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Properties

Label 40.0.80991088329...7689.1
Degree $40$
Signature $[0, 20]$
Discriminant $11^{36}\cdot 13^{30}$
Root discriminant $59.25$
Ramified primes $11, 13$
Class number $1525$ (GRH)
Class group $[1525]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 78732, 65610, 15309, -69255, -150660, -181683, -123822, 29331, 232437, 392197, 50690, -281134, -616850, -782673, -555516, 80422, 944081, 1651223, 1730608, 865161, -865292, 261981, 60913, -93393, 35971, 2592, -9698, 4482, -413, -1170, 454, 29, -121, 58, -3, -12, 7, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - x^38 + 7*x^37 - 12*x^36 - 3*x^35 + 58*x^34 - 121*x^33 + 29*x^32 + 454*x^31 - 1170*x^30 - 413*x^29 + 4482*x^28 - 9698*x^27 + 2592*x^26 + 35971*x^25 - 93393*x^24 + 60913*x^23 + 261981*x^22 - 865292*x^21 + 865161*x^20 + 1730608*x^19 + 1651223*x^18 + 944081*x^17 + 80422*x^16 - 555516*x^15 - 782673*x^14 - 616850*x^13 - 281134*x^12 + 50690*x^11 + 392197*x^10 + 232437*x^9 + 29331*x^8 - 123822*x^7 - 181683*x^6 - 150660*x^5 - 69255*x^4 + 15309*x^3 + 65610*x^2 + 78732*x + 59049)
 
gp: K = bnfinit(x^40 - x^39 - x^38 + 7*x^37 - 12*x^36 - 3*x^35 + 58*x^34 - 121*x^33 + 29*x^32 + 454*x^31 - 1170*x^30 - 413*x^29 + 4482*x^28 - 9698*x^27 + 2592*x^26 + 35971*x^25 - 93393*x^24 + 60913*x^23 + 261981*x^22 - 865292*x^21 + 865161*x^20 + 1730608*x^19 + 1651223*x^18 + 944081*x^17 + 80422*x^16 - 555516*x^15 - 782673*x^14 - 616850*x^13 - 281134*x^12 + 50690*x^11 + 392197*x^10 + 232437*x^9 + 29331*x^8 - 123822*x^7 - 181683*x^6 - 150660*x^5 - 69255*x^4 + 15309*x^3 + 65610*x^2 + 78732*x + 59049, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - x^{38} + 7 x^{37} - 12 x^{36} - 3 x^{35} + 58 x^{34} - 121 x^{33} + 29 x^{32} + 454 x^{31} - 1170 x^{30} - 413 x^{29} + 4482 x^{28} - 9698 x^{27} + 2592 x^{26} + 35971 x^{25} - 93393 x^{24} + 60913 x^{23} + 261981 x^{22} - 865292 x^{21} + 865161 x^{20} + 1730608 x^{19} + 1651223 x^{18} + 944081 x^{17} + 80422 x^{16} - 555516 x^{15} - 782673 x^{14} - 616850 x^{13} - 281134 x^{12} + 50690 x^{11} + 392197 x^{10} + 232437 x^{9} + 29331 x^{8} - 123822 x^{7} - 181683 x^{6} - 150660 x^{5} - 69255 x^{4} + 15309 x^{3} + 65610 x^{2} + 78732 x + 59049 \)

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:  \(80991088329663621512136783946706428882352994235065987550082919086927689=11^{36}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(131,·)$, $\chi_{143}(5,·)$, $\chi_{143}(129,·)$, $\chi_{143}(8,·)$, $\chi_{143}(138,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(18,·)$, $\chi_{143}(21,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(31,·)$, $\chi_{143}(34,·)$, $\chi_{143}(38,·)$, $\chi_{143}(40,·)$, $\chi_{143}(135,·)$, $\chi_{143}(47,·)$, $\chi_{143}(51,·)$, $\chi_{143}(53,·)$, $\chi_{143}(57,·)$, $\chi_{143}(60,·)$, $\chi_{143}(64,·)$, $\chi_{143}(70,·)$, $\chi_{143}(73,·)$, $\chi_{143}(79,·)$, $\chi_{143}(83,·)$, $\chi_{143}(142,·)$, $\chi_{143}(86,·)$, $\chi_{143}(90,·)$, $\chi_{143}(92,·)$, $\chi_{143}(96,·)$, $\chi_{143}(103,·)$, $\chi_{143}(105,·)$, $\chi_{143}(109,·)$, $\chi_{143}(112,·)$, $\chi_{143}(116,·)$, $\chi_{143}(118,·)$, $\chi_{143}(122,·)$, $\chi_{143}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{19} + \frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{20} + \frac{1}{3} a^{18} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{23} - \frac{1}{3} a$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{25} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{26} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{27} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{28} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{29} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{30} - \frac{1}{3} a^{8}$, $\frac{1}{2823163986336411} a^{31} + \frac{430233146757965}{2823163986336411} a^{30} - \frac{94968514623833}{941054662112137} a^{29} + \frac{365493912467710}{2823163986336411} a^{28} + \frac{43140438082871}{2823163986336411} a^{27} - \frac{381285892441771}{2823163986336411} a^{26} - \frac{211466688575045}{2823163986336411} a^{25} + \frac{16706162795647}{941054662112137} a^{24} - \frac{340695333140411}{2823163986336411} a^{23} - \frac{402605382720475}{2823163986336411} a^{22} + \frac{12253581349974}{941054662112137} a^{21} - \frac{1274163377272255}{2823163986336411} a^{20} - \frac{410205525836438}{941054662112137} a^{19} + \frac{1029560932051328}{2823163986336411} a^{18} - \frac{317366510645025}{941054662112137} a^{17} + \frac{616165475948339}{2823163986336411} a^{16} + \frac{209545029881944}{941054662112137} a^{15} - \frac{288555006427561}{2823163986336411} a^{14} - \frac{431360881979448}{941054662112137} a^{13} - \frac{285927396804634}{2823163986336411} a^{12} - \frac{55344201988806}{941054662112137} a^{11} - \frac{749614178558596}{2823163986336411} a^{10} + \frac{1029500769241673}{2823163986336411} a^{9} + \frac{309856538623986}{941054662112137} a^{8} - \frac{368938048779702}{941054662112137} a^{7} - \frac{972862107547289}{2823163986336411} a^{6} + \frac{52047907539766}{2823163986336411} a^{5} + \frac{108012949085112}{941054662112137} a^{4} - \frac{999140653506706}{2823163986336411} a^{3} + \frac{654676925460554}{2823163986336411} a^{2} - \frac{1280602600493644}{2823163986336411} a - \frac{411070585923916}{941054662112137}$, $\frac{1}{8469491959009233} a^{32} - \frac{1}{8469491959009233} a^{31} + \frac{752231988664247}{8469491959009233} a^{30} + \frac{187936101140107}{8469491959009233} a^{29} + \frac{376921302622601}{2823163986336411} a^{28} - \frac{440290713942479}{2823163986336411} a^{27} + \frac{377556750995929}{8469491959009233} a^{26} + \frac{577106888037014}{8469491959009233} a^{25} + \frac{1284655264403615}{8469491959009233} a^{24} + \frac{210810402651973}{8469491959009233} a^{23} + \frac{406266784679159}{2823163986336411} a^{22} + \frac{296347971109042}{8469491959009233} a^{21} + \frac{908273032407428}{2823163986336411} a^{20} + \frac{1058484977019283}{8469491959009233} a^{19} - \frac{556635670875184}{2823163986336411} a^{18} + \frac{1093677575139763}{8469491959009233} a^{17} - \frac{564793644492793}{2823163986336411} a^{16} - \frac{1878565239351971}{8469491959009233} a^{15} - \frac{762408491091343}{2823163986336411} a^{14} - \frac{4014200507354834}{8469491959009233} a^{13} - \frac{770652887359696}{2823163986336411} a^{12} - \frac{1642338457260680}{8469491959009233} a^{11} + \frac{2749905158984585}{8469491959009233} a^{10} + \frac{3329538629284925}{8469491959009233} a^{9} + \frac{3760803109185202}{8469491959009233} a^{8} + \frac{780613516342441}{2823163986336411} a^{7} + \frac{94591138384560}{941054662112137} a^{6} + \frac{2342759191706758}{8469491959009233} a^{5} - \frac{314118841402330}{8469491959009233} a^{4} - \frac{2345312234576173}{8469491959009233} a^{3} - \frac{1502029997520899}{8469491959009233} a^{2} + \frac{243688504093354}{2823163986336411} a - \frac{57729995407016}{941054662112137}$, $\frac{1}{25408475877027699} a^{33} - \frac{1}{25408475877027699} a^{32} - \frac{1}{25408475877027699} a^{31} - \frac{3494807606404451}{25408475877027699} a^{30} + \frac{440976549571676}{8469491959009233} a^{29} - \frac{934268798305120}{8469491959009233} a^{28} + \frac{3116700589885969}{25408475877027699} a^{27} + \frac{1326069695999750}{25408475877027699} a^{26} + \frac{2668990475449487}{25408475877027699} a^{25} - \frac{1384892241177497}{25408475877027699} a^{24} + \frac{52286695958761}{2823163986336411} a^{23} - \frac{943066189537343}{25408475877027699} a^{22} + \frac{125716636638506}{941054662112137} a^{21} - \frac{12410202950135555}{25408475877027699} a^{20} - \frac{122315745650420}{941054662112137} a^{19} + \frac{4316058459083581}{25408475877027699} a^{18} - \frac{248811696598767}{941054662112137} a^{17} - \frac{3302168666866847}{25408475877027699} a^{16} - \frac{255820082845807}{941054662112137} a^{15} - \frac{899883162319397}{25408475877027699} a^{14} - \frac{138860573481555}{941054662112137} a^{13} - \frac{12173061138105836}{25408475877027699} a^{12} + \frac{8594233318468631}{25408475877027699} a^{11} + \frac{3454596508693292}{25408475877027699} a^{10} - \frac{7270649491992545}{25408475877027699} a^{9} + \frac{734863140725407}{2823163986336411} a^{8} - \frac{3806956067231585}{8469491959009233} a^{7} - \frac{7375087989375695}{25408475877027699} a^{6} + \frac{10727910262783628}{25408475877027699} a^{5} + \frac{6253423781676617}{25408475877027699} a^{4} + \frac{2461484217365311}{25408475877027699} a^{3} - \frac{3072623491507193}{8469491959009233} a^{2} - \frac{1260513953429188}{2823163986336411} a + \frac{455973498058521}{941054662112137}$, $\frac{1}{76225427631083097} a^{34} - \frac{1}{76225427631083097} a^{33} - \frac{1}{76225427631083097} a^{32} + \frac{7}{76225427631083097} a^{31} - \frac{1586071914300472}{25408475877027699} a^{30} + \frac{3445447081029560}{25408475877027699} a^{29} - \frac{819909757285817}{76225427631083097} a^{28} + \frac{11931316053558170}{76225427631083097} a^{27} - \frac{5488436091982966}{76225427631083097} a^{26} - \frac{1845907019487284}{76225427631083097} a^{25} + \frac{1354535659498661}{8469491959009233} a^{24} - \frac{8430786975108077}{76225427631083097} a^{23} - \frac{254413877698817}{2823163986336411} a^{22} + \frac{1806325817414764}{76225427631083097} a^{21} - \frac{406853225327742}{941054662112137} a^{20} + \frac{32964220690858861}{76225427631083097} a^{19} - \frac{77389473169735}{941054662112137} a^{18} + \frac{31776966379847404}{76225427631083097} a^{17} - \frac{636812522278625}{2823163986336411} a^{16} - \frac{17875990775576462}{76225427631083097} a^{15} - \frac{376529733518418}{941054662112137} a^{14} - \frac{21630333948405833}{76225427631083097} a^{13} + \frac{34506809131778372}{76225427631083097} a^{12} + \frac{16611625062878345}{76225427631083097} a^{11} - \frac{4424426243999750}{76225427631083097} a^{10} + \frac{2407617692045317}{8469491959009233} a^{9} - \frac{11314487524738220}{25408475877027699} a^{8} + \frac{14543641125741466}{76225427631083097} a^{7} + \frac{839944406152322}{76225427631083097} a^{6} + \frac{23378004130559537}{76225427631083097} a^{5} + \frac{33313155774634867}{76225427631083097} a^{4} - \frac{6235468760086085}{25408475877027699} a^{3} + \frac{1962644684193556}{8469491959009233} a^{2} + \frac{1183904211401186}{2823163986336411} a - \frac{165207747839560}{941054662112137}$, $\frac{1}{228676282893249291} a^{35} - \frac{1}{228676282893249291} a^{34} - \frac{1}{228676282893249291} a^{33} + \frac{7}{228676282893249291} a^{32} - \frac{4}{76225427631083097} a^{31} + \frac{9193959800591822}{76225427631083097} a^{30} + \frac{11209572386347477}{228676282893249291} a^{29} + \frac{9852096441184619}{228676282893249291} a^{28} + \frac{1830848762139230}{228676282893249291} a^{27} + \frac{16783037326638139}{228676282893249291} a^{26} - \frac{1629451805024506}{25408475877027699} a^{25} - \frac{34134941987096381}{228676282893249291} a^{24} - \frac{1012821398684957}{8469491959009233} a^{23} - \frac{13393315222110032}{228676282893249291} a^{22} - \frac{100653903134264}{941054662112137} a^{21} + \frac{44265643809160003}{228676282893249291} a^{20} + \frac{202973476900999}{941054662112137} a^{19} - \frac{14955044424411230}{228676282893249291} a^{18} - \frac{2293595467452041}{8469491959009233} a^{17} - \frac{72345828302930705}{228676282893249291} a^{16} - \frac{143059985347312}{2823163986336411} a^{15} - \frac{46563661792593056}{228676282893249291} a^{14} + \frac{90388961484042341}{228676282893249291} a^{13} - \frac{55251871135842493}{228676282893249291} a^{12} - \frac{48340839669968306}{228676282893249291} a^{11} - \frac{2881937269349561}{25408475877027699} a^{10} + \frac{29265770518361416}{76225427631083097} a^{9} + \frac{7561953753254569}{228676282893249291} a^{8} - \frac{35477379813953566}{228676282893249291} a^{7} - \frac{84222968617362112}{228676282893249291} a^{6} - \frac{1740963775882127}{228676282893249291} a^{5} - \frac{23544635712018239}{76225427631083097} a^{4} + \frac{8036931394831168}{25408475877027699} a^{3} - \frac{2465626345776914}{8469491959009233} a^{2} + \frac{317549965035292}{2823163986336411} a - \frac{318416961125790}{941054662112137}$, $\frac{1}{686028848679747873} a^{36} - \frac{1}{686028848679747873} a^{35} - \frac{1}{686028848679747873} a^{34} + \frac{7}{686028848679747873} a^{33} - \frac{4}{228676282893249291} a^{32} - \frac{1}{228676282893249291} a^{31} + \frac{92741005077555316}{686028848679747873} a^{30} - \frac{94910280253158910}{686028848679747873} a^{29} - \frac{90571729901951698}{686028848679747873} a^{28} - \frac{34672537961257082}{686028848679747873} a^{27} + \frac{1700491922928608}{76225427631083097} a^{26} - \frac{92741897167300187}{686028848679747873} a^{25} - \frac{1241414906586566}{25408475877027699} a^{24} - \frac{73115245864859792}{686028848679747873} a^{23} - \frac{593246134252957}{8469491959009233} a^{22} + \frac{109760027301511243}{686028848679747873} a^{21} - \frac{2537784315783451}{8469491959009233} a^{20} + \frac{325716014056534894}{686028848679747873} a^{19} - \frac{5371563744633077}{25408475877027699} a^{18} - \frac{285864309866852816}{686028