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Properties

Label 36.0.11340841668...8125.1
Degree $36$
Signature $[0, 18]$
Discriminant $5^{27}\cdot 37^{32}$
Root discriminant $82.83$
Ramified primes $5, 37$
Class number $462217$ (GRH)
Class group $[7, 66031]$ (GRH)
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![1, -7, 56, -368, 2425, -6008, 30848, -72948, 235179, 124075, 2582197, 232605, 23034602, 9088911, 25175709, 13376744, 23851200, -3108766, 14127290, -3450543, 7334189, -1067627, 2154833, -274866, 544902, -59713, 124680, -9791, 24025, -548, 2305, -92, 217, -22, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 17*x^34 - 22*x^33 + 217*x^32 - 92*x^31 + 2305*x^30 - 548*x^29 + 24025*x^28 - 9791*x^27 + 124680*x^26 - 59713*x^25 + 544902*x^24 - 274866*x^23 + 2154833*x^22 - 1067627*x^21 + 7334189*x^20 - 3450543*x^19 + 14127290*x^18 - 3108766*x^17 + 23851200*x^16 + 13376744*x^15 + 25175709*x^14 + 9088911*x^13 + 23034602*x^12 + 232605*x^11 + 2582197*x^10 + 124075*x^9 + 235179*x^8 - 72948*x^7 + 30848*x^6 - 6008*x^5 + 2425*x^4 - 368*x^3 + 56*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 17*x^34 - 22*x^33 + 217*x^32 - 92*x^31 + 2305*x^30 - 548*x^29 + 24025*x^28 - 9791*x^27 + 124680*x^26 - 59713*x^25 + 544902*x^24 - 274866*x^23 + 2154833*x^22 - 1067627*x^21 + 7334189*x^20 - 3450543*x^19 + 14127290*x^18 - 3108766*x^17 + 23851200*x^16 + 13376744*x^15 + 25175709*x^14 + 9088911*x^13 + 23034602*x^12 + 232605*x^11 + 2582197*x^10 + 124075*x^9 + 235179*x^8 - 72948*x^7 + 30848*x^6 - 6008*x^5 + 2425*x^4 - 368*x^3 + 56*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 17 x^{34} - 22 x^{33} + 217 x^{32} - 92 x^{31} + 2305 x^{30} - 548 x^{29} + 24025 x^{28} - 9791 x^{27} + 124680 x^{26} - 59713 x^{25} + 544902 x^{24} - 274866 x^{23} + 2154833 x^{22} - 1067627 x^{21} + 7334189 x^{20} - 3450543 x^{19} + 14127290 x^{18} - 3108766 x^{17} + 23851200 x^{16} + 13376744 x^{15} + 25175709 x^{14} + 9088911 x^{13} + 23034602 x^{12} + 232605 x^{11} + 2582197 x^{10} + 124075 x^{9} + 235179 x^{8} - 72948 x^{7} + 30848 x^{6} - 6008 x^{5} + 2425 x^{4} - 368 x^{3} + 56 x^{2} - 7 x + 1 \)

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:  \(1134084166835624937413663701523229292665702790961273014545440673828125=5^{27}\cdot 37^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.83$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(7,·)$, $\chi_{185}(9,·)$, $\chi_{185}(12,·)$, $\chi_{185}(16,·)$, $\chi_{185}(149,·)$, $\chi_{185}(26,·)$, $\chi_{185}(157,·)$, $\chi_{185}(158,·)$, $\chi_{185}(33,·)$, $\chi_{185}(34,·)$, $\chi_{185}(164,·)$, $\chi_{185}(38,·)$, $\chi_{185}(44,·)$, $\chi_{185}(46,·)$, $\chi_{185}(47,·)$, $\chi_{185}(49,·)$, $\chi_{185}(181,·)$, $\chi_{185}(182,·)$, $\chi_{185}(137,·)$, $\chi_{185}(53,·)$, $\chi_{185}(71,·)$, $\chi_{185}(81,·)$, $\chi_{185}(83,·)$, $\chi_{185}(84,·)$, $\chi_{185}(86,·)$, $\chi_{185}(144,·)$, $\chi_{185}(123,·)$, $\chi_{185}(107,·)$, $\chi_{185}(108,·)$, $\chi_{185}(174,·)$, $\chi_{185}(112,·)$, $\chi_{185}(118,·)$, $\chi_{185}(121,·)$, $\chi_{185}(63,·)$, $\chi_{185}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{31} a^{22} + \frac{13}{31} a^{21} + \frac{15}{31} a^{20} - \frac{1}{31} a^{19} + \frac{13}{31} a^{18} + \frac{15}{31} a^{17} + \frac{14}{31} a^{16} - \frac{9}{31} a^{15} - \frac{14}{31} a^{14} + \frac{14}{31} a^{13} + \frac{8}{31} a^{12} + \frac{6}{31} a^{11} + \frac{12}{31} a^{10} + \frac{9}{31} a^{9} - \frac{9}{31} a^{8} - \frac{13}{31} a^{7} - \frac{13}{31} a^{6} - \frac{6}{31} a^{5} - \frac{9}{31} a^{4} - \frac{10}{31} a^{3} - \frac{7}{31} a^{2} + \frac{13}{31} a - \frac{11}{31}$, $\frac{1}{31} a^{23} + \frac{1}{31} a^{21} - \frac{10}{31} a^{20} - \frac{5}{31} a^{19} + \frac{1}{31} a^{18} + \frac{5}{31} a^{17} - \frac{5}{31} a^{16} + \frac{10}{31} a^{15} + \frac{10}{31} a^{14} + \frac{12}{31} a^{13} - \frac{5}{31} a^{12} - \frac{4}{31} a^{11} + \frac{8}{31} a^{10} - \frac{2}{31} a^{9} + \frac{11}{31} a^{8} + \frac{1}{31} a^{7} + \frac{8}{31} a^{6} + \frac{7}{31} a^{5} + \frac{14}{31} a^{4} - \frac{1}{31} a^{3} + \frac{11}{31} a^{2} + \frac{6}{31} a - \frac{12}{31}$, $\frac{1}{31} a^{24} + \frac{8}{31} a^{21} + \frac{11}{31} a^{20} + \frac{2}{31} a^{19} - \frac{8}{31} a^{18} + \frac{11}{31} a^{17} - \frac{4}{31} a^{16} - \frac{12}{31} a^{15} - \frac{5}{31} a^{14} + \frac{12}{31} a^{13} - \frac{12}{31} a^{12} + \frac{2}{31} a^{11} - \frac{14}{31} a^{10} + \frac{2}{31} a^{9} + \frac{10}{31} a^{8} - \frac{10}{31} a^{7} - \frac{11}{31} a^{6} - \frac{11}{31} a^{5} + \frac{8}{31} a^{4} - \frac{10}{31} a^{3} + \frac{13}{31} a^{2} + \frac{6}{31} a + \frac{11}{31}$, $\frac{1}{31} a^{25} + \frac{6}{31} a^{20} + \frac{5}{31} a^{15} - \frac{1}{31} a^{10} - \frac{6}{31} a^{5} - \frac{5}{31}$, $\frac{1}{31} a^{26} + \frac{6}{31} a^{21} + \frac{5}{31} a^{16} - \frac{1}{31} a^{11} - \frac{6}{31} a^{6} - \frac{5}{31} a$, $\frac{1}{31} a^{27} + \frac{15}{31} a^{21} + \frac{3}{31} a^{20} + \frac{6}{31} a^{19} + \frac{15}{31} a^{18} + \frac{8}{31} a^{17} + \frac{9}{31} a^{16} - \frac{8}{31} a^{15} - \frac{9}{31} a^{14} + \frac{9}{31} a^{13} + \frac{13}{31} a^{12} - \frac{5}{31} a^{11} - \frac{10}{31} a^{10} + \frac{8}{31} a^{9} - \frac{8}{31} a^{8} + \frac{10}{31} a^{7} - \frac{15}{31} a^{6} + \frac{5}{31} a^{5} - \frac{8}{31} a^{4} - \frac{2}{31} a^{3} + \frac{6}{31} a^{2} + \frac{15}{31} a + \frac{4}{31}$, $\frac{1}{31} a^{28} - \frac{6}{31} a^{21} - \frac{2}{31} a^{20} - \frac{1}{31} a^{19} - \frac{1}{31} a^{18} + \frac{1}{31} a^{17} - \frac{1}{31} a^{16} + \frac{2}{31} a^{15} + \frac{2}{31} a^{14} - \frac{11}{31} a^{13} - \frac{1}{31} a^{12} - \frac{7}{31} a^{11} + \frac{14}{31} a^{10} + \frac{12}{31} a^{9} - \frac{10}{31} a^{8} - \frac{6}{31} a^{7} + \frac{14}{31} a^{6} - \frac{11}{31} a^{5} + \frac{9}{31} a^{4} + \frac{1}{31} a^{3} - \frac{4}{31} a^{2} - \frac{5}{31} a + \frac{10}{31}$, $\frac{1}{31} a^{29} + \frac{14}{31} a^{21} - \frac{4}{31} a^{20} - \frac{7}{31} a^{19} - \frac{14}{31} a^{18} - \frac{4}{31} a^{17} - \frac{7}{31} a^{16} + \frac{10}{31} a^{15} - \frac{2}{31} a^{14} - \frac{10}{31} a^{13} + \frac{10}{31} a^{12} - \frac{12}{31} a^{11} - \frac{9}{31} a^{10} + \frac{13}{31} a^{9} + \frac{2}{31} a^{8} - \frac{2}{31} a^{7} + \frac{4}{31} a^{6} + \frac{4}{31} a^{5} + \frac{9}{31} a^{4} - \frac{2}{31} a^{3} + \frac{15}{31} a^{2} - \frac{5}{31} a - \frac{4}{31}$, $\frac{1}{31} a^{30} - \frac{1}{31}$, $\frac{1}{31} a^{31} - \frac{1}{31} a$, $\frac{1}{1333} a^{32} + \frac{12}{1333} a^{31} + \frac{4}{1333} a^{30} - \frac{16}{1333} a^{29} + \frac{17}{1333} a^{28} - \frac{11}{1333} a^{27} + \frac{6}{1333} a^{26} + \frac{10}{1333} a^{25} - \frac{18}{1333} a^{24} + \frac{18}{1333} a^{23} + \frac{17}{1333} a^{22} - \frac{174}{1333} a^{21} + \frac{523}{1333} a^{20} + \frac{196}{1333} a^{19} + \frac{146}{1333} a^{18} + \frac{574}{1333} a^{17} - \frac{343}{1333} a^{16} - \frac{55}{1333} a^{15} + \frac{383}{1333} a^{14} - \frac{601}{1333} a^{13} - \frac{182}{1333} a^{12} + \frac{54}{1333} a^{11} - \frac{251}{1333} a^{10} - \frac{569}{1333} a^{9} - \frac{13}{31} a^{8} - \frac{513}{1333} a^{7} - \frac{103}{1333} a^{6} + \frac{414}{1333} a^{5} + \frac{238}{1333} a^{4} + \frac{652}{1333} a^{3} + \frac{524}{1333} a^{2} - \frac{487}{1333} a - \frac{2}{43}$, $\frac{1}{3547292685552122535682087116089203664863} a^{33} - \frac{375660048111493182982647842243447964}{3547292685552122535682087116089203664863} a^{32} - \frac{52064616384137740507812277758407511559}{3547292685552122535682087116089203664863} a^{31} + \frac{46054055614353849580089912282512344146}{3547292685552122535682087116089203664863} a^{30} + \frac{32210192175279757458126998889696940624}{3547292685552122535682087116089203664863} a^{29} - \frac{51184728437266868121953983918520100224}{3547292685552122535682087116089203664863} a^{28} + \frac{41725958698895092729825185871842139965}{3547292685552122535682087116089203664863} a^{27} - \frac{54676452211740676975903277322749479245}{3547292685552122535682087116089203664863} a^{26} - \frac{34675133610118020452644460476509702798}{3547292685552122535682087116089203664863} a^{25} + \frac{45988495406703198381803390818744005862}{3547292685552122535682087116089203664863} a^{24} - \frac{14219642924073784190523691873212414300}{3547292685552122535682087116089203664863} a^{23} + \frac{709857758877067443373120114495435570}{3547292685552122535682087116089203664863} a^{22} + \frac{616068687266974183909498627857311459761}{3547292685552122535682087116089203664863} a^{21} + \frac{1377891909741041406067880232430296914262}{3547292685552122535682087116089203664863} a^{20} + \frac{14537707763474459452282568751139764946}{114428796308132985022002810196425924673} a^{19} + \frac{747599703892164978484604773257444397542}{3547292685552122535682087116089203664863} a^{18} + \frac{689826308881527726802049379358291142}{3547292685552122535682087116089203664863} a^{17} - \frac{622484449789656465402089684332440393691}{3547292685552122535682087116089203664863} a^{16} - \frac{151858841066418763925367831706766436765}{3547292685552122535682087116089203664863} a^{15} + \frac{1707491792350392887548463562048969110523}{3547292685552122535682087116089203664863} a^{14} + \frac{928480160939211189327315932551761346231}{3547292685552122535682087116089203664863} a^{13} + \frac{127021232053713803778409233346958684035}{3547292685552122535682087116089203664863} a^{12} + \frac{985713101284446654237137352056722223722}{3547292685552122535682087116089203664863} a^{11} - \frac{540037865230828478151250583147788856750}{3547292685552122535682087116089203664863} a^{10} - \frac{1350611220822795604845657900876541925589}{3547292685552122535682087116089203664863} a^{9} - \frac{1272868576554912255915296979951673439951}{3547292685552122535682087116089203664863} a^{8} + \frac{876902347195617381039133386055350765523}{3547292685552122535682087116089203664863} a^{7} + \frac{1553175084920043556517023962027309647436}{3547292685552122535682087116089203664863} a^{6} - \frac{1076722806413728549191690205242013835287}{3547292685552122535682087116089203664863} a^{5} + \frac{1560398759345844359806566169939261697101}{3547292685552122535682087116089203664863} a^{4} + \frac{762306766268950304296607512945347930487}{3547292685552122535682087116089203664863} a^{3} + \frac{1290047039124647494515240535210883991854}{3547292685552122535682087116089203664863} a^{2} - \frac{313916541002500304633067801880688764679}{3547292685552122535682087116089203664863} a - \frac{152391301740875042138490435439612048736}{3547292685552122535682087116089203664863}$, $\frac{1}{109966073252115798606144700598765313610753} a^{34} + \frac{6}{109966073252115798606144700598765313610753} a^{33} + \frac{19295063466936406505234344930427945102}{109966073252115798606144700598765313610753} a^{32} + \frac{1250712879411249682515142202049074868900}{109966073252115798606144700598765313610753} a^{31} - \frac{941991863940267178431392683162227747268}{109966073252115798606144700598765313610753} a^{30} - \frac{783749874930196692385473510933398246887}{109966073252115798606144700598765313610753} a^{29} + \frac{631371147777942247499086597585575679877}{109966073252115798606144700598765313610753} a^{28} - \frac{35942178477426216906753769973517812392}{109966073252115798606144700598765313610753} a^{27} - \frac{1573171350630277448032929975838577863400}{109966073252115798606144700598765313610753} a^{26} + \frac{339589190161886456616218056045271688251}{109966073252115798606144700598765313610753} a^{25} - \frac{1076868439759104063280857783165474801687}{109966073252115798606144700598765313610753} a^{24} - \frac{388941328341968738892596941131822610048}{109966073252115798606144700598765313610753} a^{23} - \frac{1488309750853676004618959526375047606644}{109966073252115798606144700598765313610753} a^{22} - \frac{761368582962091850490696688944718833140}{109966073252115798606144700598765313610753} a^{21} + \frac{17019056772606038483507642878357017357449}{109966073252115798606144700598765313610753} a^{20} + \frac{18551970730991022300355672409992837415893}{109966073252115798606144700598765313610753} a^{19} + \frac{51223629280811395992467287166529719850170}{109966073252115798606144700598765313610753} a^{18} + \frac{19269371969837987127055456036047493035649}{109966073252115798606144700598765313610753} a^{17} - \frac{12567102087781367350125564116905751745580}{109966073252115798606144700598765313610753} a^{16} + \frac{225620693674979148926789747856537305244}{109966073252115798606144700598765313610753} a^{15} + \frac{7340290862691280217785221796433056535935}{109966073252115798606144700598765313610753} a^{14} - \frac{47860429293638252974222488902340088435060}{109966073252115798606144700598765313610753} a^{13} - \frac{16769247116120717615001603944596934200575}{109966073252115798606144700598765313610753} a^{12} - \frac{4306992262614633636532796866302437785528}{109966073252115798606144700598765313610753} a^{11} + \frac{51971790666735983607680417084241466504959}{109966073252115798606144700598765313610753} a^{10} - \frac{28367190663304393807715197908770003587621}{109966073252115798606144700598765313610753} a^{9} - \frac{11481163689253235960509813677485043991191}{109966073252115798606144700598765313610753} a^{8} + \frac{19457670119168810828518065011543233098749}{109966073252115798606144700598765313610753} a^{7} - \frac{13971635821181051780927050074460496619006}{109966073252115798606144700598765313610753} a^{6} - \frac{18777245587544993582165270806516417820430}{109966073252115798606144700598765313610753} a^{5} + \frac{17330764494232813919749847495145708992249}{109966073252115798606144700598765313610753} a^{4} + \frac{27159843912222489444582856882489203449681}{109966073252115798606144700598765313610753} a^{3} + \frac{38179572404907805650019639838500161652803}{109966073252115798606144700598765313610753} a^{2} + \frac{22051393519320479310268003663771271168426}{109966073252115798606144700598765313610753} a + \frac{52062867786856477082255931673004518439370}{109966073252115798606144700598765313610753}$, $\frac{1}{109966073252115798606144700598765313610753} a^{35} - \frac{12}{109966073252115798606144700598765313610753} a^{33} + \frac{20362765585889794221510637396313657606}{109966073252115798606144700598765313610753} a^{32} - \frac{1350563243213358118483673445413964432696}{109966073252115798606144700598765313610753} a^{31} + \frac{1010730098313570777503647119174673651157}{109966073252115798606144700598765313610753} a^{30} + \frac{57085801171236153691761550116305226168}{3547292685552122535682087116089203664863} a^{29} - \frac{896732855394793366059297006718351592284}{109966073252115798606144700598765313610753} a^{28} - \frac{127817778145735089219596078694705262847}{109966073252115798606144700598765313610753} a^{27} + \frac{1229976424249285890813133035085978581737}{109966073252115798606144700598765313610753} a^{26} + \frac{464366263410505557359961768712863748946}{109966073252115798606144700598765313610753} a^{25} + \frac{1656917561096755145145581056295182421858}{109966073252115798606144700598765313610753} a^{24} + \frac{1046409272715473383074789853955382934635}{109966073252115798606144700598765313610753} a^{23} + \frac{325409717008173835066074167213186300791}{109966073252115798606144700598765313610753} a^{22} + \frac{47816803230309046712618446686142827831510}{109966073252115798606144700598765313610753} a^{21} - \frac{3920060701156900989091460849053389498596}{109966073252115798606144700598765313610753} a^{20} - \frac{41814546899634232069120977501950429539270}{109966073252115798606144700598765313610753} a^{19} + \frac{6169291809936781657504938050877135613074}{109966073252115798606144700598765313610753} a^{18} - \frac{10024304247004467269588391652351217314249}{109966073252115798606144700598765313610753} a^{17} + \frac{37458467084511843480915128379586170510500}{109966073252115798606144700598765313610753} a^{16} + \frac{19645149084304332808019707810718206153826}{109966073252115798606144700598765313610753} a^{15} + \frac{25536651608452863188617955202569834853860}{109966073252115798606144700598765313610753} a^{14} + \frac{26037904416577094382351839920231596405546}{109966073252115798606144700598765313610753} a^{13} - \frac{3695410303398852919862625997802220158785}{109966073252115798606144700598765313610753} a^{12} + \frac{49268631385034649072167404839114636298080}{109966073252115798606144700598765313610753} a^{11} + \frac{14091083569250299053610469708300775536242}{109966073252115798606144700598765313610753} a^{10} - \frac{18964215115862842562110049178075047942814}{109966073252115798606144700598765313610753} a^{9} - \frac{37567940045922748061443205784678588114278}{109966073252115798606144700598765313610753} a^{8} - \frac{24632229260849918654354605654328857909856}{109966073252115798606144700598765313610753} a^{7} - \frac{27279249943757998173296881637100098363371}{109966073252115798606144700598765313610753} a^{6} + \frac{38990462502499351220666577976980901328478}{109966073252115798606144700598765313610753} a^{5} - \frac{46337351638025826502836620092588625531012}{109966073252115798606144700598765313610753} a^{4} + \frac{24439659915854683809768606376289474003896}{109966073252115798606144700598765313610753} a^{3} - \frac{34783817436145984991235236358690147965957}{109966073252115798606144700598765313610753} a^{2} + \frac{53128677370565435618512840748099529143402}{109966073252115798606144700598765313610753} a - \frac{37400535510680226755025228303470518473684}{109966073252115798606144700598765313610753}$

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

Class group and class number

$C_{7}\times C_{66031}$, which has order $462217$ (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{406968674078587842386368787410359488588}{2557350540746879037352202339506170083971} a^{35} - \frac{347855718627033703917048171624948452749}{2557350540746879037352202339506170083971} a^{34} + \frac{6859816050354058386734346951668714862647}{2557350540746879037352202339506170083971} a^{33} - \frac{7949904987641604289144034799547357429398}{2557350540746879037352202339506170083971} a^{32} + \frac{87020515201398858251323134143355405632588}{2557350540746879037352202339506170083971} a^{31} - \frac{24641457230219375028958595059621196986928}{2557350540746879037352202339506170083971} a^{30} + \frac{932745914468298196266102402476638432002277}{2557350540746