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Properties

Label 36.36.8271239922...2272.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{54}\cdot 3^{48}\cdot 13^{33}$
Root discriminant $128.48$
Ramified primes $2, 3, 13$
Class number Not computed
Class group Not computed
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![-126541, 33492924, -26527284, -990979704, 1436547993, 8664074736, -14319635826, -33960132372, 60595492083, 71713506368, -139310773122, -88530910560, 194157715726, 65924502360, -174043844046, -29535704788, 104325807096, 7549082028, -43114770166, -808250712, 12569299335, -102135304, -2622544680, 49313304, 393837365, -7921692, -42416064, 703864, 3229485, -36360, -168842, 1020, 5742, -12, -114, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 114*x^34 - 12*x^33 + 5742*x^32 + 1020*x^31 - 168842*x^30 - 36360*x^29 + 3229485*x^28 + 703864*x^27 - 42416064*x^26 - 7921692*x^25 + 393837365*x^24 + 49313304*x^23 - 2622544680*x^22 - 102135304*x^21 + 12569299335*x^20 - 808250712*x^19 - 43114770166*x^18 + 7549082028*x^17 + 104325807096*x^16 - 29535704788*x^15 - 174043844046*x^14 + 65924502360*x^13 + 194157715726*x^12 - 88530910560*x^11 - 139310773122*x^10 + 71713506368*x^9 + 60595492083*x^8 - 33960132372*x^7 - 14319635826*x^6 + 8664074736*x^5 + 1436547993*x^4 - 990979704*x^3 - 26527284*x^2 + 33492924*x - 126541)
 
gp: K = bnfinit(x^36 - 114*x^34 - 12*x^33 + 5742*x^32 + 1020*x^31 - 168842*x^30 - 36360*x^29 + 3229485*x^28 + 703864*x^27 - 42416064*x^26 - 7921692*x^25 + 393837365*x^24 + 49313304*x^23 - 2622544680*x^22 - 102135304*x^21 + 12569299335*x^20 - 808250712*x^19 - 43114770166*x^18 + 7549082028*x^17 + 104325807096*x^16 - 29535704788*x^15 - 174043844046*x^14 + 65924502360*x^13 + 194157715726*x^12 - 88530910560*x^11 - 139310773122*x^10 + 71713506368*x^9 + 60595492083*x^8 - 33960132372*x^7 - 14319635826*x^6 + 8664074736*x^5 + 1436547993*x^4 - 990979704*x^3 - 26527284*x^2 + 33492924*x - 126541, 1)
 

Normalized defining polynomial

\( x^{36} - 114 x^{34} - 12 x^{33} + 5742 x^{32} + 1020 x^{31} - 168842 x^{30} - 36360 x^{29} + 3229485 x^{28} + 703864 x^{27} - 42416064 x^{26} - 7921692 x^{25} + 393837365 x^{24} + 49313304 x^{23} - 2622544680 x^{22} - 102135304 x^{21} + 12569299335 x^{20} - 808250712 x^{19} - 43114770166 x^{18} + 7549082028 x^{17} + 104325807096 x^{16} - 29535704788 x^{15} - 174043844046 x^{14} + 65924502360 x^{13} + 194157715726 x^{12} - 88530910560 x^{11} - 139310773122 x^{10} + 71713506368 x^{9} + 60595492083 x^{8} - 33960132372 x^{7} - 14319635826 x^{6} + 8664074736 x^{5} + 1436547993 x^{4} - 990979704 x^{3} - 26527284 x^{2} + 33492924 x - 126541 \)

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:  \(8271239922641569611168572203596233566179805157456691895516946594759832502272=2^{54}\cdot 3^{48}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.48$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(936=2^{3}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{936}(1,·)$, $\chi_{936}(643,·)$, $\chi_{936}(649,·)$, $\chi_{936}(529,·)$, $\chi_{936}(19,·)$, $\chi_{936}(691,·)$, $\chi_{936}(601,·)$, $\chi_{936}(25,·)$, $\chi_{936}(289,·)$, $\chi_{936}(163,·)$, $\chi_{936}(49,·)$, $\chi_{936}(811,·)$, $\chi_{936}(427,·)$, $\chi_{936}(307,·)$, $\chi_{936}(433,·)$, $\chi_{936}(115,·)$, $\chi_{936}(313,·)$, $\chi_{936}(187,·)$, $\chi_{936}(67,·)$, $\chi_{936}(673,·)$, $\chi_{936}(841,·)$, $\chi_{936}(331,·)$, $\chi_{936}(337,·)$, $\chi_{936}(931,·)$, $\chi_{936}(787,·)$, $\chi_{936}(217,·)$, $\chi_{936}(475,·)$, $\chi_{936}(739,·)$, $\chi_{936}(913,·)$, $\chi_{936}(361,·)$, $\chi_{936}(619,·)$, $\chi_{936}(625,·)$, $\chi_{936}(499,·)$, $\chi_{936}(745,·)$, $\chi_{936}(121,·)$, $\chi_{936}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} + \frac{3}{8} a^{4} + \frac{3}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} + \frac{3}{8} a^{5} + \frac{3}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{8} a^{20} - \frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{8} a^{7} + \frac{3}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{16} a^{24} - \frac{1}{8} a^{16} - \frac{1}{8} a^{12} + \frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{25} - \frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{16} a^{26} - \frac{1}{8} a^{12} - \frac{1}{16} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{5}{16} a^{2} - \frac{3}{8}$, $\frac{1}{16} a^{27} - \frac{1}{8} a^{13} - \frac{1}{16} a^{11} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{5}{16} a^{3} - \frac{3}{8} a$, $\frac{1}{16} a^{28} - \frac{1}{8} a^{14} - \frac{1}{16} a^{12} - \frac{1}{4} a^{10} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{5}{16} a^{4} - \frac{3}{8} a^{2}$, $\frac{1}{16} a^{29} - \frac{1}{8} a^{15} - \frac{1}{16} a^{13} - \frac{1}{4} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{5}{16} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{32} a^{30} - \frac{1}{32} a^{26} - \frac{1}{32} a^{24} - \frac{1}{16} a^{22} + \frac{1}{16} a^{16} + \frac{3}{32} a^{14} + \frac{1}{16} a^{12} - \frac{3}{32} a^{10} - \frac{1}{32} a^{8} + \frac{3}{32} a^{6} + \frac{1}{16} a^{4} - \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{31} - \frac{1}{32} a^{27} - \frac{1}{32} a^{25} - \frac{1}{16} a^{23} + \frac{1}{16} a^{17} + \frac{3}{32} a^{15} + \frac{1}{16} a^{13} - \frac{3}{32} a^{11} - \frac{1}{32} a^{9} + \frac{3}{32} a^{7} + \frac{1}{16} a^{5} - \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{32} a^{32} - \frac{1}{32} a^{28} - \frac{1}{32} a^{26} - \frac{1}{16} a^{18} - \frac{1}{32} a^{16} - \frac{1}{16} a^{14} - \frac{3}{32} a^{12} + \frac{3}{32} a^{10} - \frac{3}{32} a^{8} - \frac{3}{16} a^{6} + \frac{15}{32} a^{4} - \frac{5}{32} a^{2} + \frac{7}{16}$, $\frac{1}{32} a^{33} - \frac{1}{32} a^{29} - \frac{1}{32} a^{27} - \frac{1}{16} a^{19} - \frac{1}{32} a^{17} - \frac{1}{16} a^{15} - \frac{3}{32} a^{13} + \frac{3}{32} a^{11} - \frac{3}{32} a^{9} - \frac{3}{16} a^{7} + \frac{15}{32} a^{5} - \frac{5}{32} a^{3} + \frac{7}{16} a$, $\frac{1}{196957200431105531096831272597939012210899286230789387190385632} a^{34} - \frac{279436483700095797459159443958138103789381894420610298780967}{98478600215552765548415636298969506105449643115394693595192816} a^{33} + \frac{377648228625887783231664657881270569271164491156342557268469}{49239300107776382774207818149484753052724821557697346797596408} a^{32} + \frac{1887075787426227146332723647002833617156360757889900142021823}{196957200431105531096831272597939012210899286230789387190385632} a^{31} + \frac{2179156916057020071615694290162537965604318344500171729329781}{196957200431105531096831272597939012210899286230789387190385632} a^{30} + \frac{2440590714710882657588145084536791157009899404077044872905637}{98478600215552765548415636298969506105449643115394693595192816} a^{29} - \frac{3852354765098173423820500816827255414451965614402304717781263}{196957200431105531096831272597939012210899286230789387190385632} a^{28} + \frac{4005253954837948050503631375846431135275552343890033388899267}{196957200431105531096831272597939012210899286230789387190385632} a^{27} - \frac{336881618244677866151298675585522000726685714208226634946265}{49239300107776382774207818149484753052724821557697346797596408} a^{26} + \frac{3080337493737452200255393769094475917085054825696342822456807}{196957200431105531096831272597939012210899286230789387190385632} a^{25} + \frac{2842774922176142135774684726191140798383217869519143787246271}{98478600215552765548415636298969506105449643115394693595192816} a^{24} + \frac{5018256630349198999989133918424748736717232634812781729104937}{98478600215552765548415636298969506105449643115394693595192816} a^{23} - \frac{966868335666482447994651091115500031962843461536993784055353}{49239300107776382774207818149484753052724821557697346797596408} a^{22} + \frac{535542002363902892447849471169390547172675944688126462837741}{49239300107776382774207818149484753052724821557697346797596408} a^{21} - \frac{5672960531998957724848304339733951047665891979522606345276811}{98478600215552765548415636298969506105449643115394693595192816} a^{20} - \frac{424574449558488890954674908170466055516285980474885326795983}{24619650053888191387103909074742376526362410778848673398798204} a^{19} + \frac{9566325634885236867196096706720763269071845287351566299475199}{196957200431105531096831272597939012210899286230789387190385632} a^{18} + \frac{4934307716876350362228454363742846233389708930022265728528949}{49239300107776382774207818149484753052724821557697346797596408} a^{17} + \frac{8945701965336011154363351329667361866797057825025820416374969}{98478600215552765548415636298969506105449643115394693595192816} a^{16} + \frac{18512145756171365452903187670718782569245258975671978680278609}{196957200431105531096831272597939012210899286230789387190385632} a^{15} + \frac{1491394872513276142282240420009588456986271966680797540960103}{196957200431105531096831272597939012210899286230789387190385632} a^{14} - \frac{2414343963799678801141963363145533694451643735992542193751907}{24619650053888191387103909074742376526362410778848673398798204} a^{13} + \frac{8231721773045349009547273606408205735871896947873893298046997}{196957200431105531096831272597939012210899286230789387190385632} a^{12} + \frac{24325372774167431274163022285179634836488346928331470549756529}{196957200431105531096831272597939012210899286230789387190385632} a^{11} + \frac{38441239080078824169468513056057936887200009893941109462137121}{196957200431105531096831272597939012210899286230789387190385632} a^{10} - \frac{14261180805063277150677760348272126842586382881986134199544535}{196957200431105531096831272597939012210899286230789387190385632} a^{9} - \frac{864416271273172838781234123187209246050808844917377266071260}{6154912513472047846775977268685594131590602694712168349699551} a^{8} - \frac{1858292128283934832242420843148503388223032382904686805074083}{196957200431105531096831272597939012210899286230789387190385632} a^{7} - \frac{39835264236091654772087032463863704076588641325413962392708747}{196957200431105531096831272597939012210899286230789387190385632} a^{6} + \frac{24163807254120015876583593470238044748297632516788739609390649}{49239300107776382774207818149484753052724821557697346797596408} a^{5} + \frac{36068535718694137524914833667967604590274116062951808172700765}{196957200431105531096831272597939012210899286230789387190385632} a^{4} - \frac{90291249463338761193151804651857671388759374272368077656112333}{196957200431105531096831272597939012210899286230789387190385632} a^{3} - \frac{11109948916620367459560998732694224321406973137818484085786249}{98478600215552765548415636298969506105449643115394693595192816} a^{2} + \frac{97590976664813899127831926039857796648159478617818982978591287}{196957200431105531096831272597939012210899286230789387190385632} a - \frac{29694500846422358451006021563748276101086676999956494843930819}{98478600215552765548415636298969506105449643115394693595192816}$, $\frac{1}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{35} - \frac{3622549592522435}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{34} + \frac{238726245148258527117921147692122789836549216080247179214036064398773587429293}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{33} + \frac{25269583546544707432558122464011577643380804362507741060041320018122570636245}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{32} + \frac{27285336419657937155887057191452470866572433741770448169764730950503226774853}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{31} + \frac{152364573046856553190011853046551296363803748497389776421490956918075754950979}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{30} - \frac{40967879588799452625264039321914926432296595662216724198538767723047785984299}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{29} + \frac{268231642958402423962933071013353234854915275584464657438022513072641783350165}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{28} - \frac{286591848310977487344405710165141125122021959350635182004676731044568899690613}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{27} + \frac{4010510965381330006064557965858400125717944859405774667529283172587831946430}{625623109960708096271656460906773270695420747755175015754905417843251814590041} a^{26} - \frac{84969142777514727344087740341190051765554699464088219305634600299372541830483}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{25} - \frac{22160979716115141017265579159543695219647598036682114296128658643367847198917}{2502492439842832385086625843627093082781682991020700063019621671373007258360164} a^{24} + \frac{296725398856309082132266705137623949593307837748748247665481149304184316295129}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{23} + \frac{178310889806794120184503240354296507425547882359335021247240899217619053068303}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{22} - \frac{4683999013559791588550729310932465504852545267778328102269705609409436695177}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{21} + \frac{223408357720433671682286746215514402755812186610648652405060187003666056630099}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{20} + \frac{403340070370849697425864418489225739782955224169572501749496428619528093264969}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{19} + \frac{420100364455036512201587050201868512039767234087479135499695945792723246172889}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{18} + \frac{658197083812641466709508492145780243417534316852470528863208933234677446446971}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{17} - \frac{305734619608062458505281746424102177760910044867600795237231704341987402883181}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{16} + \frac{862047400233157879111434766866378886700296255906551944727475675830655746641209}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{15} + \frac{371048139879449242795301535810209712230633738993389353911317333021545779160463}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{14} + \frac{28826961719790117364347930981703852148578810387889015125655701824734417447089}{2502492439842832385086625843627093082781682991020700063019621671373007258360164} a^{13} + \frac{935537383347567341033423688028468711371710543302969756347631811284221759201413}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{12} - \frac{960218137299952619759591493078086116408725228304040247614977261969830951815051}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a^{11} + \frac{1813242522947782145628218131690692753414327652569180763003384648055560705105205}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{10} - \frac{346483933860954225683669305174027490792687283452281034196474080164747815900083}{2502492439842832385086625843627093082781682991020700063019621671373007258360164} a^{9} + \frac{719077411974874209848260642550438960824621839757534232303495723326575002188645}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{8} + \frac{32557974367297103797775099732967295896348318374146259594157276481305663467482}{625623109960708096271656460906773270695420747755175015754905417843251814590041} a^{7} - \frac{1156192369533433292524822509932918921389556969069039980382154274222666926694795}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{6} + \frac{3946855387072163350403405307021898633239086468480321178010964411214956660572875}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{5} + \frac{50360429617819389236938405607333695509051325893631348026469252944423711146587}{10009969759371329540346503374508372331126731964082800252078486685492029033440656} a^{4} + \frac{2327944962655693089154597623060844907331469635812718641087229183822983854528001}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{3} + \frac{568844197640037482681063499273281663349310388841572816852500961572384594125349}{5004984879685664770173251687254186165563365982041400126039243342746014516720328} a^{2} + \frac{7890481902602367288008568163763872119961744977270063277907981963664836361360755}{20019939518742659080693006749016744662253463928165600504156973370984058066881312} a + \frac{545515555969292672999137535292015574136366290152274772669533406403784164914555}{2502492439842832385086625843627093082781682991020700063019621671373007258360164}$

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

$C_3\times C_{12}$ (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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, 4.4.140608.1, 6.6.14414517.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 6.6.2436053373.1, 9.9.2565164201769.1, 12.12.119665832979124960100352.1, 12.12.469804094334435328.1, 12.12.20223525773472118256959488.1, 12.12.20223525773472118256959488.2, 18.18.14456408038335708501176406117.1

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$2$2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
13Data not computed