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Properties

Label 36.36.8216251102...5625.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $128.45$
Ramified primes $5, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-127399, -1560887, 14619917, 74320909, -408560825, -611541017, 3506699274, 1972347494, -14234297497, -2537946464, 32430382933, -541428095, -45930613858, 6151376608, 42972000803, -9230490750, -27502514595, 7632713374, 12253153691, -4065525074, -3815197767, 1467238032, 822660108, -365138709, -119635996, 62715081, 11035141, -7345274, -539471, 570596, 1248, -27862, 1348, 767, -65, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 65*x^34 + 767*x^33 + 1348*x^32 - 27862*x^31 + 1248*x^30 + 570596*x^29 - 539471*x^28 - 7345274*x^27 + 11035141*x^26 + 62715081*x^25 - 119635996*x^24 - 365138709*x^23 + 822660108*x^22 + 1467238032*x^21 - 3815197767*x^20 - 4065525074*x^19 + 12253153691*x^18 + 7632713374*x^17 - 27502514595*x^16 - 9230490750*x^15 + 42972000803*x^14 + 6151376608*x^13 - 45930613858*x^12 - 541428095*x^11 + 32430382933*x^10 - 2537946464*x^9 - 14234297497*x^8 + 1972347494*x^7 + 3506699274*x^6 - 611541017*x^5 - 408560825*x^4 + 74320909*x^3 + 14619917*x^2 - 1560887*x - 127399)
 
gp: K = bnfinit(x^36 - 9*x^35 - 65*x^34 + 767*x^33 + 1348*x^32 - 27862*x^31 + 1248*x^30 + 570596*x^29 - 539471*x^28 - 7345274*x^27 + 11035141*x^26 + 62715081*x^25 - 119635996*x^24 - 365138709*x^23 + 822660108*x^22 + 1467238032*x^21 - 3815197767*x^20 - 4065525074*x^19 + 12253153691*x^18 + 7632713374*x^17 - 27502514595*x^16 - 9230490750*x^15 + 42972000803*x^14 + 6151376608*x^13 - 45930613858*x^12 - 541428095*x^11 + 32430382933*x^10 - 2537946464*x^9 - 14234297497*x^8 + 1972347494*x^7 + 3506699274*x^6 - 611541017*x^5 - 408560825*x^4 + 74320909*x^3 + 14619917*x^2 - 1560887*x - 127399, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 65 x^{34} + 767 x^{33} + 1348 x^{32} - 27862 x^{31} + 1248 x^{30} + 570596 x^{29} - 539471 x^{28} - 7345274 x^{27} + 11035141 x^{26} + 62715081 x^{25} - 119635996 x^{24} - 365138709 x^{23} + 822660108 x^{22} + 1467238032 x^{21} - 3815197767 x^{20} - 4065525074 x^{19} + 12253153691 x^{18} + 7632713374 x^{17} - 27502514595 x^{16} - 9230490750 x^{15} + 42972000803 x^{14} + 6151376608 x^{13} - 45930613858 x^{12} - 541428095 x^{11} + 32430382933 x^{10} - 2537946464 x^{9} - 14234297497 x^{8} + 1972347494 x^{7} + 3506699274 x^{6} - 611541017 x^{5} - 408560825 x^{4} + 74320909 x^{3} + 14619917 x^{2} - 1560887 x - 127399 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8216251102044589607143313691057181287626972114083156077958643436431884765625=5^{27}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.45$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(4,·)$, $\chi_{455}(261,·)$, $\chi_{455}(134,·)$, $\chi_{455}(137,·)$, $\chi_{455}(394,·)$, $\chi_{455}(267,·)$, $\chi_{455}(16,·)$, $\chi_{455}(18,·)$, $\chi_{455}(148,·)$, $\chi_{455}(408,·)$, $\chi_{455}(158,·)$, $\chi_{455}(389,·)$, $\chi_{455}(288,·)$, $\chi_{455}(37,·)$, $\chi_{455}(114,·)$, $\chi_{455}(177,·)$, $\chi_{455}(179,·)$, $\chi_{455}(309,·)$, $\chi_{455}(58,·)$, $\chi_{455}(191,·)$, $\chi_{455}(64,·)$, $\chi_{455}(324,·)$, $\chi_{455}(326,·)$, $\chi_{455}(72,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(93,·)$, $\chi_{455}(102,·)$, $\chi_{455}(232,·)$, $\chi_{455}(242,·)$, $\chi_{455}(372,·)$, $\chi_{455}(123,·)$, $\chi_{455}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{19} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{7802} a^{30} + \frac{1017}{7802} a^{29} - \frac{75}{3901} a^{28} + \frac{387}{7802} a^{27} - \frac{1607}{7802} a^{26} + \frac{1359}{7802} a^{25} - \frac{159}{3901} a^{24} - \frac{341}{7802} a^{23} - \frac{17}{166} a^{22} + \frac{76}{3901} a^{21} - \frac{246}{3901} a^{20} - \frac{30}{3901} a^{19} - \frac{2521}{7802} a^{18} + \frac{2231}{7802} a^{17} + \frac{25}{83} a^{16} + \frac{1907}{3901} a^{15} - \frac{417}{7802} a^{14} - \frac{138}{3901} a^{13} + \frac{715}{3901} a^{12} - \frac{253}{3901} a^{11} - \frac{1154}{3901} a^{10} + \frac{3341}{7802} a^{9} - \frac{1543}{7802} a^{8} - \frac{1095}{3901} a^{7} - \frac{783}{7802} a^{6} + \frac{277}{7802} a^{5} - \frac{1484}{3901} a^{4} + \frac{22}{47} a^{3} - \frac{1271}{3901} a^{2} + \frac{401}{7802} a - \frac{171}{3901}$, $\frac{1}{7802} a^{31} - \frac{337}{3901} a^{29} + \frac{399}{3901} a^{28} - \frac{1185}{7802} a^{27} + \frac{1159}{7802} a^{26} - \frac{1467}{7802} a^{25} - \frac{359}{3901} a^{24} - \frac{1191}{7802} a^{23} + \frac{1327}{7802} a^{22} + \frac{482}{3901} a^{21} + \frac{488}{3901} a^{20} - \frac{8}{3901} a^{19} + \frac{3131}{7802} a^{18} - \frac{48}{3901} a^{17} + \frac{638}{3901} a^{16} - \frac{1661}{7802} a^{15} - \frac{698}{3901} a^{14} - \frac{2651}{7802} a^{13} - \frac{1822}{3901} a^{12} + \frac{1263}{7802} a^{11} + \frac{2175}{7802} a^{10} + \frac{1166}{3901} a^{9} - \frac{1161}{7802} a^{8} + \frac{2877}{7802} a^{7} - \frac{3117}{7802} a^{6} - \frac{3805}{7802} a^{5} - \frac{1167}{7802} a^{4} - \frac{1437}{3901} a^{3} + \frac{3153}{7802} a^{2} - \frac{2455}{7802} a - \frac{1638}{3901}$, $\frac{1}{15604} a^{32} - \frac{1}{15604} a^{31} + \frac{177}{7802} a^{29} - \frac{1657}{15604} a^{28} + \frac{1815}{15604} a^{27} + \frac{2635}{15604} a^{26} + \frac{3881}{15604} a^{25} + \frac{3651}{15604} a^{24} - \frac{529}{7802} a^{23} - \frac{551}{15604} a^{22} + \frac{11}{166} a^{21} + \frac{1443}{7802} a^{20} + \frac{1717}{15604} a^{19} + \frac{6257}{15604} a^{18} - \frac{4621}{15604} a^{17} - \frac{1686}{3901} a^{16} + \frac{71}{7802} a^{15} + \frac{615}{3901} a^{14} - \frac{7571}{15604} a^{13} + \frac{1279}{15604} a^{12} - \frac{745}{15604} a^{11} - \frac{3369}{7802} a^{10} + \frac{2633}{7802} a^{9} + \frac{5623}{15604} a^{8} - \frac{1869}{3901} a^{7} - \frac{1424}{3901} a^{6} + \frac{522}{3901} a^{5} - \frac{463}{7802} a^{4} - \frac{929}{7802} a^{3} - \frac{619}{3901} a^{2} + \frac{4185}{15604} a - \frac{4875}{15604}$, $\frac{1}{11136590404} a^{33} - \frac{58375}{11136590404} a^{32} + \frac{147155}{2784147601} a^{31} - \frac{118108}{2784147601} a^{30} + \frac{682948219}{11136590404} a^{29} - \frac{1326949859}{11136590404} a^{28} + \frac{79590069}{11136590404} a^{27} - \frac{2012224687}{11136590404} a^{26} + \frac{2140618219}{11136590404} a^{25} + \frac{160393935}{2784147601} a^{24} - \frac{1276836201}{11136590404} a^{23} - \frac{784583969}{5568295202} a^{22} - \frac{690179898}{2784147601} a^{21} + \frac{2226123909}{11136590404} a^{20} - \frac{5561891243}{11136590404} a^{19} - \frac{986997507}{11136590404} a^{18} - \frac{321601486}{2784147601} a^{17} + \frac{555569279}{2784147601} a^{16} - \frac{316094809}{2784147601} a^{15} + \frac{4180099107}{11136590404} a^{14} - \frac{4345632559}{11136590404} a^{13} - \frac{5386713387}{11136590404} a^{12} + \frac{942297633}{5568295202} a^{11} - \frac{788522076}{2784147601} a^{10} + \frac{4927876755}{11136590404} a^{9} + \frac{682603047}{2784147601} a^{8} + \frac{2747774509}{5568295202} a^{7} - \frac{439157177}{5568295202} a^{6} - \frac{1116746316}{2784147601} a^{5} + \frac{698108179}{2784147601} a^{4} - \frac{1385351029}{5568295202} a^{3} - \frac{1345144067}{11136590404} a^{2} - \frac{5020275379}{11136590404} a - \frac{1314840437}{2784147601}$, $\frac{1}{44546361616} a^{34} - \frac{1}{22273180808} a^{33} + \frac{246755}{11136590404} a^{32} + \frac{2190389}{44546361616} a^{31} - \frac{2071237}{44546361616} a^{30} - \frac{1115211297}{22273180808} a^{29} - \frac{2335224917}{44546361616} a^{28} + \frac{792119875}{44546361616} a^{27} + \frac{4786314127}{44546361616} a^{26} - \frac{104106016}{2784147601} a^{25} + \frac{5195053965}{22273180808} a^{24} - \frac{1238645297}{44546361616} a^{23} + \frac{6629489707}{44546361616} a^{22} + \frac{8052781173}{44546361616} a^{21} + \frac{822636155}{5568295202} a^{20} + \frac{5323104167}{44546361616} a^{19} + \frac{5178914225}{22273180808} a^{18} - \frac{16832868543}{44546361616} a^{17} - \frac{1421105411}{5568295202} a^{16} - \frac{6599839119}{44546361616} a^{15} + \frac{3251421375}{11136590404} a^{14} + \frac{6556706745}{44546361616} a^{13} + \frac{1289076555}{22273180808} a^{12} - \frac{15012372651}{44546361616} a^{11} + \frac{11743999123}{44546361616} a^{10} + \frac{16103677477}{44546361616} a^{9} + \frac{19599783905}{44546361616} a^{8} + \frac{7068711751}{22273180808} a^{7} - \frac{3318644215}{11136590404} a^{6} + \frac{3833500039}{11136590404} a^{5} - \frac{7422713707}{22273180808} a^{4} + \frac{13300405993}{44546361616} a^{3} + \frac{295916679}{11136590404} a^{2} - \frac{2782578793}{11136590404} a + \frac{19390123205}{44546361616}$, $\frac{1}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{35} + \frac{4335850082440593340164540635227416714986969397120794872269533664026477691657786104589416032730924355}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{34} + \frac{97680051774829151567538035794124509066831170947160646628056384700821795079049687135936174148787057467}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{33} + \frac{256324339825335605161489437931920150074694777781680745221339639136411377944423161468785155929389347726242265}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{32} + \frac{359347503491064283071629029073959985674955635343039814974251111685502483478587209281905069934837741998293447}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{31} - \frac{34856142568728617730778312088059203331588635273573626237916684000178994059963059019346285894226652593769625}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{30} + \frac{404882624990381886469075053211565238785346029275250044766481255803185489084610063116421148284266500034570087579}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{29} - \frac{2009800273868536862055230235504488247449546773207967401625750209298234840112578849567970805250242435881543897305}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{28} - \frac{1733063006936299000394879508034372215435389823166845603650690174032615951095217437295486536512676606379606047445}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{27} - \frac{665872406130885366182415301598947800031868791380631826873470992714724407918199589407320662386025836643522769019}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{26} - \frac{564028530918735197916092323197426265694943504410754408279882261045292664209445827773365139849513010016054959525}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{25} - \frac{1054770833120743423744753340911940370363382955137215145894574373335457208032831377632753264203334666715080944005}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{24} - \frac{1192423132126203413110322864998618017247624138529534238424106399682845443871200482435195453738356899381830156637}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{23} - \frac{1169566219394157762981218503633220891185598618618071992565581500822307088775257183040832787164731138546289998847}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{22} - \frac{171956381899293863651475474028405129215915016488656365119627758481896101335707767452071426769648835181429848665}{1470923831877954015111793548654797398499316647520315742561442091169549629401697602609877286872300700959491224462} a^{21} + \frac{8185826137612388551954723922824676016044195884377767907273688752476860885676847831979510794896525705182574093}{141775791024381109890293354087209387807163050363403926993873936498269843797753985793723111987691633827420840912} a^{20} - \frac{2427006171959470388450088565975628121604345236073173006730210055487594132470588170889985862476689983930592881381}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{19} + \frac{4422519663490506079513817480520214460332554885613345054910562493394185802455053226707175588215538429155889922989}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{18} + \frac{292732790760877130907049423122912531279240967175868112814239642628327895686151502663773266609416261848366626887}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{17} + \frac{232108753153711547620951149961355966651079421945952185614784794699197966933403317327850103006505476006918939241}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{16} - \frac{966901237677558980263834251794375735290514483960640662599333501139381821853150411340376917808487356665500545759}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{15} + \frac{1875431192334073006144782977352869891245258349246509905550983618553251446336299995236812822652466908612588673361}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{14} + \frac{2470514292367708269109657935768324153345019970536517192939906649013540369203906765693303660117409251526586856985}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{13} + \frac{924084802030529981096472427732197015595370420282186581716038103456852039174724780521850033125862837255378774441}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{12} - \frac{4899474486982815624840198230553215321575217001986059027229866782171876720504337028268191766566630464777994741193}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{11} - \frac{1391591640053559664856835688654105173069107171432471440909415615084788442143321274786872279531665507867339086075}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{10} + \frac{38939570541351458190730742228818341684097447402393725011199824558816191954366423477216866538793885021755432697}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{9} + \frac{2916432694932604238887896254352923208247072467512771762948484424400186739732216914181657996766749472837905774785}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{8} + \frac{815105324759706477061979217444312606561810344362258444340040373684056364783122581269238696240027198449499172057}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{7} - \frac{580613438357022841300263960152591846514383213318786879114927412587726932513394323635436677396264301779301321411}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{6} + \frac{327248798634774490003088471946991925085806304522484438512608448548985687935832754261699077659658743555063152877}{5883695327511816060447174194619189593997266590081262970245768364678198517606790410439509147489202803837964897848} a^{5} + \frac{3340093719524400217229775451496392524977752654157346383285443653863135829228647122463746013124938412228209403369}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a^{4} + \frac{1038250961130911436736816653145099717174429751830251846171292242249622159795980757642992954740134461288983955031}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{3} + \frac{1459563704975421139408272333963118879803875322428945339254716394227013359824107551260409416177389605225787218887}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924} a^{2} + \frac{4453891889892979199592296609025462356564501897772118826993966865360914006250762983934290182181851093412118477665}{11767390655023632120894348389238379187994533180162525940491536729356397035213580820879018294978405607675929795696} a - \frac{561201648908809262166272387221965776645486763275780635899312217177738159275166318131980728501935301110217827743}{2941847663755908030223587097309594796998633295040631485122884182339099258803395205219754573744601401918982448924}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $35$
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:  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:  \( 291218721617730900000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{65}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 4.4.274625.2, 6.6.46411625.1, 6.6.659374625.1, 6.6.111434311625.2, 6.6.111434311625.1, 9.9.567869252041.1, 12.12.3500313269603515625.1, 12.12.119400055839784712890625.1, 12.12.20178609436923616478515625.1, 12.12.20178609436923616478515625.4, 18.18.1383747355171268959993166322265625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$5$5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.12.11.8$x^{12} + 104$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.8$x^{12} + 104$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.8$x^{12} + 104$$12$$1$$11$$C_{12}$$[\ ]_{12}$