879037352202339506170083971} a^{29} - \frac{87030477734635050903047216306493642028654}{2557350540746879037352202339506170083971} a^{28} + \frac{9746165305192833056491849626700306307173828}{2557350540746879037352202339506170083971} a^{27} - \frac{2567090436153128788430983588005453321217833}{2557350540746879037352202339506170083971} a^{26} + \frac{50173505734925532399014477449443294242817081}{2557350540746879037352202339506170083971} a^{25} - \frac{16960644470709455716715068294258385485361334}{2557350540746879037352202339506170083971} a^{24} + \frac{218293635358173417748518570499178611789595653}{2557350540746879037352202339506170083971} a^{23} - \frac{79806508898327407000999822942800492820183570}{2557350540746879037352202339506170083971} a^{22} + \frac{861002517031256345811454984383759229976959075}{2557350540746879037352202339506170083971} a^{21} - \frac{9928848132779379245673805555169185714748829}{82495178733770291527490398048586131741} a^{20} + \frac{2922899263195285184506562840485339078533435284}{2557350540746879037352202339506170083971} a^{19} - \frac{973396918289334566019693899693687317895474713}{2557350540746879037352202339506170083971} a^{18} + \frac{5549664272916195848202213485147210568347184203}{2557350540746879037352202339506170083971} a^{17} - \frac{439057612044272625689817312424517195673725953}{2557350540746879037352202339506170083971} a^{16} + \frac{9532258884867320500446366074256672942139447726}{2557350540746879037352202339506170083971} a^{15} + \frac{6838595141765767871225205163448207890931485947}{2557350540746879037352202339506170083971} a^{14} + \frac{11049078055527513156471054072545035772118026488}{2557350540746879037352202339506170083971} a^{13} + \frac{5169864610779319258001703763544493767886754084}{2557350540746879037352202339506170083971} a^{12} + \frac{9906651128288439607697355119260641827880326985}{2557350540746879037352202339506170083971} a^{11} + \frac{1435007111117370999894090944327216522678356763}{2557350540746879037352202339506170083971} a^{10} + \frac{1064837801357080484067901796899274129826084701}{2557350540746879037352202339506170083971} a^{9} + \frac{180827871600770891586419513200578851954336318}{2557350540746879037352202339506170083971} a^{8} + \frac{102946936977885644014639274292573479232718672}{2557350540746879037352202339506170083971} a^{7} - \frac{16013697047820103216299214416853773958603338}{2557350540746879037352202339506170083971} a^{6} + \frac{8314915609569444218272253481129310364073247}{2557350540746879037352202339506170083971} a^{5} - \frac{651304902274271170277210551783177938355712}{2557350540746879037352202339506170083971} a^{4} + \frac{726144976602140005269575278090251220606831}{2557350540746879037352202339506170083971} a^{3} - \frac{8769308600782264830353714918237355966629}{2557350540746879037352202339506170083971} a^{2} + \frac{1394235332745020163966883441261032634771}{2557350540746879037352202339506170083971} a + \frac{407358091372925813689264430377300248803}{2557350540746879037352202339506170083971} \) (order $10$)
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:  \( 1106493498513711.8 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 3.3.1369.1, \(\Q(\zeta_{5})\), 6.6.234270125.1, 9.9.3512479453921.1, 12.0.6860311433439453125.1, 18.18.24096702957455403051316876953125.1

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 $36$ $36$ R $36$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $36$ $36$ $18^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{36}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ $36$ $18^{2}$

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
5Data not computed
37Data not computed