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Properties

Label 36.0.49997585682...0625.2
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 5^{18}\cdot 7^{30}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number $2072$ (GRH)
Class group $[2, 2, 518]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47045881, -29713188, -3518667, 6392588, -10971873, 14025420, -6371706, 260760, 3550761, -5790224, 4509732, -1673568, -646144, 2066184, 969246, -621600, -534822, -136224, 198166, 139512, -35424, -54168, -12810, 4320, 15500, 624, -2736, -344, 441, 36, -111, 36, 27, -4, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^34 - 4*x^33 + 27*x^32 + 36*x^31 - 111*x^30 + 36*x^29 + 441*x^28 - 344*x^27 - 2736*x^26 + 624*x^25 + 15500*x^24 + 4320*x^23 - 12810*x^22 - 54168*x^21 - 35424*x^20 + 139512*x^19 + 198166*x^18 - 136224*x^17 - 534822*x^16 - 621600*x^15 + 969246*x^14 + 2066184*x^13 - 646144*x^12 - 1673568*x^11 + 4509732*x^10 - 5790224*x^9 + 3550761*x^8 + 260760*x^7 - 6371706*x^6 + 14025420*x^5 - 10971873*x^4 + 6392588*x^3 - 3518667*x^2 - 29713188*x + 47045881)
 
gp: K = bnfinit(x^36 - 6*x^34 - 4*x^33 + 27*x^32 + 36*x^31 - 111*x^30 + 36*x^29 + 441*x^28 - 344*x^27 - 2736*x^26 + 624*x^25 + 15500*x^24 + 4320*x^23 - 12810*x^22 - 54168*x^21 - 35424*x^20 + 139512*x^19 + 198166*x^18 - 136224*x^17 - 534822*x^16 - 621600*x^15 + 969246*x^14 + 2066184*x^13 - 646144*x^12 - 1673568*x^11 + 4509732*x^10 - 5790224*x^9 + 3550761*x^8 + 260760*x^7 - 6371706*x^6 + 14025420*x^5 - 10971873*x^4 + 6392588*x^3 - 3518667*x^2 - 29713188*x + 47045881, 1)
 

Normalized defining polynomial

\( x^{36} - 6 x^{34} - 4 x^{33} + 27 x^{32} + 36 x^{31} - 111 x^{30} + 36 x^{29} + 441 x^{28} - 344 x^{27} - 2736 x^{26} + 624 x^{25} + 15500 x^{24} + 4320 x^{23} - 12810 x^{22} - 54168 x^{21} - 35424 x^{20} + 139512 x^{19} + 198166 x^{18} - 136224 x^{17} - 534822 x^{16} - 621600 x^{15} + 969246 x^{14} + 2066184 x^{13} - 646144 x^{12} - 1673568 x^{11} + 4509732 x^{10} - 5790224 x^{9} + 3550761 x^{8} + 260760 x^{7} - 6371706 x^{6} + 14025420 x^{5} - 10971873 x^{4} + 6392588 x^{3} - 3518667 x^{2} - 29713188 x + 47045881 \)

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:  \(4999758568289868528789868885747458073284974388119052886962890625=3^{54}\cdot 5^{18}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(134,·)$, $\chi_{315}(136,·)$, $\chi_{315}(269,·)$, $\chi_{315}(271,·)$, $\chi_{315}(16,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(286,·)$, $\chi_{315}(31,·)$, $\chi_{315}(164,·)$, $\chi_{315}(166,·)$, $\chi_{315}(299,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(179,·)$, $\chi_{315}(181,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$, $\chi_{315}(61,·)$, $\chi_{315}(194,·)$, $\chi_{315}(74,·)$, $\chi_{315}(76,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(89,·)$, $\chi_{315}(226,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(239,·)$, $\chi_{315}(241,·)$, $\chi_{315}(121,·)$, $\chi_{315}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{14} - \frac{1}{8} a^{7} + \frac{1}{8}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{15} - \frac{1}{8} a^{8} + \frac{1}{8} a$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{16} - \frac{1}{8} a^{9} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{24} - \frac{1}{8} a^{17} - \frac{1}{8} a^{10} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{25} - \frac{1}{8} a^{18} - \frac{1}{8} a^{11} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{26} - \frac{1}{8} a^{19} - \frac{1}{8} a^{12} + \frac{1}{8} a^{5}$, $\frac{1}{8} a^{27} - \frac{1}{8} a^{20} - \frac{1}{8} a^{13} + \frac{1}{8} a^{6}$, $\frac{1}{16} a^{28} - \frac{1}{8} a^{14} + \frac{1}{16}$, $\frac{1}{16} a^{29} - \frac{1}{8} a^{15} + \frac{1}{16} a$, $\frac{1}{272} a^{30} + \frac{1}{68} a^{29} - \frac{3}{136} a^{28} + \frac{3}{136} a^{27} - \frac{3}{136} a^{26} + \frac{1}{34} a^{25} + \frac{1}{17} a^{24} + \frac{1}{68} a^{23} + \frac{3}{136} a^{22} + \frac{1}{136} a^{21} + \frac{13}{136} a^{20} + \frac{13}{136} a^{19} + \frac{1}{34} a^{18} + \frac{1}{17} a^{17} - \frac{1}{136} a^{16} - \frac{9}{136} a^{15} + \frac{11}{136} a^{14} - \frac{3}{136} a^{13} + \frac{11}{136} a^{12} + \frac{1}{17} a^{10} - \frac{7}{68} a^{9} + \frac{29}{136} a^{8} + \frac{11}{136} a^{7} + \frac{43}{136} a^{6} + \frac{35}{136} a^{5} + \frac{6}{17} a^{4} + \frac{4}{17} a^{3} - \frac{135}{272} a^{2} + \frac{31}{136} a + \frac{9}{34}$, $\frac{1}{19335570443786214446818736} a^{31} + \frac{300055062974734967763}{1017661602304537602464144} a^{30} - \frac{1823826047833962077767}{9667785221893107223409368} a^{29} - \frac{17103138589559893162493}{9667785221893107223409368} a^{28} + \frac{603777440262283045154689}{9667785221893107223409368} a^{27} - \frac{15293445544822465703553}{284346624173326683041452} a^{26} - \frac{550861048122476818587823}{9667785221893107223409368} a^{25} + \frac{347721246057523301803995}{9667785221893107223409368} a^{24} - \frac{24066778635037143568139}{568693248346653366082904} a^{23} - \frac{155836654899993893204769}{4833892610946553611704684} a^{22} - \frac{22044361357182286831107}{508830801152268801232072} a^{21} - \frac{51636147423160208562555}{9667785221893107223409368} a^{20} - \frac{383002653061733046641869}{4833892610946553611704684} a^{19} + \frac{676220996866062875997413}{9667785221893107223409368} a^{18} - \frac{146469029716890526151923}{2416946305473276805852342} a^{17} - \frac{151032089753302517886242}{1208473152736638402926171} a^{16} - \frac{183992947885964554048555}{4833892610946553611704684} a^{15} + \frac{387001606252818042899187}{9667785221893107223409368} a^{14} - \frac{1790767426024437426911209}{9667785221893107223409368} a^{13} + \frac{290569717777112432682843}{4833892610946553611704684} a^{12} + \frac{676530333047645576658575}{9667785221893107223409368} a^{11} - \frac{2132741983623897504683855}{9667785221893107223409368} a^{10} + \frac{2301359658072375544836827}{9667785221893107223409368} a^{9} + \frac{888183732037056237991421}{4833892610946553611704684} a^{8} + \frac{2391740176343116854524141}{9667785221893107223409368} a^{7} - \frac{2910354697908181213974205}{9667785221893107223409368} a^{6} - \frac{212982347872852582343109}{4833892610946553611704684} a^{5} + \frac{2396695791921876827002067}{9667785221893107223409368} a^{4} + \frac{3645348787761674629203743}{19335570443786214446818736} a^{3} + \frac{444886202704956739337463}{19335570443786214446818736} a^{2} - \frac{3003509202346173242799435}{9667785221893107223409368} a + \frac{22854732969317082941}{254415400576134400616036}$, $\frac{1}{734751676863876148979111968} a^{32} + \frac{618520764079578034698249}{734751676863876148979111968} a^{30} - \frac{12404394166127051201880555}{734751676863876148979111968} a^{29} - \frac{3711124584477468208189503}{734751676863876148979111968} a^{28} + \frac{1626893883528233485080989}{45921979803992259311194498} a^{27} + \frac{637364296583267263536890}{22960989901996129655597249} a^{26} + \frac{5210229964488206046627925}{91843959607984518622388996} a^{25} + \frac{527202394919119214928357}{10805171718586413955575176} a^{24} + \frac{1099262530711188592446821}{91843959607984518622388996} a^{23} + \frac{9762178021908655203445}{2416946305473276805852342} a^{22} - \frac{775262697001958193890987}{22960989901996129655597249} a^{21} + \frac{5621346553267150605299897}{183687919215969037244777992} a^{20} - \frac{1943495825683278286150209}{22960989901996129655597249} a^{19} + \frac{7974390565203791287156749}{367375838431938074489555984} a^{18} - \frac{467579839032959806080839}{5402585859293206977787588} a^{17} + \frac{1464455842029637624132549}{367375838431938074489555984} a^{16} + \frac{41881995476520067647686295}{367375838431938074489555984} a^{15} + \frac{22477576369021790619448223}{367375838431938074489555984} a^{14} - \frac{9105828495711977820365053}{45921979803992259311194498} a^{13} - \frac{291322316047517105201813}{2701292929646603488893794} a^{12} - \frac{5695692137355239735979351}{91843959607984518622388996} a^{11} - \frac{9956676532236878432324441}{183687919215969037244777992} a^{10} - \frac{20957181342286415485254901}{91843959607984518622388996} a^{9} - \frac{3248990296458519250373943}{45921979803992259311194498} a^{8} - \frac{1017512254764054129029014}{22960989901996129655597249} a^{7} + \frac{67654693605477137075874163}{183687919215969037244777992} a^{6} + \frac{582488528497293287573240}{1350646464823301744446897} a^{5} + \frac{288931693723343372359604597}{734751676863876148979111968} a^{4} - \frac{43859972263866667852545983}{91843959607984518622388996} a^{3} - \frac{118734851122899892555418595}{734751676863876148979111968} a^{2} - \frac{17924561614094620265246801}{38671140887572428893637472} a + \frac{340922290456618595738089}{2035323204609075204928288}$, $\frac{1}{13960281860413646830603127392} a^{33} + \frac{355}{13960281860413646830603127392} a^{31} - \frac{6185875619527677808858247}{13960281860413646830603127392} a^{30} - \frac{19445887477546745811847809}{13960281860413646830603127392} a^{29} + \frac{18557626858583033426574025}{6980140930206823415301563696} a^{28} + \frac{35354706835847796526629157}{3490070465103411707650781848} a^{27} + \frac{204040600131329503929304547}{3490070465103411707650781848} a^{26} - \frac{186932621049189634509692367}{3490070465103411707650781848} a^{25} + \frac{39525904575790076047267703}{1745035232551705853825390924} a^{24} + \frac{8580142846855313144553785}{183687919215969037244777992} a^{23} + \frac{30122700898727146619594231}{3490070465103411707650781848} a^{22} - \frac{28655928583655749172390993}{872517616275852926912695462} a^{21} + \frac{312197171080838699555832159}{3490070465103411707650781848} a^{20} - \frac{6499660411103927531849185}{410596525306283730311856688} a^{19} - \frac{172601265340463859443847899}{1745035232551705853825390924} a^{18} - \frac{627649260442825999430382949}{6980140930206823415301563696} a^{17} - \frac{653546372942853425402311503}{6980140930206823415301563696} a^{16} - \frac{381108385512231068008334653}{6980140930206823415301563696} a^{15} + \frac{22682738870421766854009122}{436258808137926463456347731} a^{14} + \frac{869271319945151779272693107}{3490070465103411707650781848} a^{13} + \frac{709655903291124705381919689}{3490070465103411707650781848} a^{12} - \frac{13906122648443886856958077}{3490070465103411707650781848} a^{11} - \frac{390105848363791515552235395}{1745035232551705853825390924} a^{10} - \frac{541787274522377377291776395}{3490070465103411707650781848} a^{9} - \frac{851014319468136466095379823}{3490070465103411707650781848} a^{8} - \frac{26553804847234679464554547}{436258808137926463456347731} a^{7} + \frac{978551594696532825706644461}{3490070465103411707650781848} a^{6} - \frac{2060734357106945974513727535}{13960281860413646830603127392} a^{5} - \frac{15611926673904194064283155}{1745035232551705853825390924} a^{4} + \frac{1216910280716627947651577335}{13960281860413646830603127392} a^{3} - \frac{38725246840739616047840025}{734751676863876148979111968} a^{2} + \frac{569247365301306587186803}{38671140887572428893637472} a - \frac{262363108894145243373979}{1017661602304537602464144}$, $\frac{1}{265245355347859289781459420448} a^{34} - \frac{3}{132622677673929644890729710224} a^{32} + \frac{6855}{265245355347859289781459420448} a^{31} - \frac{137593543209312350931149413}{132622677673929644890729710224} a^{30} - \frac{396544991072897970305427743}{265245355347859289781459420448} a^{29} + \frac{1651122518511748211173765571}{265245355347859289781459420448} a^{28} - \frac{648681286438139613177717067}{66311338836964822445364855112} a^{27} + \frac{2683490835057482635570234653}{66311338836964822445364855112} a^{26} - \frac{185705988023317396168690275}{66311338836964822445364855112} a^{25} + \frac{11142569180779842032251809}{3490070465103411707650781848} a^{24} + \frac{2650807321417944404728939851}{66311338836964822445364855112} a^{23} - \frac{776290786074941467682812699}{33155669418482411222682427556} a^{22} - \frac{900800726959902805545164061}{33155669418482411222682427556} a^{21} - \frac{10569275496797904665806802335}{132622677673929644890729710224} a^{20} - \frac{2924711440708953845352142781}{33155669418482411222682427556} a^{19} + \frac{119369103725955354585238259}{16577834709241205611341213778} a^{18} + \frac{14243433510519129789904891085}{132622677673929644890729710224} a^{17} - \frac{5874489405339910150013656479}{66311338836964822445364855112} a^{16} - \frac{7818045142509897002796238131}{132622677673929644890729710224} a^{15} + \frac{14571252762099478110814981053}{132622677673929644890729710224} a^{14} + \frac{15396493962685342545327753683}{66311338836964822445364855112} a^{13} - \frac{11717494817162615024942764557}{66311338836964822445364855112} a^{12} - \frac{10247416616319038488545683061}{66311338836964822445364855112} a^{11} + \frac{3964798487210375734163454921}{66311338836964822445364855112} a^{10} - \frac{1332693668910264433861546051}{66311338836964822445364855112} a^{9} - \frac{2223428097056178923568471207}{33155669418482411222682427556} a^{8} - \frac{36717383429819869209490765}{1950333495204847718981319268} a^{7} - \frac{7627334922835061569117225395}{265245355347859289781459420448} a^{6} + \frac{9308500393725115485516090751}{33155669418482411222682427556} a^{5} - \frac{38920819323191712327418481613}{132622677673929644890729710224} a^{4} + \frac{853963901560999046539407125}{13960281860413646830603127392} a^{3} - \frac{123834971503259353407422085}{367375838431938074489555984} a^{2} + \frac{8313119664284480799325679}{38671140887572428893637472} a - \frac{821516080359489898341373}{2035323204609075204928288}$, $\frac{1}{5039661751609326505847728988512} a^{35} - \frac{3}{2519830875804663252923864494256} a^{33} - \frac{1}{1259915437902331626461932247128} a^{32} - \frac{65147}{2519830875804663252923864494256} a^{31} - \frac{119326978677049519468443455}{74112672817784213321290132184} a^{30} + \frac{27249638840808643019894484861}{2519830875804663252923864494256} a^{29} + \frac{142077672117818992529180165185}{5039661751609326505847728988512} a^{28} - \frac{73634681972386561735829006441}{1259915437902331626461932247128} a^{27} - \frac{13124613514070018393274681647}{1259915437902331626461932247128} a^{26} + \frac{578425247963001749474871637}{16577834709241205611341213778} a^{25} - \frac{40152146275165437807607651601}{1259915437902331626461932247128} a^{24} + \frac{69623554877758062457293971019}{1259915437902331626461932247128} a^{23} - \frac{34261041769462562188749467023}{1259915437902331626461932247128} a^{22} - \frac{154761421314500864748219062433}{2519830875804663252923864494256} a^{21} - \frac{147204251530486635784815730071}{1259915437902331626461932247128} a^{20} - \frac{31862503079738042525465223307}{629957718951165813230966123564} a^{19} - \frac{36597112236445840079079968055}{314978859475582906615483061782} a^{18} - \frac{11625703520653157845961125651}{629957718951165813230966123564} a^{17} + \frac{128947290256284254646880019357}{1259915437902331626461932247128} a^{16} - \frac{38715466621445842563255898085}{314978859475582906615483061782} a^{15} + \frac{94763951204460407508359628771}{2519830875804663252923864494256} a^{14} + \frac{282803181147145464458866541925}{1259915437902331626461932247128} a^{13} + \frac{187664827897464707977599995055}{1259915437902331626461932247128} a^{12} - \frac{5821660708104948507832081667}{314978859475582906615483061782} a^{11} - \frac{1455314643533870524967457271}{1259915437902331626461932247128} a^{10} + \frac{131962936795682276508078166521}{1259915437902331626461932247128} a^{9} - \frac{283645451913846365747706008885}{1259915437902331626461932247128} a^{8} + \frac{17652267158554783727431981937}{296450691271136853285160528736} a^{7} - \frac{51710633379349107019189581781}{1259915437902331626461932247128} a^{6} + \frac{543342825930126857757871506367}{2519830875804663252923864494256} a^{5} - \frac{29884959731246194774832828489}{66311338836964822445364855112} a^{4} + \frac{2766751096928713850798308831}{6980140930206823415301563696} a^{3} + \frac{36610311385995239815703213}{91843959607984518622388996} a^{2} + \frac{52012894014194929733603}{1137386496693306732165808} a + \frac{330199879046290475914355}{2035323204609075204928288}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{518}$, which has order $2072$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{3901545553853697}{284346624173326683041452} a^{34} - \frac{1300515184617899}{142173312086663341520726} a^{33} + \frac{35113909984683273}{568693248346653366082904} a^{32} + \frac{11704636661561091}{142173312086663341520726} a^{31} - \frac{144357185492586789}{568693248346653366082904} a^{30} - \frac{33834737569295871}{71086656043331670760363} a^{29} + \frac{573527196416493459}{568693248346653366082904} a^{28} - \frac{55922152938569657}{71086656043331670760363} a^{27} - \frac{23409273323122182}{3741402949649035303177} a^{26} + \frac{101440184400196122}{71086656043331670760363} a^{25} + \frac{5039496340394358625}{142173312086663341520726} a^{24} + \frac{702278199693665460}{71086656043331670760363} a^{23} - \frac{47468011697105704563}{284346624173326683041452} a^{22} - \frac{8805788315047794129}{71086656043331670760363} a^{21} - \frac{5758681237488056772}{71086656043331670760363} a^{20} + \frac{22679684304551540661}{71086656043331670760363} a^{19} + \frac{128858946037495286617}{284346624173326683041452} a^{18} - \frac{22145172563673584172}{71086656043331670760363} a^{17} - \frac{347772066033856989489}{284346624173326683041452} a^{16} + \frac{244158077825752179801}{142173312086663341520726} a^{15} + \frac{630259570315080067077}{284346624173326683041452} a^{14} + \frac{335887958276818628427}{71086656043331670760363} a^{13} - \frac{105040010431218466432}{71086656043331670760363} a^{12} - \frac{272062574561325999204}{71086656043331670760363} a^{11} + \frac{1466243736139311723267}{142173312086663341520726} a^{10} - \frac{941284279292373702422}{71086656043331670760363} a^{9} - \frac{36528088324739885202069}{284346624173326683041452} a^{8} + \frac{42390292442620417905}{71086656043331670760363} a^{7} - \frac{4143250202460487382847}{284346624173326683041452} a^{6} + \frac{240003574745310170955}{7482805899298070606354} a^{5} - \frac{751004602115744281833}{29931223597192282425416} a^{4} + \frac{109390233723765338587}{7482805899298070606354} a^{3} - \frac{240846308584942569507}{29931223597192282425416} a^{2} + \frac{82914742573123582203699}{142173312086663341520726} a + \frac{3220204348117195096001}{29931223597192282425416} \) (order $14$)
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:  \( 1227150026460930.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-7}, \sqrt{-15})\), 6.6.843908625.1, 6.6.41351522625.1, 6.6.41351522625.2, 6.6.56723625.1, 6.0.2250423.1, 6.0.2460375.1, 6.0.110270727.2, 6.0.5907360375.1, 6.0.110270727.1, 6.0.5907360375.2, \(\Q(\zeta_{7})\), 6.0.8103375.1, 9.9.62523502209.1, 12.0.712181767349390625.2, 12.0.1709948423405886890625.1, 12.0.1709948423405886890625.4, 12.0.3217569633140625.3, 18.18.70708970918051611315870587890625.1, 18.0.1340851596668237962730583.1, 18.0.206148603259625688967552734375.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{12}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed