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Properties

Label 36.0.86704309487...4197.1
Degree $36$
Signature $[0, 18]$
Discriminant $13^{18}\cdot 37^{35}$
Root discriminant $120.67$
Ramified primes $13, 37$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25311050383009, -25296715824916, 25296715824916, -25024359221149, 25024359221149, -23481005133136, 23481005133136, -19365394231768, 19365394231768, -13077655354678, 13077655354678, -6942346510972, 6942346510972, -2852140615168, 2852140615168, -904423521928, 904423521928, -221767776013, 221767776013, -42121527088, 42121527088, -6192277303, 6192277303, -701113963, 701113963, -60478240, 60478240, -3897877, 3897877, -181597, 181597, -5773, 5773, -112, 112, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 112*x^34 - 112*x^33 + 5773*x^32 - 5773*x^31 + 181597*x^30 - 181597*x^29 + 3897877*x^28 - 3897877*x^27 + 60478240*x^26 - 60478240*x^25 + 701113963*x^24 - 701113963*x^23 + 6192277303*x^22 - 6192277303*x^21 + 42121527088*x^20 - 42121527088*x^19 + 221767776013*x^18 - 221767776013*x^17 + 904423521928*x^16 - 904423521928*x^15 + 2852140615168*x^14 - 2852140615168*x^13 + 6942346510972*x^12 - 6942346510972*x^11 + 13077655354678*x^10 - 13077655354678*x^9 + 19365394231768*x^8 - 19365394231768*x^7 + 23481005133136*x^6 - 23481005133136*x^5 + 25024359221149*x^4 - 25024359221149*x^3 + 25296715824916*x^2 - 25296715824916*x + 25311050383009)
 
gp: K = bnfinit(x^36 - x^35 + 112*x^34 - 112*x^33 + 5773*x^32 - 5773*x^31 + 181597*x^30 - 181597*x^29 + 3897877*x^28 - 3897877*x^27 + 60478240*x^26 - 60478240*x^25 + 701113963*x^24 - 701113963*x^23 + 6192277303*x^22 - 6192277303*x^21 + 42121527088*x^20 - 42121527088*x^19 + 221767776013*x^18 - 221767776013*x^17 + 904423521928*x^16 - 904423521928*x^15 + 2852140615168*x^14 - 2852140615168*x^13 + 6942346510972*x^12 - 6942346510972*x^11 + 13077655354678*x^10 - 13077655354678*x^9 + 19365394231768*x^8 - 19365394231768*x^7 + 23481005133136*x^6 - 23481005133136*x^5 + 25024359221149*x^4 - 25024359221149*x^3 + 25296715824916*x^2 - 25296715824916*x + 25311050383009, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 112 x^{34} - 112 x^{33} + 5773 x^{32} - 5773 x^{31} + 181597 x^{30} - 181597 x^{29} + 3897877 x^{28} - 3897877 x^{27} + 60478240 x^{26} - 60478240 x^{25} + 701113963 x^{24} - 701113963 x^{23} + 6192277303 x^{22} - 6192277303 x^{21} + 42121527088 x^{20} - 42121527088 x^{19} + 221767776013 x^{18} - 221767776013 x^{17} + 904423521928 x^{16} - 904423521928 x^{15} + 2852140615168 x^{14} - 2852140615168 x^{13} + 6942346510972 x^{12} - 6942346510972 x^{11} + 13077655354678 x^{10} - 13077655354678 x^{9} + 19365394231768 x^{8} - 19365394231768 x^{7} + 23481005133136 x^{6} - 23481005133136 x^{5} + 25024359221149 x^{4} - 25024359221149 x^{3} + 25296715824916 x^{2} - 25296715824916 x + 25311050383009 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(867043094875691333993605639958113203169536870438724249863522795358139084197=13^{18}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.67$
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:  $13, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(481=13\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{481}(1,·)$, $\chi_{481}(389,·)$, $\chi_{481}(129,·)$, $\chi_{481}(142,·)$, $\chi_{481}(144,·)$, $\chi_{481}(402,·)$, $\chi_{481}(404,·)$, $\chi_{481}(27,·)$, $\chi_{481}(157,·)$, $\chi_{481}(287,·)$, $\chi_{481}(417,·)$, $\chi_{481}(40,·)$, $\chi_{481}(298,·)$, $\chi_{481}(391,·)$, $\chi_{481}(300,·)$, $\chi_{481}(51,·)$, $\chi_{481}(53,·)$, $\chi_{481}(311,·)$, $\chi_{481}(415,·)$, $\chi_{481}(196,·)$, $\chi_{481}(118,·)$, $\chi_{481}(326,·)$, $\chi_{481}(456,·)$, $\chi_{481}(207,·)$, $\chi_{481}(376,·)$, $\chi_{481}(467,·)$, $\chi_{481}(469,·)$, $\chi_{481}(220,·)$, $\chi_{481}(350,·)$, $\chi_{481}(272,·)$, $\chi_{481}(443,·)$, $\chi_{481}(103,·)$, $\chi_{481}(168,·)$, $\chi_{481}(116,·)$, $\chi_{481}(246,·)$, $\chi_{481}(248,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7020022316929} a^{19} - \frac{776174064480}{7020022316929} a^{18} + \frac{57}{7020022316929} a^{17} + \frac{206734419654}{7020022316929} a^{16} + \frac{1368}{7020022316929} a^{15} - \frac{2368497874714}{7020022316929} a^{14} + \frac{17955}{7020022316929} a^{13} + \frac{278318030110}{7020022316929} a^{12} + \frac{140049}{7020022316929} a^{11} - \frac{1040474922906}{7020022316929} a^{10} + \frac{660231}{7020022316929} a^{9} + \frac{1078044410182}{7020022316929} a^{8} + \frac{1828332}{7020022316929} a^{7} + \frac{175429518115}{7020022316929} a^{6} + \frac{2742498}{7020022316929} a^{5} + \frac{387385730704}{7020022316929} a^{4} + \frac{1869885}{7020022316929} a^{3} - \frac{1229680884569}{7020022316929} a^{2} + \frac{373977}{7020022316929} a + \frac{3288923272257}{7020022316929}$, $\frac{1}{7020022316929} a^{20} + \frac{60}{7020022316929} a^{18} + \frac{2328522193440}{7020022316929} a^{17} + \frac{1530}{7020022316929} a^{16} - \frac{585747522353}{7020022316929} a^{15} + \frac{21600}{7020022316929} a^{14} + \frac{1739346664445}{7020022316929} a^{13} + \frac{184275}{7020022316929} a^{12} + \frac{3335526107978}{7020022316929} a^{11} + \frac{972972}{7020022316929} a^{10} + \frac{647696604991}{7020022316929} a^{9} + \frac{3127410}{7020022316929} a^{8} - \frac{476301148804}{7020022316929} a^{7} + \frac{5773680}{7020022316929} a^{6} + \frac{2919800888790}{7020022316929} a^{5} + \frac{5412825}{7020022316929} a^{4} + \frac{496965814126}{7020022316929} a^{3} + \frac{1968300}{7020022316929} a^{2} - \frac{3385769704933}{7020022316929} a + \frac{118098}{7020022316929}$, $\frac{1}{7020022316929} a^{21} - \frac{241190156263}{7020022316929} a^{18} - \frac{1890}{7020022316929} a^{17} + \frac{1050231932265}{7020022316929} a^{16} - \frac{60480}{7020022316929} a^{15} + \frac{3448772808705}{7020022316929} a^{14} - \frac{893025}{7020022316929} a^{13} + \frac{676488935236}{7020022316929} a^{12} - \frac{7429968}{7020022316929} a^{11} - \frac{104008873010}{7020022316929} a^{10} - \frac{36486450}{7020022316929} a^{9} - \frac{1978764907363}{7020022316929} a^{8} - \frac{103926240}{7020022316929} a^{7} - \frac{585947881181}{7020022316929} a^{6} - \frac{159137055}{7020022316929} a^{5} - \frac{1686111077327}{7020022316929} a^{4} - \frac{110224800}{7020022316929} a^{3} + \frac{194860199917}{7020022316929} a^{2} - \frac{22320522}{7020022316929} a - \frac{774771461408}{7020022316929}$, $\frac{1}{7020022316929} a^{22} - \frac{2079}{7020022316929} a^{18} + \frac{758026205398}{7020022316929} a^{17} - \frac{70686}{7020022316929} a^{16} + \frac{3455857680826}{7020022316929} a^{15} - \frac{1122660}{7020022316929} a^{14} - \frac{108024907792}{7020022316929} a^{13} - \frac{10216206}{7020022316929} a^{12} - \frac{2011203458471}{7020022316929} a^{11} - \frac{56189133}{7020022316929} a^{10} - \frac{2946942448046}{7020022316929} a^{9} - \frac{185768154}{7020022316929} a^{8} - \frac{1647049766858}{7020022316929} a^{7} - \frac{350101521}{7020022316929} a^{6} + \frac{3181469214}{96164689273} a^{5} - \frac{333430020}{7020022316929} a^{4} - \frac{3283547063933}{7020022316929} a^{3} - \frac{122762871}{7020022316929} a^{2} - \frac{1470452914178}{7020022316929} a - \frac{7440174}{7020022316929}$, $\frac{1}{7020022316929} a^{23} + \frac{1697279045148}{7020022316929} a^{18} + \frac{47817}{7020022316929} a^{17} - \frac{1984667508106}{7020022316929} a^{16} + \frac{1721412}{7020022316929} a^{15} - \frac{3179462270969}{7020022316929} a^{14} + \frac{27112239}{7020022316929} a^{13} + \frac{970151152041}{7020022316929} a^{12} + \frac{234972738}{7020022316929} a^{11} + \frac{3092588761441}{7020022316929} a^{10} + \frac{1186852095}{7020022316929} a^{9} + \frac{220159901169}{7020022316929} a^{8} + \frac{3451000707}{7020022316929} a^{7} - \frac{90945066601}{7020022316929} a^{6} + \frac{5368223322}{7020022316929} a^{5} + \frac{1808842939777}{7020022316929} a^{4} + \frac{3764728044}{7020022316929} a^{3} - \frac{2688888570973}{7020022316929} a^{2} + \frac{770058009}{7020022316929} a + \frac{169746333457}{7020022316929}$, $\frac{1}{7020022316929} a^{24} + \frac{54648}{7020022316929} a^{18} - \frac{449260644536}{7020022316929} a^{17} + \frac{2090286}{7020022316929} a^{16} - \frac{1429809129934}{7020022316929} a^{15} + \frac{35411904}{7020022316929} a^{14} + \frac{3311963130}{96164689273} a^{13} + \frac{335675340}{7020022316929} a^{12} - \frac{1184753954871}{7020022316929} a^{11} + \frac{1898963352}{7020022316929} a^{10} + \frac{3121331851322}{7020022316929} a^{9} + \frac{6409001313}{7020022316929} a^{8} - \frac{2856964769145}{7020022316929} a^{7} + \frac{12270224736}{7020022316929} a^{6} - \frac{2340186614039}{7020022316929} a^{5} + \frac{11832002424}{7020022316929} a^{4} - \frac{2326853122698}{7020022316929} a^{3} + \frac{4400331480}{7020022316929} a^{2} + \frac{242153423112}{7020022316929} a + \frac{268909146}{7020022316929}$, $\frac{1}{7020022316929} a^{25} + \frac{936176173486}{7020022316929} a^{18} - \frac{1024650}{7020022316929} a^{17} + \frac{3183555873964}{7020022316929} a^{16} - \frac{39346560}{7020022316929} a^{15} - \frac{1257848857740}{7020022316929} a^{14} - \frac{645529500}{7020022316929} a^{13} + \frac{1679897378992}{7020022316929} a^{12} - \frac{5754434400}{7020022316929} a^{11} + \frac{814151693510}{7020022316929} a^{10} - \frac{29671302375}{7020022316929} a^{9} + \frac{3219413590016}{7020022316929} a^{8} - \frac{87644462400}{7020022316929} a^{7} + \frac{1890306335}{96164689273} a^{6} - \frac{138040028280}{7020022316929} a^{5} + \frac{205043222974}{7020022316929} a^{4} - \frac{97785144000}{7020022316929} a^{3} - \frac{2830506611493}{7020022316929} a^{2} - \frac{20168185950}{7020022316929} a + \frac{552398032651}{7020022316929}$, $\frac{1}{7020022316929} a^{26} - \frac{1210950}{7020022316929} a^{18} - \frac{1038329796235}{7020022316929} a^{17} - \frac{49406760}{7020022316929} a^{16} + \frac{2717229811419}{7020022316929} a^{15} - \frac{871884000}{7020022316929} a^{14} - \frac{1429870834112}{7020022316929} a^{13} - \frac{8500869000}{7020022316929} a^{12} + \frac{3234044435629}{7020022316929} a^{11} - \frac{49092518475}{7020022316929} a^{10} + \frac{2593155402413}{7020022316929} a^{9} - \frac{168317206200}{7020022316929} a^{8} + \frac{2183750936670}{7020022316929} a^{7} - \frac{326276430480}{7020022316929} a^{6} - \frac{236330074168}{7020022316929} a^{5} - \frac{317801718000}{7020022316929} a^{4} + \frac{3250395519482}{7020022316929} a^{3} - \frac{119175644250}{7020022316929} a^{2} + \frac{1768578458846}{7020022316929} a - \frac{7333885800}{7020022316929}$, $\frac{1}{7020022316929} a^{27} + \frac{1766301771575}{7020022316929} a^{18} + \frac{19617390}{7020022316929} a^{17} - \frac{273156499279}{7020022316929} a^{16} + \frac{784695600}{7020022316929} a^{15} + \frac{1486660344473}{7020022316929} a^{14} + \frac{13241738250}{7020022316929} a^{13} + \frac{1181170378839}{7020022316929} a^{12} + \frac{120499818075}{7020022316929} a^{11} + \frac{110727115562}{7020022316929} a^{10} + \frac{631189523250}{7020022316929} a^{9} + \frac{2672160078872}{7020022316929} a^{8} + \frac{1887742204920}{7020022316929} a^{7} + \frac{3243298696613}{7020022316929} a^{6} + \frac{3003226235100}{7020022316929} a^{5} + \frac{2029685064786}{7020022316929} a^{4} + \frac{2145161596500}{7020022316929} a^{3} - \frac{184745709153}{7020022316929} a^{2} + \frac{445533562350}{7020022316929} a + \frac{3235320066077}{7020022316929}$, $\frac{1}{7020022316929} a^{28} + \frac{23882040}{7020022316929} a^{18} - \frac{2672045042048}{7020022316929} a^{17} + \frac{1014986700}{7020022316929} a^{16} + \frac{73513853449}{7020022316929} a^{15} + \frac{18423288000}{7020022316929} a^{14} - \frac{3326332681993}{7020022316929} a^{13} + \frac{183369288375}{7020022316929} a^{12} + \frac{2860323752489}{7020022316929} a^{11} + \frac{1075766491800}{7020022316929} a^{10} + \frac{1594499590527}{7020022316929} a^{9} - \frac{3285575781109}{7020022316929} a^{8} + \frac{2959016696938}{7020022316929} a^{7} + \frac{292180690271}{7020022316929} a^{6} - \frac{1906749093191}{7020022316929} a^{5} + \frac{161605636571}{7020022316929} a^{4} - \frac{1273218472108}{7020022316929} a^{3} + \frac{2711943423000}{7020022316929} a^{2} + \frac{3017625513486}{7020022316929} a + \frac{167882211900}{7020022316929}$, $\frac{1}{7020022316929} a^{29} + \frac{2754196780137}{7020022316929} a^{18} - \frac{346289580}{7020022316929} a^{17} - \frac{1750365059579}{7020022316929} a^{16} - \frac{14247342720}{7020022316929} a^{15} - \frac{2243474323549}{7020022316929} a^{14} - \frac{245432739825}{7020022316929} a^{13} + \frac{3362964997804}{7020022316929} a^{12} - \frac{2268889328160}{7020022316929} a^{11} - \frac{2372561228598}{7020022316929} a^{10} + \frac{2006828018438}{7020022316929} a^{9} - \frac{19500711487}{7020022316929} a^{8} - \frac{1251983365435}{7020022316929} a^{7} + \frac{2843414148699}{7020022316929} a^{6} - \frac{2154640446988}{7020022316929} a^{5} - \frac{1738241603890}{7020022316929} a^{4} + \frac{175408959174}{7020022316929} a^{3} - \frac{3509654981052}{7020022316929} a^{2} - \frac{1743429144251}{7020022316929} a - \frac{2823509697831}{7020022316929}$, $\frac{1}{7020022316929} a^{30} - \frac{432861975}{7020022316929} a^{18} + \frac{2720931761979}{7020022316929} a^{17} - \frac{18922252050}{7020022316929} a^{16} - \frac{232685360092}{7020022316929} a^{15} - \frac{350618199750}{7020022316929} a^{14} + \frac{796978085845}{7020022316929} a^{13} + \frac{3474882741679}{7020022316929} a^{12} - \frac{2731196654477}{7020022316929} a^{11} + \frac{1937873802}{7020022316929} a^{10} - \frac{37165354895}{96164689273} a^{9} + \frac{3380052618869}{7020022316929} a^{8} - \frac{1915694733292}{7020022316929} a^{7} + \frac{1633421199459}{7020022316929} a^{6} + \frac{1653128548517}{7020022316929} a^{5} + \frac{3235476666009}{7020022316929} a^{4} + \frac{1036286314612}{7020022316929} a^{3} + \frac{1388606903057}{7020022316929} a^{2} - \frac{3318325902084}{7020022316929} a - \frac{3408008901570}{7020022316929}$, $\frac{1}{7020022316929} a^{31} - \frac{3491908355431}{7020022316929} a^{18} + \frac{5750880525}{7020022316929} a^{17} + \frac{1235019512360}{7020022316929} a^{16} + \frac{241536982050}{7020022316929} a^{15} - \frac{1279773868630}{7020022316929} a^{14} - \frac{2793125131054}{7020022316929} a^{13} - \frac{2189038540034}{7020022316929} a^{12} - \frac{2556376241784}{7020022316929} a^{11} + \frac{1622191638864}{7020022316929} a^{10} + \frac{1348032241005}{7020022316929} a^{9} + \frac{2052274286292}{7020022316929} a^{8} - \frac{213700137818}{7020022316929} a^{7} + \frac{1551693171641}{7020022316929} a^{6} - \frac{3045216498371}{7020022316929} a^{5} - \frac{2820208476465}{7020022316929} a^{4} + \frac{3488154579097}{7020022316929} a^{3} + \frac{913270411321}{7020022316929} a^{2} - \frac{2988099366362}{7020022316929} a + \frac{2913459844300}{7020022316929}$, $\frac{1}{7020022316929} a^{32} + \frac{7361127072}{7020022316929} a^{18} - \frac{3306851419014}{7020022316929} a^{17} + \frac{328490295588}{7020022316929} a^{16} + \frac{2035680849258}{7020022316929} a^{15} - \frac{836675576449}{7020022316929} a^{14} - \frac{793829269328}{7020022316929} a^{13} + \frac{121811403303}{7020022316929} a^{12} - \frac{2939225146873}{7020022316929} a^{11} + \frac{730868419818}{7020022316929} a^{10} + \frac{1608519247176}{7020022316929} a^{9} + \frac{2082711747433}{7020022316929} a^{8} + \frac{3022841646754}{7020022316929} a^{7} + \frac{2313077900548}{7020022316929} a^{6} + \frac{2896525394669}{7020022316929} a^{5} + \frac{2731073767398}{7020022316929} a^{4} - \frac{2208978811653}{7020022316929} a^{3} + \frac{3340235469456}{7020022316929} a^{2} - \frac{3327007853838}{7020022316929} a + \frac{208764716841}{7020022316929}$, $\frac{1}{7020022316929} a^{33} - \frac{2552258101055}{7020022316929} a^{18} - \frac{91093947516}{7020022316929} a^{17} + \frac{663586117470}{7020022316929} a^{16} + \frac{3133347222913}{7020022316929} a^{15} + \frac{3434150850087}{7020022316929} a^{14} + \frac{1333198847194}{7020022316929} a^{13} + \frac{1560118189169}{7020022316929} a^{12} + \frac{1755663701853}{7020022316929} a^{11} - \frac{671946987325}{7020022316929} a^{10} - \frac{106132811331}{7020022316929} a^{9} + \frac{570158066525}{7020022316929} a^{8} + \frac{1111677649537}{7020022316929} a^{7} + \frac{1853611918754}{7020022316929} a^{6} - \frac{2581037767583}{7020022316929} a^{5} + \frac{674641856262}{7020022316929} a^{4} - \frac{1877118376424}{7020022316929} a^{3} - \frac{274372974044}{7020022316929} a^{2} - \frac{834706052335}{7020022316929} a - \frac{3413890469534}{7020022316929}$, $\frac{1}{7020022316929} a^{34} - \frac{119122854444}{7020022316929} a^{18} - \frac{1278170777904}{7020022316929} a^{17} + \frac{1619786248801}{7020022316929} a^{16} - \frac{1047880737315}{7020022316929} a^{15} + \frac{2378188514319}{7020022316929} a^{14} + \frac{648637719182}{7020022316929} a^{13} + \frac{2734834468088}{7020022316929} a^{12} + \frac{2046536590477}{7020022316929} a^{11} - \frac{1706309492083}{7020022316929} a^{10} + \frac{1351542389999}{7020022316929} a^{9} + \frac{1484812215650}{7020022316929} a^{8} - \frac{302568303582}{7020022316929} a^{7} + \frac{2969624431300}{7020022316929} a^{6} - \frac{2559628477242}{7020022316929} a^{5} + \frac{2969624431300}{7020022316929} a^{4} + \frac{73178095632}{7020022316929} a^{3} - \frac{57676252245}{7020022316929} a^{2} - \frac{1940375792213}{7020022316929} a - \frac{1242446369795}{7020022316929}$, $\frac{1}{7020022316929} a^{35} + \frac{1428025163599}{7020022316929} a^{18} + \frac{1389766635180}{7020022316929} a^{17} - \frac{2333768551662}{7020022316929} a^{16} - \frac{3142282212585}{7020022316929} a^{15} + \frac{3306262322555}{7020022316929} a^{14} + \frac{478879346763}{7020022316929} a^{13} + \frac{2000279120196}{7020022316929} a^{12} + \frac{1757307512369}{7020022316929} a^{11} + \frac{523005827517}{7020022316929} a^{10} - \frac{2243914240302}{7020022316929} a^{9} + \frac{575832016526}{7020022316929} a^{8} + \frac{2903953016483}{7020022316929} a^{7} + \frac{1829979708653}{7020022316929} a^{6} - \frac{638893849390}{7020022316929} a^{5} - \frac{83920055684}{7020022316929} a^{4} + \frac{672889609525}{7020022316929} a^{3} - \frac{3415074423691}{7020022316929} a^{2} - \frac{1096333197441}{7020022316929} a + \frac{3270773809523}{7020022316929}$

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:  $17$
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

$C_{36}$ (as 36T1):

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

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.8560357.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.858774378063695536612117.1, \(\Q(\zeta_{37})^+\)

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 $36$ $18^{2}$ $36$ $18^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ R $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$ $36$

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
13Data not computed
37Data not computed