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Properties

Label 36.36.1139470288...8125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 5^{18}\cdot 37^{35}$
Root discriminant $129.62$
Ramified primes $3, 5, 37$
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![20763315901, 2521857323331, -2521857323331, -33710486785725, 33710486785725, 120276975677763, -120276975677763, -187697949249213, 187697949249213, 165189985562947, -165189985562947, -93059821276861, 93059821276861, 36065082143043, -36065082143043, -10050954792637, 10050954792637, 2071459328323, -2071459328323, -321122406077, 321122406077, 37764854083, -37764854083, -3372420797, 3372420797, 227090755, -227090755, -11338429, 11338429, 406851, -406851, -9917, 9917, 147, -147, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 147*x^34 + 147*x^33 + 9917*x^32 - 9917*x^31 - 406851*x^30 + 406851*x^29 + 11338429*x^28 - 11338429*x^27 - 227090755*x^26 + 227090755*x^25 + 3372420797*x^24 - 3372420797*x^23 - 37764854083*x^22 + 37764854083*x^21 + 321122406077*x^20 - 321122406077*x^19 - 2071459328323*x^18 + 2071459328323*x^17 + 10050954792637*x^16 - 10050954792637*x^15 - 36065082143043*x^14 + 36065082143043*x^13 + 93059821276861*x^12 - 93059821276861*x^11 - 165189985562947*x^10 + 165189985562947*x^9 + 187697949249213*x^8 - 187697949249213*x^7 - 120276975677763*x^6 + 120276975677763*x^5 + 33710486785725*x^4 - 33710486785725*x^3 - 2521857323331*x^2 + 2521857323331*x + 20763315901)
 
gp: K = bnfinit(x^36 - x^35 - 147*x^34 + 147*x^33 + 9917*x^32 - 9917*x^31 - 406851*x^30 + 406851*x^29 + 11338429*x^28 - 11338429*x^27 - 227090755*x^26 + 227090755*x^25 + 3372420797*x^24 - 3372420797*x^23 - 37764854083*x^22 + 37764854083*x^21 + 321122406077*x^20 - 321122406077*x^19 - 2071459328323*x^18 + 2071459328323*x^17 + 10050954792637*x^16 - 10050954792637*x^15 - 36065082143043*x^14 + 36065082143043*x^13 + 93059821276861*x^12 - 93059821276861*x^11 - 165189985562947*x^10 + 165189985562947*x^9 + 187697949249213*x^8 - 187697949249213*x^7 - 120276975677763*x^6 + 120276975677763*x^5 + 33710486785725*x^4 - 33710486785725*x^3 - 2521857323331*x^2 + 2521857323331*x + 20763315901, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 147 x^{34} + 147 x^{33} + 9917 x^{32} - 9917 x^{31} - 406851 x^{30} + 406851 x^{29} + 11338429 x^{28} - 11338429 x^{27} - 227090755 x^{26} + 227090755 x^{25} + 3372420797 x^{24} - 3372420797 x^{23} - 37764854083 x^{22} + 37764854083 x^{21} + 321122406077 x^{20} - 321122406077 x^{19} - 2071459328323 x^{18} + 2071459328323 x^{17} + 10050954792637 x^{16} - 10050954792637 x^{15} - 36065082143043 x^{14} + 36065082143043 x^{13} + 93059821276861 x^{12} - 93059821276861 x^{11} - 165189985562947 x^{10} + 165189985562947 x^{9} + 187697949249213 x^{8} - 187697949249213 x^{7} - 120276975677763 x^{6} + 120276975677763 x^{5} + 33710486785725 x^{4} - 33710486785725 x^{3} - 2521857323331 x^{2} + 2521857323331 x + 20763315901 \)

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:  \(11394702880804821835353664975608445751870049996765358066064135272979736328125=3^{18}\cdot 5^{18}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.62$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(555=3\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{555}(256,·)$, $\chi_{555}(1,·)$, $\chi_{555}(389,·)$, $\chi_{555}(134,·)$, $\chi_{555}(391,·)$, $\chi_{555}(136,·)$, $\chi_{555}(524,·)$, $\chi_{555}(14,·)$, $\chi_{555}(271,·)$, $\chi_{555}(16,·)$, $\chi_{555}(406,·)$, $\chi_{555}(151,·)$, $\chi_{555}(29,·)$, $\chi_{555}(286,·)$, $\chi_{555}(46,·)$, $\chi_{555}(179,·)$, $\chi_{555}(181,·)$, $\chi_{555}(314,·)$, $\chi_{555}(59,·)$, $\chi_{555}(449,·)$, $\chi_{555}(451,·)$, $\chi_{555}(196,·)$, $\chi_{555}(464,·)$, $\chi_{555}(209,·)$, $\chi_{555}(211,·)$, $\chi_{555}(89,·)$, $\chi_{555}(479,·)$, $\chi_{555}(224,·)$, $\chi_{555}(226,·)$, $\chi_{555}(361,·)$, $\chi_{555}(494,·)$, $\chi_{555}(239,·)$, $\chi_{555}(119,·)$, $\chi_{555}(121,·)$, $\chi_{555}(254,·)$, $\chi_{555}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{70770402683} a^{19} + \frac{1087911260}{70770402683} a^{18} - \frac{76}{70770402683} a^{17} - \frac{7559208037}{70770402683} a^{16} + \frac{2432}{70770402683} a^{15} + \frac{14465033061}{70770402683} a^{14} - \frac{42560}{70770402683} a^{13} - \frac{12264828669}{70770402683} a^{12} + \frac{442624}{70770402683} a^{11} - \frac{15790934675}{70770402683} a^{10} - \frac{2782208}{70770402683} a^{9} - \frac{5088426847}{70770402683} a^{8} + \frac{10272768}{70770402683} a^{7} + \frac{103905150}{70770402683} a^{6} - \frac{20545536}{70770402683} a^{5} - \frac{13948372185}{70770402683} a^{4} + \frac{18677760}{70770402683} a^{3} + \frac{8369023311}{70770402683} a^{2} - \frac{4980736}{70770402683} a + \frac{30626942100}{70770402683}$, $\frac{1}{70770402683} a^{20} - \frac{80}{70770402683} a^{18} + \frac{4351645040}{70770402683} a^{17} + \frac{2720}{70770402683} a^{16} - \frac{12830251988}{70770402683} a^{15} - \frac{51200}{70770402683} a^{14} + \frac{5395042249}{70770402683} a^{13} + \frac{582400}{70770402683} a^{12} - \frac{29604625783}{70770402683} a^{11} - \frac{4100096}{70770402683} a^{10} + \frac{10970086006}{70770402683} a^{9} + \frac{17571840}{70770402683} a^{8} - \frac{10194171219}{70770402683} a^{7} - \frac{43253760}{70770402683} a^{6} + \frac{6647780553}{70770402683} a^{5} + \frac{54067200}{70770402683} a^{4} + \frac{2512594037}{70770402683} a^{3} - \frac{26214400}{70770402683} a^{2} + \frac{22752602882}{70770402683} a + \frac{2097152}{70770402683}$, $\frac{1}{70770402683} a^{21} + \frac{20614143157}{70770402683} a^{18} - \frac{3360}{70770402683} a^{17} + \frac{19366729199}{70770402683} a^{16} + \frac{143360}{70770402683} a^{15} + \frac{30271244201}{70770402683} a^{14} - \frac{2822400}{70770402683} a^{13} - \frac{20005281741}{70770402683} a^{12} + \frac{31309824}{70770402683} a^{11} + \frac{21562560300}{70770402683} a^{10} - \frac{205004800}{70770402683} a^{9} + \frac{7354097119}{70770402683} a^{8} + \frac{778567680}{70770402683} a^{7} + \frac{14960192553}{70770402683} a^{6} - \frac{1589575680}{70770402683} a^{5} + \frac{18969262165}{70770402683} a^{4} + \frac{1468006400}{70770402683} a^{3} - \frac{15429559068}{70770402683} a^{2} - \frac{396361728}{70770402683} a - \frac{26808725905}{70770402683}$, $\frac{1}{70770402683} a^{22} - \frac{3696}{70770402683} a^{18} + \frac{29092750105}{70770402683} a^{17} + \frac{167552}{70770402683} a^{16} + \frac{2120185941}{70770402683} a^{15} - \frac{3548160}{70770402683} a^{14} - \frac{22754580972}{70770402683} a^{13} + \frac{43051008}{70770402683} a^{12} - \frac{6461049844}{70770402683} a^{11} - \frac{315707392}{70770402683} a^{10} + \frac{11631525794}{70770402683} a^{9} + \frac{1391689728}{70770402683} a^{8} - \frac{833390247}{70770402683} a^{7} - \frac{3497066496}{70770402683} a^{6} - \frac{276903766}{969457571} a^{5} + \frac{4440719360}{70770402683} a^{4} - \frac{11976799303}{70770402683} a^{3} - \frac{2179989504}{70770402683} a^{2} + \frac{19450806613}{70770402683} a + \frac{176160768}{70770402683}$, $\frac{1}{70770402683} a^{23} + \frac{16099814134}{70770402683} a^{18} - \frac{113344}{70770402683} a^{17} + \frac{17596340974}{70770402683} a^{16} + \frac{5440512}{70770402683} a^{15} + \frac{8353586819}{70770402683} a^{14} - \frac{114250752}{70770402683} a^{13} + \frac{26560309335}{70770402683} a^{12} + \frac{1320230912}{70770402683} a^{11} + \frac{33919180469}{70770402683} a^{10} - \frac{8891351040}{70770402683} a^{9} + \frac{17268096919}{70770402683} a^{8} + \frac{34471084032}{70770402683} a^{7} + \frac{9967446067}{70770402683} a^{6} - \frac{725179013}{70770402683} a^{5} + \frac{26463160844}{70770402683} a^{4} - \frac{3917391227}{70770402683} a^{3} + \frac{24694991598}{70770402683} a^{2} - \frac{18232639488}{70770402683} a + \frac{35304111483}{70770402683}$, $\frac{1}{70770402683} a^{24} - \frac{129536}{70770402683} a^{18} - \frac{32685033136}{70770402683} a^{17} + \frac{6606336}{70770402683} a^{16} - \frac{10361703370}{70770402683} a^{15} - \frac{149225472}{70770402683} a^{14} - \frac{481634618}{969457571} a^{13} + \frac{1886044160}{70770402683} a^{12} + \frac{24715694855}{70770402683} a^{11} - \frac{14226161664}{70770402683} a^{10} - \frac{15871939814}{70770402683} a^{9} - \frac{6752675195}{70770402683} a^{8} + \frac{7121484642}{70770402683} a^{7} - \frac{21877667082}{70770402683} a^{6} + \frac{32155083694}{70770402683} a^{5} - \frac{2201743473}{70770402683} a^{4} + \frac{25954189251}{70770402683} a^{3} - \frac{33416108677}{70770402683} a^{2} + \frac{6660164369}{70770402683} a + \frac{8489271296}{70770402683}$, $\frac{1}{70770402683} a^{25} - \frac{12883799629}{70770402683} a^{18} - \frac{3238400}{70770402683} a^{17} - \frac{20642462214}{70770402683} a^{16} + \frac{165806080}{70770402683} a^{15} - \frac{9818172526}{70770402683} a^{14} - \frac{3627008000}{70770402683} a^{13} + \frac{12639057938}{70770402683} a^{12} - \frac{27660821883}{70770402683} a^{11} - \frac{33437253865}{70770402683} a^{10} - \frac{13296757268}{70770402683} a^{9} + \frac{28192021112}{70770402683} a^{8} + \frac{34948360272}{70770402683} a^{7} - \frac{349785183}{969457571} a^{6} + \frac{25687007185}{70770402683} a^{5} - \frac{22004669919}{70770402683} a^{4} - \frac{20167480539}{70770402683} a^{3} - \frac{35334922812}{70770402683} a^{2} + \frac{238276947}{70770402683} a - \frac{26432140697}{70770402683}$, $\frac{1}{70770402683} a^{26} - \frac{3827200}{70770402683} a^{18} - \frac{9025596456}{70770402683} a^{17} + \frac{208199680}{70770402683} a^{16} - \frac{27705863367}{70770402683} a^{15} - \frac{4898816000}{70770402683} a^{14} + \frac{7206835582}{70770402683} a^{13} - \frac{7085794683}{70770402683} a^{12} - \frac{33558463509}{70770402683} a^{11} + \frac{5021337181}{70770402683} a^{10} - \frac{31706434854}{70770402683} a^{9} - \frac{22954684256}{70770402683} a^{8} - \frac{350365299}{70770402683} a^{7} + \frac{9245360486}{70770402683} a^{6} - \frac{1602173356}{70770402683} a^{5} + \frac{22918691602}{70770402683} a^{4} + \frac{23321428694}{70770402683} a^{3} - \frac{11459345801}{70770402683} a^{2} + \frac{19260514560}{70770402683} a + \frac{25619163668}{70770402683}$, $\frac{1}{70770402683} a^{27} + \frac{9847626605}{70770402683} a^{18} - \frac{82667520}{70770402683} a^{17} + \frac{28830129901}{70770402683} a^{16} + \frac{4408934400}{70770402683} a^{15} + \frac{9616701934}{70770402683} a^{14} - \frac{28430621317}{70770402683} a^{13} - \frac{30082504216}{70770402683} a^{12} + \frac{542245589}{70770402683} a^{11} - \frac{3819420174}{70770402683} a^{10} + \frac{15309663277}{70770402683} a^{9} + \frac{30290298875}{70770402683} a^{8} - \frac{23160841662}{70770402683} a^{7} + \frac{5295230867}{70770402683} a^{6} + \frac{16960693215}{70770402683} a^{5} - \frac{10405176161}{70770402683} a^{4} - \frac{6042983631}{70770402683} a^{3} - \frac{31273925210}{70770402683} a^{2} + \frac{584666195}{70770402683} a - \frac{28209873874}{70770402683}$, $\frac{1}{70770402683} a^{28} - \frac{100638720}{70770402683} a^{18} - \frac{1224677632}{70770402683} a^{17} + \frac{5702860800}{70770402683} a^{16} - \frac{19415094572}{70770402683} a^{15} + \frac{3521989366}{70770402683} a^{14} - \frac{17418884142}{70770402683} a^{13} - \frac{8405765758}{70770402683} a^{12} + \frac{20173816959}{70770402683} a^{11} - \frac{31754120434}{70770402683} a^{10} - \frac{19423759271}{70770402683} a^{9} + \frac{3413397709}{70770402683} a^{8} + \frac{20094639162}{70770402683} a^{7} - \frac{26397827782}{70770402683} a^{6} + \frac{30766409932}{70770402683} a^{5} - \frac{29461780332}{70770402683} a^{4} + \frac{2067762843}{70770402683} a^{3} + \frac{481863679}{70770402683} a^{2} - \frac{18228883306}{70770402683} a + \frac{9620948761}{70770402683}$, $\frac{1}{70770402683} a^{29} + \frac{7050950271}{70770402683} a^{18} - \frac{1945681920}{70770402683} a^{17} + \frac{21673320559}{70770402683} a^{16} - \frac{34806254326}{70770402683} a^{15} + \frac{23795910222}{70770402683} a^{14} + \frac{25404874705}{70770402683} a^{13} + \frac{4497737265}{70770402683} a^{12} - \frac{1224606761}{70770402683} a^{11} - \frac{3051469996}{70770402683} a^{10} - \frac{26725482103}{70770402683} a^{9} - \frac{21587761119}{70770402683} a^{8} - \frac{2217844086}{70770402683} a^{7} + \frac{18904183218}{70770402683} a^{6} - \frac{7051345041}{70770402683} a^{5} - \frac{26319553146}{70770402683} a^{4} - \frac{26324932284}{70770402683} a^{3} + \frac{12957586824}{70770402683} a^{2} + \frac{21487454530}{70770402683} a - \frac{20554318700}{70770402683}$, $\frac{1}{70770402683} a^{30} - \frac{2432102400}{70770402683} a^{18} - \frac{8617680309}{70770402683} a^{17} + \frac{216020234}{70770402683} a^{16} + \frac{2322300436}{70770402683} a^{15} - \frac{34477724533}{70770402683} a^{14} + \frac{26433895105}{70770402683} a^{13} + \frac{11356002439}{70770402683} a^{12} - \frac{18876303483}{70770402683} a^{11} + \frac{6308208332}{70770402683} a^{10} - \frac{179560232}{969457571} a^{9} - \frac{81845632}{70770402683} a^{8} - \frac{28737715923}{70770402683} a^{7} + \frac{17756838739}{70770402683} a^{6} - \frac{1806157547}{70770402683} a^{5} + \frac{33356249969}{70770402683} a^{4} - \frac{9327318266}{70770402683} a^{3} + \frac{28140986788}{70770402683} a^{2} + \frac{8978456885}{70770402683} a - \frac{32110651292}{70770402683}$, $\frac{1}{70770402683} a^{31} - \frac{9139480255}{70770402683} a^{18} + \frac{27687445883}{70770402683} a^{17} - \frac{30878506135}{70770402683} a^{16} + \frac{6451889578}{70770402683} a^{15} + \frac{31672454673}{70770402683} a^{14} - \frac{32593419015}{70770402683} a^{13} + \frac{26907520010}{70770402683} a^{12} + \frac{24605694819}{70770402683} a^{11} - \frac{30271588441}{70770402683} a^{10} + \frac{26446187530}{70770402683} a^{9} - \frac{9891550477}{70770402683} a^{8} + \frac{12353089034}{70770402683} a^{7} + \frac{18195108491}{70770402683} a^{6} - \frac{26606653304}{70770402683} a^{5} - \frac{23035295845}{70770402683} a^{4} + \frac{5448641382}{70770402683} a^{3} + \frac{31357323969}{70770402683} a^{2} + \frac{6966828735}{70770402683} a - \frac{24926684030}{70770402683}$, $\frac{1}{70770402683} a^{32} + \frac{15624217979}{70770402683} a^{18} - \frac{17774978685}{70770402683} a^{17} + \frac{25759466470}{70770402683} a^{16} - \frac{33788410312}{70770402683} a^{15} + \frac{25112898372}{70770402683} a^{14} + \frac{4761012978}{70770402683} a^{13} - \frac{17247210436}{70770402683} a^{12} + \frac{16049037716}{70770402683} a^{11} - \frac{3563121878}{70770402683} a^{10} + \frac{2251953749}{70770402683} a^{9} - \frac{33752713063}{70770402683} a^{8} + \frac{11464694332}{70770402683} a^{7} + \frac{22473082976}{70770402683} a^{6} - \frac{9574467527}{70770402683} a^{5} + \frac{13517585450}{70770402683} a^{4} - \frac{25915503482}{70770402683} a^{3} - \frac{26182698779}{70770402683} a^{2} - \frac{217876352}{70770402683} a - \frac{3715641992}{70770402683}$, $\frac{1}{70770402683} a^{33} + \frac{17304247175}{70770402683} a^{18} + \frac{10103187263}{70770402683} a^{17} - \frac{11758175644}{70770402683} a^{16} + \frac{30721014215}{70770402683} a^{15} - \frac{19212402915}{70770402683} a^{14} - \frac{9233633664}{70770402683} a^{13} + \frac{21725458119}{70770402683} a^{12} + \frac{26328323986}{70770402683} a^{11} - \frac{31048466427}{70770402683} a^{10} - \frac{9330593302}{70770402683} a^{9} - \frac{32745048746}{70770402683} a^{8} + \frac{3069505003}{70770402683} a^{7} + \frac{10016284244}{70770402683} a^{6} - \frac{18445661287}{70770402683} a^{5} + \frac{10647472925}{70770402683} a^{4} + \frac{15742144197}{70770402683} a^{3} + \frac{6152498877}{70770402683} a^{2} - \frac{24331653810}{70770402683} a - \frac{17229637689}{70770402683}$, $\frac{1}{70770402683} a^{34} + \frac{13211860267}{70770402683} a^{18} + \frac{29497361362}{70770402683} a^{17} + \frac{19207765976}{70770402683} a^{16} + \frac{5248063870}{70770402683} a^{15} - \frac{17688199909}{70770402683} a^{14} - \frac{17095495862}{70770402683} a^{13} + \frac{29426990615}{70770402683} a^{12} + \frac{32991522097}{70770402683} a^{11} + \frac{22113752210}{70770402683} a^{10} + \frac{9560411682}{70770402683} a^{9} - \frac{2323602204}{70770402683} a^{8} + \frac{6240066904}{70770402683} a^{7} + \frac{6196272544}{70770402683} a^{6} + \frac{18199907605}{70770402683} a^{5} - \frac{31851830953}{70770402683} a^{4} - \frac{32881715688}{70770402683} a^{3} + \frac{4224731280}{70770402683} a^{2} - \frac{6820513805}{70770402683} a + \frac{2421868338}{70770402683}$, $\frac{1}{70770402683} a^{35} - \frac{30182450078}{70770402683} a^{18} + \frac{32523508706}{70770402683} a^{17} - \frac{14800076113}{70770402683} a^{16} - \frac{19169551171}{70770402683} a^{15} + \frac{18185482164}{70770402683} a^{14} - \frac{15419764983}{70770402683} a^{13} + \frac{22096218813}{70770402683} a^{12} - \frac{35180969425}{70770402683} a^{11} + \frac{31407096722}{70770402683} a^{10} - \frac{6147422868}{70770402683} a^{9} - \frac{33098800991}{70770402683} a^{8} - \frac{23236022040}{70770402683} a^{7} - \frac{24543442982}{70770402683} a^{6} + \frac{27012758215}{70770402683} a^{5} - \frac{30605431498}{70770402683} a^{4} + \frac{21481871083}{70770402683} a^{3} - \frac{8186284319}{70770402683} a^{2} - \frac{6898062455}{70770402683} a - \frac{26076369338}{70770402683}$

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:  $35$
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:  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, 4.4.11396925.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.2026592910581666266828125.1, \(\Q(\zeta_{37})^+\)

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 $36$ R R $18^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $36$ $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$ $36$

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
3Data not computed
5Data not computed
37Data not computed