848679747873} a^{17} - \frac{3069641946672029}{8469491959009233} a^{16} - \frac{50878460496089144}{686028848679747873} a^{15} - \frac{264624740775131236}{686028848679747873} a^{14} + \frac{229410600646130576}{686028848679747873} a^{13} + \frac{156219183281330494}{686028848679747873} a^{12} - \frac{16538474314039268}{76225427631083097} a^{11} + \frac{58622276604704230}{228676282893249291} a^{10} + \frac{58470639025524382}{686028848679747873} a^{9} + \frac{261826458231936923}{686028848679747873} a^{8} - \frac{283464671743648879}{686028848679747873} a^{7} + \frac{152122670619282364}{686028848679747873} a^{6} - \frac{28452369546768602}{228676282893249291} a^{5} + \frac{16504059407091967}{76225427631083097} a^{4} + \frac{1210888877279008}{25408475877027699} a^{3} - \frac{2627881282225454}{8469491959009233} a^{2} + \frac{341492664569656}{2823163986336411} a - \frac{275412672028124}{941054662112137}$, $\frac{1}{2058086546039243619} a^{37} - \frac{1}{2058086546039243619} a^{36} - \frac{1}{2058086546039243619} a^{35} + \frac{7}{2058086546039243619} a^{34} - \frac{4}{686028848679747873} a^{33} - \frac{1}{686028848679747873} a^{32} + \frac{58}{2058086546039243619} a^{31} - \frac{156453204033498559}{2058086546039243619} a^{30} - \frac{322076378402797291}{2058086546039243619} a^{29} - \frac{51046062209953223}{2058086546039243619} a^{28} + \frac{7709555875728155}{228676282893249291} a^{27} + \frac{287915167558779202}{2058086546039243619} a^{26} + \frac{12420656038766407}{76225427631083097} a^{25} + \frac{205522998671569729}{2058086546039243619} a^{24} + \frac{830422676858036}{25408475877027699} a^{23} - \frac{624884658206024}{2058086546039243619} a^{22} - \frac{3924502882056739}{25408475877027699} a^{21} + \frac{657651302730050257}{2058086546039243619} a^{20} + \frac{33763025946333367}{76225427631083097} a^{19} + \frac{238553848777781452}{2058086546039243619} a^{18} - \frac{6612152897025164}{25408475877027699} a^{17} - \frac{326156964921915374}{2058086546039243619} a^{16} + \frac{884369104577653718}{2058086546039243619} a^{15} + \frac{1026118832375166776}{2058086546039243619} a^{14} - \frac{434645201201777237}{2058086546039243619} a^{13} + \frac{93363144832062340}{228676282893249291} a^{12} - \frac{192565477118848085}{686028848679747873} a^{11} + \frac{254482160564543716}{2058086546039243619} a^{10} - \frac{608340118409610112}{2058086546039243619} a^{9} + \frac{431322658984051076}{2058086546039243619} a^{8} + \frac{91137938201218354}{2058086546039243619} a^{7} + \frac{217203909752281003}{686028848679747873} a^{6} + \frac{73148537718501256}{228676282893249291} a^{5} + \frac{25612345217988166}{76225427631083097} a^{4} + \frac{4010198797592764}{25408475877027699} a^{3} + \frac{1410193922134309}{8469491959009233} a^{2} - \frac{276003515974591}{2823163986336411} a + \frac{365177554088002}{941054662112137}$, $\frac{1}{6174259638117730857} a^{38} - \frac{1}{6174259638117730857} a^{37} - \frac{1}{6174259638117730857} a^{36} + \frac{7}{6174259638117730857} a^{35} - \frac{4}{2058086546039243619} a^{34} - \frac{1}{2058086546039243619} a^{33} + \frac{58}{6174259638117730857} a^{32} - \frac{121}{6174259638117730857} a^{31} + \frac{95573358714194039}{6174259638117730857} a^{30} + \frac{754402487618488270}{6174259638117730857} a^{29} - \frac{105061022782986334}{686028848679747873} a^{28} - \frac{478865085161699123}{6174259638117730857} a^{27} + \frac{36469639907716180}{228676282893249291} a^{26} + \frac{101905247889819409}{6174259638117730857} a^{25} + \frac{11209975526254136}{76225427631083097} a^{24} + \frac{147324760033733737}{6174259638117730857} a^{23} - \frac{4859105483631475}{76225427631083097} a^{22} - \frac{690918741134923400}{6174259638117730857} a^{21} - \frac{24319673454133250}{228676282893249291} a^{20} - \frac{2150665019868624125}{6174259638117730857} a^{19} - \frac{30925222890256010}{76225427631083097} a^{18} + \frac{78244946088633403}{6174259638117730857} a^{17} - \frac{2491578836716567957}{6174259638117730857} a^{16} - \frac{1232688212297773714}{6174259638117730857} a^{15} + \frac{1695135556182191956}{6174259638117730857} a^{14} + \frac{324189995501849461}{686028848679747873} a^{13} + \frac{803420742398436937}{2058086546039243619} a^{12} + \frac{2232924715393487857}{6174259638117730857} a^{11} - \frac{468016000264104751}{6174259638117730857} a^{10} + \frac{3064345237826167691}{6174259638117730857} a^{9} + \frac{549160324647233080}{6174259638117730857} a^{8} + \frac{325426603777596205}{2058086546039243619} a^{7} - \frac{84632309418414608}{686028848679747873} a^{6} + \frac{32721384431915707}{228676282893249291} a^{5} + \frac{10363653274320376}{76225427631083097} a^{4} - \frac{9908451067847570}{25408475877027699} a^{3} - \frac{2310730496798114}{8469491959009233} a^{2} - \frac{468644220906539}{2823163986336411} a + \frac{72692268861993}{941054662112137}$, $\frac{1}{18522778914353192571} a^{39} - \frac{1}{18522778914353192571} a^{38} - \frac{1}{18522778914353192571} a^{37} + \frac{7}{18522778914353192571} a^{36} - \frac{4}{6174259638117730857} a^{35} - \frac{1}{6174259638117730857} a^{34} + \frac{58}{18522778914353192571} a^{33} - \frac{121}{18522778914353192571} a^{32} + \frac{29}{18522778914353192571} a^{31} + \frac{276875148564448654}{18522778914353192571} a^{30} + \frac{221324393649018938}{2058086546039243619} a^{29} - \frac{785486043759199760}{18522778914353192571} a^{28} - \frac{77438979543160985}{686028848679747873} a^{27} - \frac{1549642015380379793}{18522778914353192571} a^{26} + \frac{34418564756401154}{228676282893249291} a^{25} + \frac{478688264226209869}{18522778914353192571} a^{24} + \frac{2392014380605613}{228676282893249291} a^{23} + \frac{2300892057698177170}{18522778914353192571} a^{22} - \frac{109744330671822095}{686028848679747873} a^{21} - \frac{2299739320619568770}{18522778914353192571} a^{20} - \frac{100272171249191810}{228676282893249291} a^{19} + \frac{6980802394846018264}{18522778914353192571} a^{18} - \frac{4185349895932797067}{18522778914353192571} a^{17} + \frac{1680337412067705551}{18522778914353192571} a^{16} + \frac{3411372829675993756}{18522778914353192571} a^{15} - \frac{823366288152695864}{2058086546039243619} a^{14} + \frac{447390451912765366}{6174259638117730857} a^{13} + \frac{3560121999157895254}{18522778914353192571} a^{12} - \frac{9074211744453134680}{18522778914353192571} a^{11} - \frac{7492014879493068250}{18522778914353192571} a^{10} - \frac{1191145530890354810}{18522778914353192571} a^{9} + \frac{2691278932034290840}{6174259638117730857} a^{8} + \frac{1002128402944583902}{2058086546039243619} a^{7} + \frac{212279519286681970}{686028848679747873} a^{6} - \frac{79306832234735327}{228676282893249291} a^{5} + \frac{2640377979882334}{76225427631083097} a^{4} + \frac{2607073440680209}{25408475877027699} a^{3} + \frac{1518171999171319}{8469491959009233} a^{2} + \frac{1301995542244957}{2823163986336411} a + \frac{144441030578683}{941054662112137}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{1525}$, which has order $1525$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{6179798381201}{2058086546039243619} a^{39} - \frac{6179798381201}{2058086546039243619} a^{38} + \frac{43258588668407}{2058086546039243619} a^{37} - \frac{24719193524804}{686028848679747873} a^{36} - \frac{6179798381201}{686028848679747873} a^{35} + \frac{358428306109658}{2058086546039243619} a^{34} - \frac{747755604125321}{2058086546039243619} a^{33} + \frac{179214153054829}{2058086546039243619} a^{32} + \frac{2805628465065254}{2058086546039243619} a^{31} - \frac{803373789556130}{228676282893249291} a^{30} + \frac{1526289632852126}{686028848679747873} a^{29} + \frac{1025846531279366}{76225427631083097} a^{28} - \frac{59931684700887298}{2058086546039243619} a^{27} + \frac{197753548198432}{25408475877027699} a^{26} + \frac{222293527570181171}{2058086546039243619} a^{25} - \frac{7125307533524753}{25408475877027699} a^{24} + \frac{376430058794096513}{2058086546039243619} a^{23} + \frac{59962583692793303}{76225427631083097} a^{22} - \frac{5347330100866175692}{2058086546039243619} a^{21} + \frac{66006426509607881}{25408475877027699} a^{20} + \frac{10694808516893500208}{2058086546039243619} a^{19} - \frac{588070668286187429}{25408475877027699} a^{18} + \frac{5834230235522621281}{2058086546039243619} a^{17} + \frac{496991745412946822}{2058086546039243619} a^{16} - \frac{381441875281250524}{228676282893249291} a^{15} - \frac{1612253779469910091}{686028848679747873} a^{14} - \frac{3812008631443836850}{2058086546039243619} a^{13} - \frac{1737351438100561934}{2058086546039243619} a^{12} + \frac{313253979943078690}{2058086546039243619} a^{11} + \frac{2423698385711888597}{2058086546039243619} a^{10} + \frac{478804598777072279}{686028848679747873} a^{9} + \frac{20139962924334059}{228676282893249291} a^{8} + \frac{2327256198942633311}{2058086546039243619} a^{7} - \frac{13861287769033843}{25408475877027699} a^{6} - \frac{3831474996344620}{8469491959009233} a^{5} - \frac{587080846214095}{2823163986336411} a^{4} + \frac{43258588668407}{941054662112137} a^{3} + \frac{185393951436030}{941054662112137} a^{2} + \frac{222472741723236}{941054662112137} a + \frac{166854556292427}{941054662112137} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44099133482673080 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-11}, \sqrt{13})\), 4.0.2197.1, 4.4.265837.1, \(\Q(\zeta_{11})^+\), 8.0.70669310569.1, \(\Q(\zeta_{11})\), 10.10.79589952003133.1, 10.0.875489472034463.1, 20.0.766481815643182771348259698369.1, 20.0.2351977956823175708448011472615877.1, 20.20.284589332775604260722209388186521117.1

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 $20^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ $20^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ $20^{2}$

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
13Data not computed