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Properties

Label 40.0.35962420425...7073.1
Degree $40$
Signature $[0, 20]$
Discriminant $11^{36}\cdot 17^{35}$
Root discriminant $103.25$
Ramified primes $11, 17$
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![152206781, 29191834, 655492636, 41392946, 2653892568, -237856327, 5463109687, 1107007524, 5822116569, 3389499394, 3767218658, 3192048789, 1932414069, 965778567, 817022219, -251868115, 171913213, -193658014, -6008611, -12003364, -1044252, 10713834, 2079843, 2363548, -499093, -45151, -314455, -161288, 121256, -7695, 49835, 7452, -2514, 1082, -1648, -94, -25, -23, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 18*x^38 - 23*x^37 - 25*x^36 - 94*x^35 - 1648*x^34 + 1082*x^33 - 2514*x^32 + 7452*x^31 + 49835*x^30 - 7695*x^29 + 121256*x^28 - 161288*x^27 - 314455*x^26 - 45151*x^25 - 499093*x^24 + 2363548*x^23 + 2079843*x^22 + 10713834*x^21 - 1044252*x^20 - 12003364*x^19 - 6008611*x^18 - 193658014*x^17 + 171913213*x^16 - 251868115*x^15 + 817022219*x^14 + 965778567*x^13 + 1932414069*x^12 + 3192048789*x^11 + 3767218658*x^10 + 3389499394*x^9 + 5822116569*x^8 + 1107007524*x^7 + 5463109687*x^6 - 237856327*x^5 + 2653892568*x^4 + 41392946*x^3 + 655492636*x^2 + 29191834*x + 152206781)
 
gp: K = bnfinit(x^40 - x^39 + 18*x^38 - 23*x^37 - 25*x^36 - 94*x^35 - 1648*x^34 + 1082*x^33 - 2514*x^32 + 7452*x^31 + 49835*x^30 - 7695*x^29 + 121256*x^28 - 161288*x^27 - 314455*x^26 - 45151*x^25 - 499093*x^24 + 2363548*x^23 + 2079843*x^22 + 10713834*x^21 - 1044252*x^20 - 12003364*x^19 - 6008611*x^18 - 193658014*x^17 + 171913213*x^16 - 251868115*x^15 + 817022219*x^14 + 965778567*x^13 + 1932414069*x^12 + 3192048789*x^11 + 3767218658*x^10 + 3389499394*x^9 + 5822116569*x^8 + 1107007524*x^7 + 5463109687*x^6 - 237856327*x^5 + 2653892568*x^4 + 41392946*x^3 + 655492636*x^2 + 29191834*x + 152206781, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 18 x^{38} - 23 x^{37} - 25 x^{36} - 94 x^{35} - 1648 x^{34} + 1082 x^{33} - 2514 x^{32} + 7452 x^{31} + 49835 x^{30} - 7695 x^{29} + 121256 x^{28} - 161288 x^{27} - 314455 x^{26} - 45151 x^{25} - 499093 x^{24} + 2363548 x^{23} + 2079843 x^{22} + 10713834 x^{21} - 1044252 x^{20} - 12003364 x^{19} - 6008611 x^{18} - 193658014 x^{17} + 171913213 x^{16} - 251868115 x^{15} + 817022219 x^{14} + 965778567 x^{13} + 1932414069 x^{12} + 3192048789 x^{11} + 3767218658 x^{10} + 3389499394 x^{9} + 5822116569 x^{8} + 1107007524 x^{7} + 5463109687 x^{6} - 237856327 x^{5} + 2653892568 x^{4} + 41392946 x^{3} + 655492636 x^{2} + 29191834 x + 152206781 \)

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:  \(359624204259227998212313764863527746816862563620018205460931204658277030572367073=11^{36}\cdot 17^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.25$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(128,·)$, $\chi_{187}(1,·)$, $\chi_{187}(2,·)$, $\chi_{187}(4,·)$, $\chi_{187}(134,·)$, $\chi_{187}(135,·)$, $\chi_{187}(8,·)$, $\chi_{187}(137,·)$, $\chi_{187}(138,·)$, $\chi_{187}(16,·)$, $\chi_{187}(145,·)$, $\chi_{187}(19,·)$, $\chi_{187}(151,·)$, $\chi_{187}(152,·)$, $\chi_{187}(157,·)$, $\chi_{187}(32,·)$, $\chi_{187}(161,·)$, $\chi_{187}(162,·)$, $\chi_{187}(38,·)$, $\chi_{187}(169,·)$, $\chi_{187}(43,·)$, $\chi_{187}(172,·)$, $\chi_{187}(174,·)$, $\chi_{187}(47,·)$, $\chi_{187}(178,·)$, $\chi_{187}(64,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(76,·)$, $\chi_{187}(81,·)$, $\chi_{187}(83,·)$, $\chi_{187}(86,·)$, $\chi_{187}(87,·)$, $\chi_{187}(89,·)$, $\chi_{187}(94,·)$, $\chi_{187}(166,·)$, $\chi_{187}(103,·)$, $\chi_{187}(115,·)$, $\chi_{187}(117,·)$, $\chi_{187}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{67} a^{34} - \frac{30}{67} a^{33} + \frac{14}{67} a^{32} + \frac{9}{67} a^{31} - \frac{19}{67} a^{30} + \frac{30}{67} a^{29} + \frac{27}{67} a^{28} - \frac{32}{67} a^{26} + \frac{12}{67} a^{25} + \frac{20}{67} a^{24} + \frac{21}{67} a^{23} - \frac{4}{67} a^{22} + \frac{2}{67} a^{21} + \frac{13}{67} a^{20} + \frac{5}{67} a^{19} - \frac{20}{67} a^{18} + \frac{28}{67} a^{17} + \frac{18}{67} a^{16} - \frac{23}{67} a^{15} + \frac{17}{67} a^{14} + \frac{3}{67} a^{13} + \frac{13}{67} a^{12} - \frac{15}{67} a^{11} - \frac{4}{67} a^{10} + \frac{12}{67} a^{9} - \frac{7}{67} a^{8} + \frac{29}{67} a^{7} - \frac{27}{67} a^{6} + \frac{1}{67} a^{5} - \frac{2}{67} a^{4} - \frac{24}{67} a^{3} - \frac{8}{67} a^{2} + \frac{7}{67} a$, $\frac{1}{2602124359} a^{35} + \frac{9307941}{2602124359} a^{34} - \frac{17753563}{38837677} a^{33} + \frac{1092183234}{2602124359} a^{32} + \frac{731137780}{2602124359} a^{31} + \frac{794307283}{2602124359} a^{30} + \frac{992058321}{2602124359} a^{29} + \frac{1068240723}{2602124359} a^{28} + \frac{1220153249}{2602124359} a^{27} + \frac{996208601}{2602124359} a^{26} - \frac{380623477}{2602124359} a^{25} - \frac{589009923}{2602124359} a^{24} - \frac{822759083}{2602124359} a^{23} - \frac{1248108946}{2602124359} a^{22} + \frac{707950547}{2602124359} a^{21} + \frac{109350519}{2602124359} a^{20} + \frac{325863170}{2602124359} a^{19} + \frac{681037085}{2602124359} a^{18} - \frac{86106953}{2602124359} a^{17} - \frac{381841001}{2602124359} a^{16} - \frac{381624588}{2602124359} a^{15} - \frac{803601683}{2602124359} a^{14} - \frac{679624750}{2602124359} a^{13} + \frac{6714869}{38837677} a^{12} - \frac{682227980}{2602124359} a^{11} - \frac{925658304}{2602124359} a^{10} - \frac{579754740}{2602124359} a^{9} + \frac{1071239138}{2602124359} a^{8} + \frac{464060952}{2602124359} a^{7} - \frac{358106047}{2602124359} a^{6} - \frac{116436291}{2602124359} a^{5} - \frac{878593729}{2602124359} a^{4} - \frac{1172494686}{2602124359} a^{3} + \frac{1096578520}{2602124359} a^{2} + \frac{394490009}{2602124359} a + \frac{5199009}{38837677}$, $\frac{1}{2602124359} a^{36} + \frac{18190367}{2602124359} a^{34} + \frac{610106907}{2602124359} a^{33} - \frac{344097790}{2602124359} a^{32} - \frac{471594636}{2602124359} a^{31} - \frac{629424847}{2602124359} a^{30} - \frac{199689530}{2602124359} a^{29} - \frac{170177188}{2602124359} a^{28} - \frac{1176870822}{2602124359} a^{27} - \frac{61913894}{2602124359} a^{26} + \frac{1243020658}{2602124359} a^{25} - \frac{230655130}{2602124359} a^{24} - \frac{671907875}{2602124359} a^{23} + \frac{656445909}{2602124359} a^{22} + \frac{550439463}{2602124359} a^{21} - \frac{1162309163}{2602124359} a^{20} + \frac{746964887}{2602124359} a^{19} - \frac{1163262574}{2602124359} a^{18} - \frac{1296500893}{2602124359} a^{17} - \frac{1111282820}{2602124359} a^{16} - \frac{760986726}{2602124359} a^{15} - \frac{43361824}{2602124359} a^{14} + \frac{1217759861}{2602124359} a^{13} - \frac{1100995568}{2602124359} a^{12} + \frac{620041035}{2602124359} a^{11} + \frac{504247716}{2602124359} a^{10} - \frac{1241008293}{2602124359} a^{9} - \frac{231711198}{2602124359} a^{8} + \frac{868355299}{2602124359} a^{7} - \frac{116879760}{2602124359} a^{6} + \frac{297699625}{2602124359} a^{5} + \frac{949915391}{2602124359} a^{4} - \frac{1274633943}{2602124359} a^{3} + \frac{137303852}{2602124359} a^{2} + \frac{421215913}{2602124359} a - \frac{12787053}{38837677}$, $\frac{1}{2602124359} a^{37} - \frac{1659568}{2602124359} a^{34} - \frac{595704194}{2602124359} a^{33} - \frac{518637356}{2602124359} a^{32} - \frac{1192925529}{2602124359} a^{31} + \frac{1026075963}{2602124359} a^{30} - \frac{828185270}{2602124359} a^{29} - \frac{609435137}{2602124359} a^{28} + \frac{238946020}{2602124359} a^{27} - \frac{378955858}{2602124359} a^{26} + \frac{1140305865}{2602124359} a^{25} - \frac{825263537}{2602124359} a^{24} - \frac{361275010}{2602124359} a^{23} - \frac{214466144}{2602124359} a^{22} - \frac{756398300}{2602124359} a^{21} + \frac{1296420046}{2602124359} a^{20} + \frac{1137703720}{2602124359} a^{19} + \frac{57235631}{2602124359} a^{18} + \frac{102651300}{2602124359} a^{17} - \frac{920450753}{2602124359} a^{16} + \frac{800838577}{2602124359} a^{15} - \frac{335555418}{2602124359} a^{14} + \frac{223400114}{2602124359} a^{13} + \frac{910114893}{2602124359} a^{12} + \frac{447461292}{2602124359} a^{11} + \frac{434705207}{2602124359} a^{10} + \frac{293193522}{2602124359} a^{9} + \frac{81464089}{2602124359} a^{8} - \frac{1111302469}{2602124359} a^{7} + \frac{816309752}{2602124359} a^{6} - \frac{900432274}{2602124359} a^{5} - \frac{711146317}{2602124359} a^{4} - \frac{134435437}{2602124359} a^{3} + \frac{255525232}{2602124359} a^{2} + \frac{847502128}{2602124359} a - \frac{2179068}{38837677}$, $\frac{1}{2602124359} a^{38} + \frac{5591177}{2602124359} a^{34} - \frac{1266848155}{2602124359} a^{33} + \frac{106807605}{2602124359} a^{32} + \frac{154901466}{2602124359} a^{31} - \frac{19058928}{2602124359} a^{30} + \frac{55496306}{2602124359} a^{29} - \frac{189338014}{2602124359} a^{28} + \frac{570245236}{2602124359} a^{27} - \frac{93757390}{2602124359} a^{26} + \frac{534434097}{2602124359} a^{25} - \frac{539496912}{2602124359} a^{24} - \frac{1270874658}{2602124359} a^{23} + \frac{1189719052}{2602124359} a^{22} - \frac{304979611}{2602124359} a^{21} - \frac{936985357}{2602124359} a^{20} - \frac{1300307167}{2602124359} a^{19} - \frac{870943569}{2602124359} a^{18} + \frac{220598978}{2602124359} a^{17} - \frac{1252065395}{2602124359} a^{16} - \frac{1009050330}{2602124359} a^{15} + \frac{817732910}{2602124359} a^{14} - \frac{628574913}{2602124359} a^{13} - \frac{865987482}{2602124359} a^{12} - \frac{727298343}{2602124359} a^{11} - \frac{1072323538}{2602124359} a^{10} - \frac{637119661}{2602124359} a^{9} + \frac{748055535}{2602124359} a^{8} - \frac{1145997926}{2602124359} a^{7} + \frac{1179018833}{2602124359} a^{6} - \frac{1248368358}{2602124359} a^{5} - \frac{701186827}{2602124359} a^{4} + \frac{945078272}{2602124359} a^{3} - \frac{49983957}{2602124359} a^{2} - \frac{1075617700}{2602124359} a + \frac{8321146}{38837677}$, $\frac{1}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{39} - \frac{965131001982153398339881844555597024591527579125767938992618596432745697814970994771465495886388453942909652922608845082964869379209457984739915095453754235792857694244758}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{38} + \frac{1546947406324480274766752250863713353373851792871918209932718708205627993808507606898846099116582056900879606271731008148920042539576200323861890437095202361383637481852212}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{37} - \frac{1563006709979939579522157083484609003326155727495471148985632038128508202164920974472116521787256180280842176651146934478389226182927590199194330369354014626940503421318992}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{36} - \frac{1713649600417685590743921796398380612188914892326965661548337641120679116719117597681354074791787688947774125823544734890074540422317925250111855653147254764431237909862031}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{35} + \frac{57158817297708909619409795596548510568645928348974512233868405855393480381611469979209411674563979880352925636930128733924912166518103779075071947947708791007767969123927632718160}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{34} + \frac{3411408066619574170818019417392209819496976072940444965986003597829757834923126434555193577954175127677021903031352080868682746865986170921518173979845385047272044595316402291462996}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{33} - \frac{1659528478326174451656854025156375685781837272917524020203606055963056760842758307059430610967312194077566607376046487774093508928191875264645214416669664190950863896562486261591705}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{32} - \frac{4245822297890764843265516458092113717090677459869528331456819450835177433166759419826831369749003720842995394736430496943960630938454198933939743531968760837790561880801814302491811}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{31} - \frac{2476419167786582140643766677548665541583061883705124161412249414863109390764402416805845605857243932815349389572488165060668571006380320852170622487080208980719358043011251619667173}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{30} + \frac{3632034365609357815682371213059406948282805631129391189477600680961376902449784196822415225984571034255546565745837436664304490730998445923297536465820023339693677048767417295215838}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{29} - \frac{3864573403270524529316703277066936822502486864890348648178789199147643781438460154729709141941782880587304976390232274079769723143872928844811571063108988659749448930515372149425545}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{28} - \frac{2361753159606124190587031574094271545256572994628621734449988585606177753022877342517272449413859214413105808057588085185192080296944523197350977874676846942783646353666084510867926}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{27} - \frac{4190815699863724924433435490252071555974161987486942039910341579443627376549067339663141688939091777383099174627001175330107880605968670924904476889185020826731768825597916331663471}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{26} - \frac{1434526478188354399032654514016202558895035120279346797422034162261215498658084134553955454733567783052230974404814974247304862623532062697439087371842030868749737185270148880402828}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{25} + \frac{133427154758737084752805348159569668917155152364513615413484161388153062656616603100434273456471860570167894184370301151914765348261020597142268833672163137652699566058451968088846}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{24} - \frac{3646866258426924038865369199240325060534391482865990747769076872524481137438974441949973540732860430107451000799446156819698362168311037810881326554922795588760350545058947987105284}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{23} - \frac{120143122911337164522255989091537195468823358998359332780480188430515563526751370235152920647699664097844508812404343312558255633228689904589748100148602049515677066915341937686346}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{22} + \frac{3546361661164326057703659642043082536007977720876101979380554882254031980841856027863226682546716591350527592203466014987506504614007619611060727218175970497345277858731233653097879}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{21} + \frac{4252453867868316272804372805408112341116202353902576765493997632576608695151547255282582360118649146941528648106798160924393862001528790282058479241587928860119970442291463742661516}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{20} + \frac{1967510655380940145971194586913900697168106365157830339488941576781894228861398414963789779751684000391996094981931806879371513439537483769199948761453526303709644239221150817914681}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{19} + \frac{1431294043785921190195038420631925380172130813012626578248095385781397922638083965573988238980225019716670805295095204551229157803587763803777598728780073670045075785118830187256723}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{18} + \frac{2658916004959606409227254545236227518985105204381525380409262339116544649586195003580242878662815550796790098838659104798540227899638812905551292658860503423477796320827262524822662}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{17} - \frac{130256932649918800140070506711027698382264870283437860671798409128935129544478706887158805502838649498364094792378176764945676747878290477105720816543664469103827944384518151298714}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{16} + \frac{1529524024751658415696008764706894786430606485203002717657360455958292363885158740082862939216178890222790734281253272167362758937328147612839404438539276792958273888576193497908000}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{15} + \frac{1603032534899568031182306100724381153137910194848829422701544058265973642244775443352391940985061414145440495796034400945629172556558487200694498256794143953841386010313790269113437}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{14} + \frac{2129521448549870708749307209333042821508855373328790866673994369567965111450336882698802882396595320362525153279099197237227066983009454364716050476097254611030302271058172108551657}