{324469625563302}{2029056410309741} a^{10} - \frac{658576841163323}{2029056410309741} a^{9} + \frac{80168655934661}{2029056410309741} a^{8} + \frac{828410254861104}{2029056410309741} a^{7} + \frac{614364781442805}{2029056410309741} a^{6} - \frac{480524176015273}{2029056410309741} a^{5} + \frac{412295944806828}{2029056410309741} a^{4} + \frac{627859281660714}{2029056410309741} a^{3} + \frac{756843239650603}{2029056410309741} a^{2} + \frac{820237606531592}{2029056410309741} a + \frac{80710338979843}{2029056410309741}$, $\frac{1}{2029056410309741} a^{35} + \frac{423219066074708}{2029056410309741} a^{20} - \frac{39531139845120}{2029056410309741} a^{19} - \frac{986223886796489}{2029056410309741} a^{18} - \frac{263123395487962}{2029056410309741} a^{17} + \frac{128905258159954}{2029056410309741} a^{16} - \frac{627281361825700}{2029056410309741} a^{15} - \frac{821146649696958}{2029056410309741} a^{14} - \frac{212377695830474}{2029056410309741} a^{13} + \frac{60149205440905}{2029056410309741} a^{12} + \frac{184051946195757}{2029056410309741} a^{11} - \frac{120768229052215}{2029056410309741} a^{10} + \frac{409123283654377}{2029056410309741} a^{9} - \frac{889739614159098}{2029056410309741} a^{8} + \frac{986288913405457}{2029056410309741} a^{7} + \frac{400569620302324}{2029056410309741} a^{6} + \frac{759745262164858}{2029056410309741} a^{5} + \frac{751232772586243}{2029056410309741} a^{4} + \frac{821689867099401}{2029056410309741} a^{3} - \frac{376246610674106}{2029056410309741} a^{2} + \frac{22595871627332}{2029056410309741} a - \frac{586882201646823}{2029056410309741}$, $\frac{1}{2029056410309741} a^{36} - \frac{50825751229440}{2029056410309741} a^{20} - \frac{416088229436547}{2029056410309741} a^{19} - \frac{936826439679303}{2029056410309741} a^{18} - \frac{222574084630799}{2029056410309741} a^{17} - \frac{762385083332906}{2029056410309741} a^{16} + \frac{332410239867008}{2029056410309741} a^{15} - \frac{455797203305316}{2029056410309741} a^{14} + \frac{50189042664102}{2029056410309741} a^{13} + \frac{25652420577049}{2029056410309741} a^{12} - \frac{207139965594594}{2029056410309741} a^{11} - \frac{515369931008535}{2029056410309741} a^{10} - \frac{932699016284899}{2029056410309741} a^{9} - \frac{317170457724898}{2029056410309741} a^{8} - \frac{859384328711307}{2029056410309741} a^{7} - \frac{202488381466171}{2029056410309741} a^{6} - \frac{956280390530505}{2029056410309741} a^{5} + \frac{850453589886557}{2029056410309741} a^{4} - \frac{958967597589625}{2029056410309741} a^{3} + \frac{739515495167488}{2029056410309741} a^{2} + \frac{267881310172353}{2029056410309741} a - \frac{248354867649598}{2029056410309741}$, $\frac{1}{2029056410309741} a^{37} + \frac{865544503031864}{2029056410309741} a^{20} + \frac{775354546650221}{2029056410309741} a^{19} + \frac{189785202700197}{2029056410309741} a^{18} + \frac{116436529120884}{2029056410309741} a^{17} - \frac{407349310916378}{2029056410309741} a^{16} - \frac{187104510035505}{2029056410309741} a^{15} - \frac{665965421455771}{2029056410309741} a^{14} - \frac{894952310567614}{2029056410309741} a^{13} - \frac{796173011629501}{2029056410309741} a^{12} - \frac{92482269195657}{2029056410309741} a^{11} + \frac{985890359987068}{2029056410309741} a^{10} + \frac{639766176266812}{2029056410309741} a^{9} + \frac{1010221695994235}{2029056410309741} a^{8} + \frac{175612610414888}{2029056410309741} a^{7} + \frac{881109593429455}{2029056410309741} a^{6} - \frac{611998616293247}{2029056410309741} a^{5} - \frac{345960996895068}{2029056410309741} a^{4} - \frac{115495443900191}{2029056410309741} a^{3} - \frac{546861019480005}{2029056410309741} a^{2} + \frac{326602372090332}{2029056410309741} a + \frac{990553900094569}{2029056410309741}$, $\frac{1}{2029056410309741} a^{38} + \frac{386274657928301}{2029056410309741} a^{20} - \frac{320653572010953}{2029056410309741} a^{19} + \frac{137725738627241}{2029056410309741} a^{18} + \frac{778611189459089}{2029056410309741} a^{17} + \frac{756202215450029}{2029056410309741} a^{16} + \frac{240138887250980}{2029056410309741} a^{15} - \frac{674336776059649}{2029056410309741} a^{14} - \frac{159738972653288}{2029056410309741} a^{13} + \frac{328194019526031}{2029056410309741} a^{12} - \frac{170313024074427}{2029056410309741} a^{11} + \frac{417861900810048}{2029056410309741} a^{10} + \frac{20116597666407}{2029056410309741} a^{9} - \frac{716781516288476}{2029056410309741} a^{8} - \frac{261784311313090}{2029056410309741} a^{7} - \frac{594711426658100}{2029056410309741} a^{6} - \frac{464144503921225}{2029056410309741} a^{5} + \frac{105723126822626}{2029056410309741} a^{4} - \frac{190095299416526}{2029056410309741} a^{3} + \frac{534635486166418}{2029056410309741} a^{2} + \frac{209739384699697}{2029056410309741} a - \frac{998911050637806}{2029056410309741}$, $\frac{1}{2029056410309741} a^{39} + \frac{639557598262278}{2029056410309741} a^{20} + \frac{110978790159383}{2029056410309741} a^{19} - \frac{819570265983135}{2029056410309741} a^{18} + \frac{171481022404620}{2029056410309741} a^{17} + \frac{585136702414764}{2029056410309741} a^{16} + \frac{313884351210397}{2029056410309741} a^{15} - \frac{591065518209554}{2029056410309741} a^{14} - \frac{437592026034708}{2029056410309741} a^{13} + \frac{848394877261421}{2029056410309741} a^{12} + \frac{652616069538808}{2029056410309741} a^{11} + \frac{787601570119181}{2029056410309741} a^{10} + \frac{943448596883519}{2029056410309741} a^{9} + \frac{601687752375480}{2029056410309741} a^{8} - \frac{128325698227888}{2029056410309741} a^{7} + \frac{874007560842708}{2029056410309741} a^{6} - \frac{868284106657523}{2029056410309741} a^{5} - \frac{923706061827877}{2029056410309741} a^{4} - \frac{863092089234211}{2029056410309741} a^{3} + \frac{743274484635225}{2029056410309741} a^{2} + \frac{675960911133699}{2029056410309741} a - \frac{145186468404653}{2029056410309741}$

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:  $39$
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, 8.8.54500230757132921.1, 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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ $40$ $40$ $40$ $40$ $40$ R $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ 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
$23$23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed