Normalized defining polynomial

\( x^{36} - x^{35} + 3 x^{34} - 4 x^{33} + 9 x^{32} - 14 x^{31} + 28 x^{30} - 47 x^{29} + 89 x^{28} + \cdots + 1 \) x^36 - x^35 + 3*x^34 - 4*x^33 + 9*x^32 - 14*x^31 + 28*x^30 - 47*x^29 + 89*x^28 - 155*x^27 + 286*x^26 - 507*x^25 + 924*x^24 + 442*x^23 + 899*x^22 + 909*x^21 + 1331*x^20 + 1386*x^19 + 2185*x^18 + 1918*x^17 + 3838*x^16 + 2183*x^15 + 7411*x^14 + 793*x^13 + 16212*x^12 - 7215*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1

Invariants

Degree:		$36$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 18]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(1102766555593920971763188134004988790509406708671598011853\) 1102766555593920971763188134004988790509406708671598011853 \(\medspace = 7^{24}\cdot 13^{33}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(38.42\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$7^{2/3}13^{11/12}\approx 38.416114130176354$
Ramified primes:		\(7\), \(13\) 7, 13	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:		$36$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(91=7\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(2,·)$, $\chi_{91}(4,·)$, $\chi_{91}(8,·)$, $\chi_{91}(9,·)$, $\chi_{91}(11,·)$, $\chi_{91}(15,·)$, $\chi_{91}(16,·)$, $\chi_{91}(18,·)$, $\chi_{91}(22,·)$, $\chi_{91}(23,·)$, $\chi_{91}(25,·)$, $\chi_{91}(29,·)$, $\chi_{91}(30,·)$, $\chi_{91}(32,·)$, $\chi_{91}(36,·)$, $\chi_{91}(37,·)$, $\chi_{91}(43,·)$, $\chi_{91}(44,·)$, $\chi_{91}(46,·)$, $\chi_{91}(50,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(57,·)$, $\chi_{91}(58,·)$, $\chi_{91}(60,·)$, $\chi_{91}(64,·)$, $\chi_{91}(67,·)$, $\chi_{91}(71,·)$, $\chi_{91}(72,·)$, $\chi_{91}(74,·)$, $\chi_{91}(79,·)$, $\chi_{91}(81,·)$, $\chi_{91}(85,·)$, $\chi_{91}(86,·)$, $\chi_{91}(88,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{8}$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{9}$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{10}$, $\frac{1}{3}a^{24}-\frac{1}{3}a^{11}$, $\frac{1}{1257987}a^{25}-\frac{35565}{419329}a^{24}-\frac{120667}{1257987}a^{23}-\frac{92722}{1257987}a^{22}+\frac{54674}{419329}a^{21}-\frac{50804}{1257987}a^{20}-\frac{44401}{419329}a^{19}+\frac{195617}{1257987}a^{18}-\frac{31166}{419329}a^{17}-\frac{22600}{419329}a^{16}+\frac{76421}{1257987}a^{15}+\frac{113810}{1257987}a^{14}-\frac{28768}{1257987}a^{13}-\frac{461542}{1257987}a^{12}+\frac{107525}{419329}a^{11}+\frac{546037}{1257987}a^{10}-\frac{362429}{1257987}a^{9}+\frac{33254}{419329}a^{8}+\frac{560075}{1257987}a^{7}-\frac{101217}{419329}a^{6}+\frac{265576}{1257987}a^{5}+\frac{175285}{419329}a^{4}+\frac{40325}{419329}a^{3}-\frac{61676}{1257987}a^{2}+\frac{410152}{1257987}a-\frac{412529}{1257987}$, $\frac{1}{3773961}a^{26}+\frac{463636}{3773961}a^{13}+\frac{86830}{3773961}$, $\frac{1}{3773961}a^{27}+\frac{463636}{3773961}a^{14}+\frac{86830}{3773961}a$, $\frac{1}{3773961}a^{28}+\frac{463636}{3773961}a^{15}+\frac{86830}{3773961}a^{2}$, $\frac{1}{3773961}a^{29}+\frac{463636}{3773961}a^{16}+\frac{86830}{3773961}a^{3}$, $\frac{1}{3773961}a^{30}+\frac{463636}{3773961}a^{17}+\frac{86830}{3773961}a^{4}$, $\frac{1}{3773961}a^{31}+\frac{463636}{3773961}a^{18}+\frac{86830}{3773961}a^{5}$, $\frac{1}{3773961}a^{32}+\frac{463636}{3773961}a^{19}+\frac{86830}{3773961}a^{6}$, $\frac{1}{3773961}a^{33}+\frac{463636}{3773961}a^{20}+\frac{86830}{3773961}a^{7}$, $\frac{1}{3773961}a^{34}+\frac{463636}{3773961}a^{21}+\frac{86830}{3773961}a^{8}$, $\frac{1}{3773961}a^{35}+\frac{463636}{3773961}a^{22}+\frac{86830}{3773961}a^{9}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/3*a^13 - 1/3, 1/3*a^14 - 1/3*a, 1/3*a^15 - 1/3*a^2, 1/3*a^16 - 1/3*a^3, 1/3*a^17 - 1/3*a^4, 1/3*a^18 - 1/3*a^5, 1/3*a^19 - 1/3*a^6, 1/3*a^20 - 1/3*a^7, 1/3*a^21 - 1/3*a^8, 1/3*a^22 - 1/3*a^9, 1/3*a^23 - 1/3*a^10, 1/3*a^24 - 1/3*a^11, 1/1257987*a^25 - 35565/419329*a^24 - 120667/1257987*a^23 - 92722/1257987*a^22 + 54674/419329*a^21 - 50804/1257987*a^20 - 44401/419329*a^19 + 195617/1257987*a^18 - 31166/419329*a^17 - 22600/419329*a^16 + 76421/1257987*a^15 + 113810/1257987*a^14 - 28768/1257987*a^13 - 461542/1257987*a^12 + 107525/419329*a^11 + 546037/1257987*a^10 - 362429/1257987*a^9 + 33254/419329*a^8 + 560075/1257987*a^7 - 101217/419329*a^6 + 265576/1257987*a^5 + 175285/419329*a^4 + 40325/419329*a^3 - 61676/1257987*a^2 + 410152/1257987*a - 412529/1257987, 1/3773961*a^26 + 463636/3773961*a^13 + 86830/3773961, 1/3773961*a^27 + 463636/3773961*a^14 + 86830/3773961*a, 1/3773961*a^28 + 463636/3773961*a^15 + 86830/3773961*a^2, 1/3773961*a^29 + 463636/3773961*a^16 + 86830/3773961*a^3, 1/3773961*a^30 + 463636/3773961*a^17 + 86830/3773961*a^4, 1/3773961*a^31 + 463636/3773961*a^18 + 86830/3773961*a^5, 1/3773961*a^32 + 463636/3773961*a^19 + 86830/3773961*a^6, 1/3773961*a^33 + 463636/3773961*a^20 + 86830/3773961*a^7, 1/3773961*a^34 + 463636/3773961*a^21 + 86830/3773961*a^8, 1/3773961*a^35 + 463636/3773961*a^22 + 86830/3773961*a^9

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (assuming GRH)

Unit group

Rank:		$17$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( \frac{332648}{3773961} a^{35} - \frac{332648}{1257987} a^{34} + \frac{1330592}{3773961} a^{33} - \frac{332648}{419329} a^{32} + \frac{4657072}{3773961} a^{31} - \frac{9314144}{3773961} a^{30} + \frac{15634456}{3773961} a^{29} - \frac{29605672}{3773961} a^{28} + \frac{51560440}{3773961} a^{27} - \frac{95137328}{3773961} a^{26} + \frac{56217512}{1257987} a^{25} - \frac{102455584}{1257987} a^{24} + \frac{183178099}{1257987} a^{23} - \frac{299050552}{3773961} a^{22} - \frac{33597448}{419329} a^{21} - \frac{442754488}{3773961} a^{20} - \frac{51227792}{419329} a^{19} - \frac{726835880}{3773961} a^{18} - \frac{638018864}{3773961} a^{17} - \frac{1276703024}{3773961} a^{16} - \frac{726170584}{3773961} a^{15} - \frac{2465254328}{3773961} a^{14} - \frac{263789864}{3773961} a^{13} - \frac{1797629792}{1257987} a^{12} + \frac{800018440}{1257987} a^{11} - \frac{4483351483}{1257987} a^{10} + \frac{475353992}{3773961} a^{9} - \frac{70521376}{1257987} a^{8} + \frac{94139384}{3773961} a^{7} - \frac{4657072}{419329} a^{6} + \frac{18628288}{3773961} a^{5} - \frac{8316200}{3773961} a^{4} + \frac{3659128}{3773961} a^{3} - \frac{1663240}{3773961} a^{2} + \frac{665296}{3773961} a - \frac{332648}{3773961} \) (332648)/(3773961)*a^(35) - (332648)/(1257987)*a^(34) + (1330592)/(3773961)*a^(33) - (332648)/(419329)*a^(32) + (4657072)/(3773961)*a^(31) - (9314144)/(3773961)*a^(30) + (15634456)/(3773961)*a^(29) - (29605672)/(3773961)*a^(28) + (51560440)/(3773961)*a^(27) - (95137328)/(3773961)*a^(26) + (56217512)/(1257987)*a^(25) - (102455584)/(1257987)*a^(24) + (183178099)/(1257987)*a^(23) - (299050552)/(3773961)*a^(22) - (33597448)/(419329)*a^(21) - (442754488)/(3773961)*a^(20) - (51227792)/(419329)*a^(19) - (726835880)/(3773961)*a^(18) - (638018864)/(3773961)*a^(17) - (1276703024)/(3773961)*a^(16) - (726170584)/(3773961)*a^(15) - (2465254328)/(3773961)*a^(14) - (263789864)/(3773961)*a^(13) - (1797629792)/(1257987)*a^(12) + (800018440)/(1257987)*a^(11) - (4483351483)/(1257987)*a^(10) + (475353992)/(3773961)*a^(9) - (70521376)/(1257987)*a^(8) + (94139384)/(3773961)*a^(7) - (4657072)/(419329)*a^(6) + (18628288)/(3773961)*a^(5) - (8316200)/(3773961)*a^(4) + (3659128)/(3773961)*a^(3) - (1663240)/(3773961)*a^(2) + (665296)/(3773961)*a - (332648)/(3773961)  (order $26$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$\frac{65870}{3773961}a^{34}-\frac{1202}{3773961}a^{29}+\frac{137931491}{3773961}a^{21}-\frac{2518205}{3773961}a^{16}-\frac{2452391452}{3773961}a^{8}+\frac{42814819}{3773961}a^{3}$, $\frac{1494974}{3773961}a^{35}-\frac{747487}{3773961}a^{34}+\frac{3737435}{3773961}a^{33}-\frac{3737435}{3773961}a^{32}+\frac{10464818}{3773961}a^{31}-\frac{14202253}{3773961}a^{30}+\frac{10464818}{1257987}a^{29}-\frac{16444714}{1257987}a^{28}+\frac{32640163}{1257987}a^{27}-\frac{165194627}{3773961}a^{26}+\frac{103900693}{1257987}a^{25}-\frac{181390225}{1257987}a^{24}+\frac{111375563}{419329}a^{23}+\frac{1351456496}{3773961}a^{22}+\frac{1674370880}{3773961}a^{21}+\frac{2030922179}{3773961}a^{20}+\frac{2669276077}{3773961}a^{19}+\frac{3066939161}{3773961}a^{18}+\frac{4302535172}{3773961}a^{17}+\frac{500068803}{419329}a^{16}+\frac{796821142}{419329}a^{15}+\frac{681303999}{419329}a^{14}+\frac{12711016435}{3773961}a^{13}+\frac{2241713513}{1257987}a^{12}+\frac{8276329204}{1257987}a^{11}+\frac{148002426}{419329}a^{10}-\frac{592757191}{3773961}a^{9}+\frac{263862911}{3773961}a^{8}-\frac{117355459}{3773961}a^{7}+\frac{52324090}{3773961}a^{6}-\frac{23172097}{3773961}a^{5}+\frac{10464818}{3773961}a^{4}-\frac{1494974}{1257987}a^{3}+\frac{747487}{1257987}a^{2}+\frac{2577503}{1257987}a+\frac{747487}{3773961}$, $\frac{29344}{3773961}a^{33}-\frac{1447}{419329}a^{32}+\frac{649}{419329}a^{31}+\frac{61446742}{3773961}a^{20}-\frac{9089914}{1257987}a^{19}+\frac{4077245}{1257987}a^{18}-\frac{1091416871}{3773961}a^{7}+\frac{161908735}{1257987}a^{6}-\frac{72056165}{1257987}a^{5}$, $\frac{148061}{3773961}a^{35}+\frac{649}{419329}a^{31}+\frac{310039982}{3773961}a^{22}+\frac{4077245}{1257987}a^{18}-\frac{5510473570}{3773961}a^{9}-\frac{72056165}{1257987}a^{5}$, $\frac{148061}{3773961}a^{35}-\frac{1447}{419329}a^{32}+\frac{308}{3773961}a^{27}+\frac{310039982}{3773961}a^{22}-\frac{9089914}{1257987}a^{19}+\frac{647357}{3773961}a^{14}-\frac{5510473570}{3773961}a^{9}+\frac{161908735}{1257987}a^{6}-\frac{8479996}{3773961}a$, $\frac{27397}{1257987}a^{35}+\frac{57369497}{1257987}a^{22}-\frac{339786902}{419329}a^{9}$, $\frac{139}{3773961}a^{26}+\frac{288067}{3773961}a^{13}-\frac{3026474}{3773961}$, $\frac{332648}{3773961}a^{35}-\frac{332648}{1257987}a^{34}+\frac{1330592}{3773961}a^{33}-\frac{332648}{419329}a^{32}+\frac{4657072}{