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{13} - \frac{4005077047876799955087861228374257155589004939503428927259749522755797741414822280926158437192533850324774011322763980289302525339384018123463194063403661001723592489907155757267796}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{12} - \frac{3990116214051768957790606444371579135511099830279546405805530441547982953552245318372152273722965993732913897229703572206383976837575220511118159591369012987857402240701698802850356}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{11} + \frac{2034588284211992691864185535746392284831171132772878544377408694375557689332938947186443315461447418128280672539210151094314099068811506565017393921207370059545191145600809779754607}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{10} - \frac{3669584717629360327338478462772441336065915062842097004352753242210705226224770725595827249184389491657818247679028449423854266966021075538187531386512147367038797618856541413165443}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{9} - \frac{2239587171158803749440761892075651904952113450875039454468571894525989836985944881920793720371439413227279878162392564914871219861785238563192521523431425859368859134528830739543265}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{8} - \frac{3449810969624198700781952598749739885838554880913423952688592572584018795219692641419180615159557734611764315740074853632326884224477074192594566728876728595611505534585334984432204}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{7} - \frac{1256348707434147358114638966216112210422147218758823034489843597823761739053040795595991255392260682433716973448352901625305211536224530646146709583246182368075725950791967738763346}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{6} + \frac{131652691274699852595254609884922265244068248822285118876341376864757946839077178654166522191872303860649113564558383027451358339482790875783841863653622720779766134549818818265632}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{5} - \frac{495625305377624163679253577871226737816467038856692089427000022415941051080259517046657357482370253277550688665174439495524966393823600765250861139452813322617344627640661583977951}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{4} + \frac{3829792756129573786921600502313362759266101528823407123844064866854146244383139006881836853252277831119822932586779073482763255591514359556032019085621888520227057390228963950257996}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{3} - \frac{3938243835371402353067209039203782051691380369249079565572949218885134190699660437720041941740007207409291980489181445076240569495461423883637424648080601513923660412605151691415858}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a^{2} - \frac{3660545444886134756526136398927794494282215214927227954261485359270991067469899322795648136138326131126454133905454887855023843245373019472897329105955736458927496939632607105285083}{9186403151492326039511929711246943207751031711068949219101304016924936713835622108020832801280258739004864313225192950843934526700272853725716489720127115858999485013600950733564473} a - \frac{39155469358780178205450938463325208556691163733985991992091294063677253064994222650766996630703165203189252631677984778694988152472505529663949761452175365468713721921419991438198}{137110494798392925962864622555924525488821368821924615210467224133208010654263016537624370168362070731415885272017805236476634726869744085458455070449658445656708731546282846769619}$

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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), 8.0.6007768511393.1, 10.10.304358957700017.1, 20.20.131527565972137936816816034072938673.1

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 $20^{2}$ $40$ $40$ $40$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{8}$ R $20^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{5}$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$

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
11Data not computed
17Data not computed