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Properties

Label 40.40.2632664960...5625.1
Degree $40$
Signature $[40, 0]$
Discriminant $3^{20}\cdot 5^{20}\cdot 41^{39}$
Root discriminant $144.71$
Ramified primes $3, 5, 41$
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![-3538957955699, 48618934694515, -48618934694515, -740280658234765, 740280658234765, 3381719714820723, -3381719714820723, -6776066918780301, 6776066918780301, 7614130812154483, -7614130812154483, -5467867125058957, 5467867125058957, 2708381585699443, -2708381585699443, -970930334141837, 970930334141837, 260015933158003, -260015933158003, -53119520804237, 53119520804237, 8389229081203, -8389229081203, -1031577472397, 1031577472397, 98919314035, -98919314035, -7366708621, 7366708621, 421491315, -421491315, -18165133, 18165133, 570227, -570227, -12301, 12301, 163, -163, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 163*x^38 + 163*x^37 + 12301*x^36 - 12301*x^35 - 570227*x^34 + 570227*x^33 + 18165133*x^32 - 18165133*x^31 - 421491315*x^30 + 421491315*x^29 + 7366708621*x^28 - 7366708621*x^27 - 98919314035*x^26 + 98919314035*x^25 + 1031577472397*x^24 - 1031577472397*x^23 - 8389229081203*x^22 + 8389229081203*x^21 + 53119520804237*x^20 - 53119520804237*x^19 - 260015933158003*x^18 + 260015933158003*x^17 + 970930334141837*x^16 - 970930334141837*x^15 - 2708381585699443*x^14 + 2708381585699443*x^13 + 5467867125058957*x^12 - 5467867125058957*x^11 - 7614130812154483*x^10 + 7614130812154483*x^9 + 6776066918780301*x^8 - 6776066918780301*x^7 - 3381719714820723*x^6 + 3381719714820723*x^5 + 740280658234765*x^4 - 740280658234765*x^3 - 48618934694515*x^2 + 48618934694515*x - 3538957955699)
 
gp: K = bnfinit(x^40 - x^39 - 163*x^38 + 163*x^37 + 12301*x^36 - 12301*x^35 - 570227*x^34 + 570227*x^33 + 18165133*x^32 - 18165133*x^31 - 421491315*x^30 + 421491315*x^29 + 7366708621*x^28 - 7366708621*x^27 - 98919314035*x^26 + 98919314035*x^25 + 1031577472397*x^24 - 1031577472397*x^23 - 8389229081203*x^22 + 8389229081203*x^21 + 53119520804237*x^20 - 53119520804237*x^19 - 260015933158003*x^18 + 260015933158003*x^17 + 970930334141837*x^16 - 970930334141837*x^15 - 2708381585699443*x^14 + 2708381585699443*x^13 + 5467867125058957*x^12 - 5467867125058957*x^11 - 7614130812154483*x^10 + 7614130812154483*x^9 + 6776066918780301*x^8 - 6776066918780301*x^7 - 3381719714820723*x^6 + 3381719714820723*x^5 + 740280658234765*x^4 - 740280658234765*x^3 - 48618934694515*x^2 + 48618934694515*x - 3538957955699, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 163 x^{38} + 163 x^{37} + 12301 x^{36} - 12301 x^{35} - 570227 x^{34} + 570227 x^{33} + 18165133 x^{32} - 18165133 x^{31} - 421491315 x^{30} + 421491315 x^{29} + 7366708621 x^{28} - 7366708621 x^{27} - 98919314035 x^{26} + 98919314035 x^{25} + 1031577472397 x^{24} - 1031577472397 x^{23} - 8389229081203 x^{22} + 8389229081203 x^{21} + 53119520804237 x^{20} - 53119520804237 x^{19} - 260015933158003 x^{18} + 260015933158003 x^{17} + 970930334141837 x^{16} - 970930334141837 x^{15} - 2708381585699443 x^{14} + 2708381585699443 x^{13} + 5467867125058957 x^{12} - 5467867125058957 x^{11} - 7614130812154483 x^{10} + 7614130812154483 x^{9} + 6776066918780301 x^{8} - 6776066918780301 x^{7} - 3381719714820723 x^{6} + 3381719714820723 x^{5} + 740280658234765 x^{4} - 740280658234765 x^{3} - 48618934694515 x^{2} + 48618934694515 x - 3538957955699 \)

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:  \(263266496024621374471745700545497393800039466828408289234757666722767978763580322265625=3^{20}\cdot 5^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.71$
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:  $3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(615=3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{615}(256,·)$, $\chi_{615}(1,·)$, $\chi_{615}(134,·)$, $\chi_{615}(44,·)$, $\chi_{615}(14,·)$, $\chi_{615}(271,·)$, $\chi_{615}(16,·)$, $\chi_{615}(404,·)$, $\chi_{615}(149,·)$, $\chi_{615}(406,·)$, $\chi_{615}(539,·)$, $\chi_{615}(284,·)$, $\chi_{615}(29,·)$, $\chi_{615}(286,·)$, $\chi_{615}(31,·)$, $\chi_{615}(166,·)$, $\chi_{615}(299,·)$, $\chi_{615}(556,·)$, $\chi_{615}(46,·)$, $\chi_{615}(541,·)$, $\chi_{615}(434,·)$, $\chi_{615}(179,·)$, $\chi_{615}(314,·)$, $\chi_{615}(61,·)$, $\chi_{615}(194,·)$, $\chi_{615}(196,·)$, $\chi_{615}(464,·)$, $\chi_{615}(89,·)$, $\chi_{615}(346,·)$, $\chi_{615}(91,·)$, $\chi_{615}(479,·)$, $\chi_{615}(224,·)$, $\chi_{615}(226,·)$, $\chi_{615}(104,·)$, $\chi_{615}(361,·)$, $\chi_{615}(239,·)$, $\chi_{615}(496,·)$, $\chi_{615}(241,·)$, $\chi_{615}(121,·)$, $\chi_{615}(509,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{674220028459} a^{21} - \frac{167907233177}{674220028459} a^{20} - \frac{84}{674220028459} a^{19} - \frac{51821915020}{674220028459} a^{18} + \frac{3024}{674220028459} a^{17} - \frac{260714974697}{674220028459} a^{16} - \frac{60928}{674220028459} a^{15} - \frac{129244218309}{674220028459} a^{14} + \frac{752640}{674220028459} a^{13} - \frac{299674591440}{674220028459} a^{12} - \frac{5870592}{674220028459} a^{11} - \frac{182638973023}{674220028459} a^{10} + \frac{28700672}{674220028459} a^{9} - \frac{276750160337}{674220028459} a^{8} - \frac{84344832}{674220028459} a^{7} - \frac{96715022777}{674220028459} a^{6} + \frac{136249344}{674220028459} a^{5} + \frac{289448785586}{674220028459} a^{4} - \frac{100925440}{674220028459} a^{3} - \frac{99476985226}{674220028459} a^{2} + \frac{22020096}{674220028459} a - \frac{72948244597}{674220028459}$, $\frac{1}{674220028459} a^{22} - \frac{88}{674220028459} a^{20} + \frac{2591095751}{674220028459} a^{19} + \frac{3344}{674220028459} a^{18} - \frac{196923277076}{674220028459} a^{17} - \frac{71808}{674220028459} a^{16} + \frac{233564610301}{674220028459} a^{15} + \frac{957440}{674220028459} a^{14} + \frac{295049504716}{674220028459} a^{13} - \frac{8200192}{674220028459} a^{12} + \frac{32897281865}{674220028459} a^{11} + \frac{45101056}{674220028459} a^{10} - \frac{206782914580}{674220028459} a^{9} - \frac{154632192}{674220028459} a^{8} + \frac{193012275907}{674220028459} a^{7} + \frac{309264384}{674220028459} a^{6} + \frac{288195476645}{674220028459} a^{5} - \frac{317194240}{674220028459} a^{4} - \frac{323288617719}{674220028459} a^{3} + \frac{126877696}{674220028459} a^{2} - \frac{318321719017}{674220028459} a - \frac{8388608}{674220028459}$, $\frac{1}{674220028459} a^{23} + \frac{59595202273}{674220028459} a^{20} - \frac{4048}{674220028459} a^{19} - \frac{37711599623}{674220028459} a^{18} + \frac{194304}{674220028459} a^{17} + \frac{214127804571}{674220028459} a^{16} - \frac{4404224}{674220028459} a^{15} - \frac{290921251132}{674220028459} a^{14} + \frac{58032128}{674220028459} a^{13} - \frac{43885654954}{674220028459} a^{12} - \frac{471511040}{674220028459} a^{11} - \frac{97731857588}{674220028459} a^{10} + \frac{2371026944}{674220028459} a^{9} + \frac{110919190775}{674220028459} a^{8} - \frac{7113080832}{674220028459} a^{7} - \frac{132086186223}{674220028459} a^{6} + \frac{11672748032}{674220028459} a^{5} + \frac{202063460866}{674220028459} a^{4} - \frac{8754561024}{674220028459} a^{3} - \frac{307436048938}{674220028459} a^{2} + \frac{1929379840}{674220028459} a + \frac{322754760054}{674220028459}$, $\frac{1}{674220028459} a^{24} - \frac{4416}{674220028459} a^{20} + \frac{248745192096}{674220028459} a^{19} + \frac{223744}{674220028459} a^{18} + \frac{14983729572}{674220028459} a^{17} - \frac{5405184}{674220028459} a^{16} + \frac{50709586497}{674220028459} a^{15} + \frac{76873728}{674220028459} a^{14} + \frac{58908886219}{674220028459} a^{13} - \frac{685834240}{674220028459} a^{12} + \frac{179222766797}{674220028459} a^{11} + \frac{3879862272}{674220028459} a^{10} - \frac{194294484171}{674220028459} a^{9} - \frac{13579517952}{674220028459} a^{8} + \frac{228244182804}{674220028459} a^{7} + \frac{27590131712}{674220028459} a^{6} + \frac{1045289618}{8123132873} a^{5} - \frac{28651290624}{674220028459} a^{4} + \frac{2714643935}{674220028459} a^{3} + \frac{11576279040}{674220028459} a^{2} - \frac{2343947439}{674220028459} a - \frac{771751936}{674220028459}$, $\frac{1}{674220028459} a^{25} - \frac{261785241095}{674220028459} a^{20} - \frac{147200}{674220028459} a^{19} - \frac{270003351147}{674220028459} a^{18} + \frac{7948800}{674220028459} a^{17} + \frac{301189932517}{674220028459} a^{16} - \frac{192184320}{674220028459} a^{15} - \frac{293415090011}{674220028459} a^{14} + \frac{2637824000}{674220028459} a^{13} + \frac{310142832774}{674220028459} a^{12} - \frac{22044672000}{674220028459} a^{11} + \frac{313374711684}{674220028459} a^{10} + \frac{113162649600}{674220028459} a^{9} - \frac{213772297680}{674220028459} a^{8} + \frac{329343382059}{674220028459} a^{7} - \frac{225503530391}{674220028459} a^{6} - \frac{101194215979}{674220028459} a^{5} - \frac{112622166553}{674220028459} a^{4} + \frac{240109564459}{674220028459} a^{3} + \frac{298747849813}{674220028459} a^{2} + \frac{96468992000}{674220028459} a + \frac{137725463050}{674220028459}$, $\frac{1}{674220028459} a^{26} - \frac{166400}{674220028459} a^{20} - \frac{10702663980}{674220028459} a^{19} + \frac{9484800}{674220028459} a^{18} - \frac{268774435528}{674220028459} a^{17} - \frac{244408320}{674220028459} a^{16} - \frac{321371271608}{674220028459} a^{15} + \frac{3620864000}{674220028459} a^{14} - \frac{336016142291}{674220028459} a^{13} - \frac{33226752000}{674220028459} a^{12} - \frac{40345510481}{674220028459} a^{11} + \frac{191884492800}{674220028459} a^{10} - \frac{132566836662}{674220028459} a^{9} - \frac{8035945941}{674220028459} a^{8} + \frac{223831470351}{674220028459} a^{7} + \frac{55057947562}{674220028459} a^{6} - \frac{60524827222}{674220028459} a^{5} - \frac{123760647082}{674220028459} a^{4} + \frac{35417218600}{674220028459} a^{3} - \frac{74434556459}{674220028459} a^{2} + \frac{207437964923}{674220028459} a - \frac{40265318400}{674220028459}$, $\frac{1}{674220028459} a^{27} - \frac{96323975820}{674220028459} a^{20} - \frac{4492800}{674220028459} a^{19} - \frac{161269772918}{674220028459} a^{18} + \frac{258785280}{674220028459} a^{17} + \frac{68790370406}{674220028459} a^{16} - \frac{6517555200}{674220028459} a^{15} - \frac{303474974709}{674220028459} a^{14} + \frac{92012544000}{674220028459} a^{13} + \frac{95163729618}{674220028459} a^{12} - \frac{110761987541}{674220028459} a^{11} - \frac{115675045978}{674220028459} a^{10} + \frac{48215675646}{674220028459} a^{9} + \frac{247755228628}{674220028459} a^{8} + \frac{178698500401}{674220028459} a^{7} + \frac{191764396308}{674220028459} a^{6} + \frac{298869255371}{674220028459} a^{5} + \frac{57165703417}{674220028459} a^{4} - \frac{12927060984}{674220028459} a^{3} + \frac{13015055432}{674220028459} a^{2} + \frac{252778513705}{674220028459} a + \frac{69491435036}{674220028459}$, $\frac{1}{674220028459} a^{28} - \frac{5241600}{674220028459} a^{20} - \frac{161843400290}{674220028459} a^{19} + \frac{318689280}{674220028459} a^{18} + \frac{89440955798}{674220028459} a^{17} - \frac{8554291200}{674220028459} a^{16} - \frac{45326000074}{674220028459} a^{15} + \frac{130351104000}{674220028459} a^{14} - \frac{158895244934}{674220028459} a^{13} + \frac{127356920918}{674220028459} a^{12} + \frac{247856887226}{674220028459} a^{11} - \frac{252732581849}{674220028459} a^{10} + \frac{43668066035}{674220028459} a^{9} - \frac{168914750878}{674220028459} a^{8} + \frac{186108685312}{674220028459} a^{7} + \frac{324723377339}{674220028459} a^{6} + \frac{66301851949}{674220028459} a^{5} - \frac{45244713444}{674220028459} a^{4} - \frac{143579134990}{674220028459} a^{3} + \frac{329740408394}{674220028459} a^{2} + \frac{91398894247}{674220028459} a - \frac{221907360682}{674220028459}$, $\frac{1}{674220028459} a^{29} - \frac{162034609414}{674220028459} a^{20} - \frac{121605120}{674220028459} a^{19} + \frac{104737685718}{674220028459} a^{18} + \frac{7296307200}{674220028459} a^{17} + \frac{108805211105}{674220028459} a^{16} - \frac{189009100800}{674220028459} a^{15} + \frac{191931505440}{674220028459} a^{14} + \frac{27074574164}{674220028459} a^{13} + \frac{133967765361}{674220028459} a^{12} - \frac{9906299935}{674220028459} a^{11} - \frac{98460555878}{674220028459} a^{10} - \frac{82538742035}{674220028459} a^{9} - \frac{71622971651}{674220028459} a^{8} - \frac{163029393216}{674220028459} a^{7} - \frac{77227948364}{674220028459} a^{6} + \frac{120306658875}{674220028459} a^{5} + \frac{204388122516}{674220028459} a^{4} - \frac{92543583750}{674220028459} a^{3} - \frac{302052512818}{674220028459} a^{2} - \frac{92997033571}{674220028459} a + \frac{166320118257}{674220028459}$, $\frac{1}{674220028459} a^{30} - \frac{145926144}{674220028459} a^{20} - \frac{21768935878}{674220028459} a^{19} + \frac{9241989120}{674220028459} a^{18} - \frac{56496610652}{674220028459} a^{17} - \frac{255162286080}{674220028459} a^{16} - \frac{323094174074}{674220028459} a^{15} - \frac{76129053954}{674220028459} a^{14} + \frac{269429425942}{674220028459} a^{13} - \frac{15850079896}{674220028459} a^{12} + \frac{328222554200}{674220028459} a^{11} - \frac{148569735663}{674220028459} a^{10} + \frac{14857470780}{674220028459} a^{9} - \frac{72673714370}{674220028459} a^{8} - \frac{180019038099}{674220028459} a^{7} + \frac{312797313075}{674220028459} a^{6} - \frac{200058602762}{674220028459} a^{5} - \frac{333156901500}{674220028459} a^{4} - \frac{102346794606}{674220028459} a^{3} + \frac{330127618274}{674220028459} a^{2} + \frac{257591967248}{674220028459} a + \frac{235838039236}{674220028459}$, $\frac{1}{674220028459} a^{31} + \frac{134720608940}{674220028459} a^{20} - \frac{3015806976}{674220028459} a^{19} + \frac{234520344629}{674220028459} a^{18} + \frac{186118373376}{674220028459} a^{17} - \frac{104370226743}{674220028459} a^{16} - \frac{202256785619}{674220028459} a^{15} - \frac{238667077361}{674220028459} a^{14} - \frac{83861698553}{674220028459} a^{13} + \frac{124117526698}{674220028459} a^{12} + \frac{112232878478}{674220028459} a^{11} + \frac{22410404061}{674220028459} a^{10} - \frac{149095332910}{674220028459} a^{9} - \frac{19754458315}{674220028459} a^{8} + \frac{83316744312}{674220028459} a^{7} - \frac{231514476137}{674220028459} a^{6} - \frac{66183679415}{674220028459} a^{5} - \frac{231581960002}{674220028459} a^{4} + \frac{332138573310}{674220028459} a^{3} - \frac{313106767173}{674220028459} a^{2} + \frac{210882193466}{674220028459} a - \frac{304938516045}{674220028459}$, $\frac{1}{674220028459} a^{32} - \frac{3711762432}{674220028459} a^{20} + \frac{89311011786}{674220028459} a^{19} + \frac{241794809856}{674220028459} a^{18} - \frac{270594472067}{674220028459} a^{17} - \frac{72595540562}{674220028459} a^{16} + \frac{63967959093}{674220028459} a^{15} - \frac{184603859680}{674220028459} a^{14} - \frac{44915125892}{674220028459} a^{13} - \frac{5987592063}{674220028459} a^{12} - \frac{7574962655}{674220028459} a^{11} + \frac{35925552378}{674220028459} a^{10} + \frac{276667340843}{674220028459} a^{9} - \frac{131727025386}{674220028459} a^{8} + \frac{94452940132}{674220028459} a^{7} + \frac{48250197713}{674220028459} a^{6} + \frac{279116872885}{674220028459} a^{5} + \frac{189096226900}{674220028459} a^{4} + \frac{7788941683}{674220028459} a^{3} - \frac{128382140674}{674220028459} a^{2} + \frac{336043541125}{674220028459} a + \frac{161985152423}{674220028459}$, $\frac{1}{674220028459} a^{33} - \frac{183009463636}{674220028459} a^{20} - \frac{69993234432}{674220028459} a^{19} - \frac{59997348995}{674220028459} a^{18} - \frac{309966429997}{674220028459} a^{17} + \frac{320086087694}{674220028459} a^{16} + \frac{203064245648}{674220028459} a^{15} - \frac{106762051612}{674220028459} a^{14} + \frac{321311322780}{674220028459} a^{13} + \frac{21713788951}{674220028459} a^{12} - \frac{89813880945}{674220028459} a^{11} + \frac{313193739782}{674220028459} a^{10} - \frac{191220935377}{674220028459} a^{9} - \frac{254723063452}{674220028459} a^{8} + \frac{71733926808}{674220028459} a^{7} - \frac{54223784028}{674220028459} a^{6} - \frac{315606739802}{674220028459} a^{5} - \frac{81200258520}{674220028459} a^{4} + \frac{95645235744}{674220028459} a^{3} - \frac{315461118401}{674220028459} a^{2} - \frac{144323013298}{674220028459} a - \frac{28786030653}{674220028459}$, $\frac{1}{674220028459} a^{34} - \frac{88139628544}{674220028459} a^{20} + \frac{74268360138}{674220028459} a^{19} - \frac{206694957955}{674220028459} a^{18} + \frac{206060758119}{674220028459} a^{17} + \frac{62934338515}{674220028459} a^{16} - \frac{256531810878}{674220028459} a^{15} - \frac{332729387781}{674220028459} a^{14} - \frac{190509271873}{674220028459} a^{13} - \frac{150798614920}{674220028459} a^{12} + \frac{80717534524}{674220028459} a^{11} + \frac{238111591807}{674220028459} a^{10} + \frac{57780571915}{674220028459} a^{9} + \frac{158080320188}{674220028459} a^{8} + \frac{279521095501}{674220028459} a^{7} + \frac{146068320337}{674220028459} a^{6} - \frac{49832732140}{674220028459} a^{5} - \frac{312893582315}{674220028459} a^{4} - \frac{86330460}{4136319193} a^{3} + \frac{74187323593}{674220028459} a^{2} + \frac{3524681831}{674220028459} a - \frac{71418429228}{674220028459}$, $\frac{1}{674220028459} a^{35} - \frac{302878227769}{674220028459} a^{20} - \frac{194003442602}{674220028459} a^{19} + \frac{37886510024}{674220028459} a^{18} + \frac{280259814266}{674220028459} a^{17} + \frac{67632396680}{674220028459} a^{16} + \frac{332729387781}{674220028459} a^{15} + \frac{53176500767}{674220028459} a^{14} + \frac{76408631771}{674220028459} a^{13} + \frac{5671984580}{674220028459} a^{12} - \frac{50820869773}{674220028459} a^{11} + \frac{154600425307}{674220028459} a^{10} - \frac{77454381654}{674220028459} a^{9} + \frac{249847454369}{674220028459} a^{8} - \frac{56121573226}{674220028459} a^{7} + \frac{178799314137}{674220028459} a^{6} - \frac{193731455316}{674220028459} a^{5} + \frac{205803098518}{674220028459} a^{4} + \frac{235630203023}{674220028459} a^{3} + \frac{39935008596}{674220028459} a^{2} + \frac{199821319105}{674220028459} a - \frac{137756831833}{674220028459}$, $\frac{1}{674220028459} a^{36} + \frac{39518443137}{674220028459} a^{20} + \frac{216476458870}{674220028459} a^{19} - \frac{197550276461}{674220028459} a^{18} - \frac{293625505645}{674220028459} a^{17} - \frac{277228238976}{674220028459} a^{16} - \frac{309306086035}{674220028459} a^{15} - \frac{300376541582}{674220028459} a^{14} - \frac{236142142373}{674220028459} a^{13} + \frac{105512802806}{674220028459} a^{12} + \frac{287778541547}{674220028459} a^{11} + \frac{332209427138}{674220028459} a^{10} - \frac{251418885369}{674220028459} a^{9} + \frac{210836474471}{674220028459} a^{8} + \frac{757467345}{674220028459} a^{7} + \frac{134539545793}{674220028459} a^{6} - \frac{284537027963}{674220028459} a^{5} + \frac{330956472981}{674220028459} a^{4} + \frac{328236580605}{674220028459} a^{3} + \frac{112740701643}{674220028459} a^{2} + \frac{69029018385}{674220028459} a - \frac{289690363125}{674220028459}$, $\frac{1}{674220028459} a^{37} + \frac{158428586700}{674220028459} a^{20} - \frac{249101195248}{674220028459} a^{19} + \frac{304884097201}{674220028459} a^{18} + \frac{230164780438}{674220028459} a^{17} + \frac{242188299430}{674220028459} a^{16} - \frac{160394717535}{674220028459} a^{15} - \frac{146972485645}{674220028459} a^{14} + \frac{161025639911}{674220028459} a^{13} - \frac{52306275275}{674220028459} a^{12} - \frac{100790702281}{674220028459} a^{11} - \frac{95632269426}{674220028459} a^{10} + \frac{305065910698}{674220028459} a^{9} + \frac{7320570334}{674220028459} a^{8} - \frac{32302874391}{674220028459} a^{7} - \frac{196966200173}{674220028459} a^{6} - \frac{21885797607}{674220028459} a^{5} - \frac{118336968362}{674220028459} a^{4} - \frac{300114681936}{674220028459} a^{3} - \frac{123068672091}{674220028459} a^{2} + \frac{82333732466}{674220028459} a - \frac{65538588177}{674220028459}$, $\frac{1}{674220028459} a^{38} - \frac{256661563243}{674220028459} a^{20} + \frac{128484810821}{674220028459} a^{19} + \frac{160774191579}{674220028459} a^{18} - \frac{1802863560}{8123132873} a^{17} + \frac{176225218294}{674220028459} a^{16} - \frac{218189475548}{674220028459} a^{15} + \frac{249570648445}{674220028459} a^{14} + \frac{113552981629}{674220028459} a^{13} + \frac{71894875448}{674220028459} a^{12} - \frac{175738422051}{674220028459} a^{11} - \frac{74607721225}{674220028459} a^{10} - \frac{275269378625}{674220028459} a^{9} + \frac{53795483087}{674220028459} a^{8} - \frac{54477399619}{674220028459} a^{7} + \frac{77358258659}{674220028459} a^{6} + \frac{221982685593}{674220028459} a^{5} - \frac{87059738039}{674220028459} a^{4} + \frac{174207162983}{674220028459} a^{3} + \frac{244636195121}{674220028459} a^{2} - \frac{105881678431}{674220028459} a - \frac{282672208396}{674220028459}$, $\frac{1}{674220028459} a^{39} + \frac{107287494585}{674220028459} a^{20} + \frac{176243789855}{674220028459} a^{19} + \frac{197707566482}{674220028459} a^{18} + \frac{293539708817}{674220028459} a^{17} + \frac{108032810006}{674220028459} a^{16} + \frac{233185456987}{674220028459} a^{15} - \frac{78101554161}{674220028459} a^{14} - \frac{320599843417}{674220028459} a^{13} + \frac{150226844298}{674220028459} a^{12} + \frac{289970972086}{674220028459} a^{11} - \frac{250747955823}{674220028459} a^{10} - \frac{80495836867}{674220028459} a^{9} + \frac{322016522986}{674220028459} a^{8} + \frac{114260898117}{674220028459} a^{7} + \frac{125037680361}{674220028459} a^{6} - \frac{250397852311}{674220028459} a^{5} + \frac{295388419523}{674220028459} a^{4} + \frac{190829605722}{674220028459} a^{3} + \frac{203120152103}{674220028459} a^{2} - \frac{111571491255}{674220028459} a + \frac{29217455834}{674220028459}$

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, 8.8.9859435115225625.1, 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 $20^{2}$ R R $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
41Data not computed