3773961}a^{31}-\frac{9314144}{3773961}a^{30}+\frac{15633254}{3773961}a^{29}-\frac{29605672}{3773961}a^{28}+\frac{51560440}{3773961}a^{27}-\frac{95137159}{3773961}a^{26}+\frac{56217512}{1257987}a^{25}-\frac{102455584}{1257987}a^{24}+\frac{183178099}{1257987}a^{23}-\frac{299050552}{3773961}a^{22}-\frac{33597448}{419329}a^{21}-\frac{442754488}{3773961}a^{20}-\frac{51227792}{419329}a^{19}-\frac{726835880}{3773961}a^{18}-\frac{638018864}{3773961}a^{17}-\frac{1279221229}{3773961}a^{16}-\frac{726170584}{3773961}a^{15}-\frac{2465254328}{3773961}a^{14}-\frac{263430574}{3773961}a^{13}-\frac{1797629792}{1257987}a^{12}+\frac{800018440}{1257987}a^{11}-\frac{4483351483}{1257987}a^{10}+\frac{475353992}{3773961}a^{9}-\frac{70521376}{1257987}a^{8}+\frac{94139384}{3773961}a^{7}-\frac{4657072}{419329}a^{6}+\frac{18628288}{3773961}a^{5}-\frac{8316200}{3773961}a^{4}+\frac{46473947}{3773961}a^{3}-\frac{1663240}{3773961}a^{2}+\frac{665296}{3773961}a-\frac{2012209}{3773961}$, $\frac{414839}{3773961}a^{35}-\frac{414839}{1257987}a^{34}+\frac{1659356}{3773961}a^{33}-\frac{414839}{419329}a^{32}+\frac{5813587}{3773961}a^{31}-\frac{11615492}{3773961}a^{30}+\frac{19497433}{3773961}a^{29}-\frac{36921118}{3773961}a^{28}+\frac{64300045}{3773961}a^{27}-\frac{118643954}{3773961}a^{26}+\frac{70107791}{1257987}a^{25}-\frac{127770412}{1257987}a^{24}+\frac{228438122}{1257987}a^{23}-\frac{372940261}{3773961}a^{22}-\frac{41898739}{419329}a^{21}-\frac{552150709}{3773961}a^{20}-\frac{63885206}{419329}a^{19}-\frac{894191480}{3773961}a^{18}-\frac{795661202}{3773961}a^{17}-\frac{1592152082}{3773961}a^{16}-\frac{906528961}{3773961}a^{15}-\frac{3074371829}{3773961}a^{14}-\frac{328967327}{3773961}a^{13}-\frac{2241789956}{1257987}a^{12}+\frac{997687795}{1257987}a^{11}-\frac{5590683956}{1257987}a^{10}+\frac{592804931}{3773961}a^{9}-\frac{87945868}{1257987}a^{8}+\frac{117399437}{3773961}a^{7}-\frac{5807746}{419329}a^{6}-\frac{192937511}{3773961}a^{5}-\frac{10370975}{3773961}a^{4}+\frac{4563229}{3773961}a^{3}+\frac{16980197}{3773961}a^{2}+\frac{829678}{3773961}a-\frac{414839}{3773961}$, $\frac{1679561}{3773961}a^{35}-\frac{1679561}{3773961}a^{34}+\frac{5068027}{3773961}a^{33}-\frac{6718244}{3773961}a^{32}+\frac{1679561}{419329}a^{31}-\frac{23513854}{3773961}a^{30}+\frac{47029804}{3773961}a^{29}-\frac{78939367}{3773961}a^{28}+\frac{149480929}{3773961}a^{27}-\frac{260331955}{3773961}a^{26}+\frac{53372735}{419329}a^{25}-\frac{283845809}{1257987}a^{24}+\frac{517304788}{1257987}a^{23}+\frac{742365962}{3773961}a^{22}+\frac{1509925339}{3773961}a^{21}+\frac{1588167691}{3773961}a^{20}+\frac{2235495691}{3773961}a^{19}+\frac{258652394}{419329}a^{18}+\frac{3669840785}{3773961}a^{17}+\frac{3225787051}{3773961}a^{16}+\frac{6446155118}{3773961}a^{15}+\frac{3666481663}{3773961}a^{14}+\frac{12447226571}{3773961}a^{13}+\frac{148027907}{419329}a^{12}+\frac{9076347644}{1257987}a^{11}-\frac{4039344205}{1257987}a^{10}+\frac{5393070371}{3773961}a^{9}-\frac{2400092669}{3773961}a^{8}-\frac{23216075}{3773961}a^{7}-\frac{475315763}{3773961}a^{6}+\frac{23513854}{419329}a^{5}-\frac{94055416}{3773961}a^{4}-\frac{35160617}{3773961}a^{3}-\frac{18475171}{3773961}a^{2}+\frac{8397805}{3773961}a-\frac{3359122}{3773961}$, $\frac{61529}{1257987}a^{35}-\frac{649}{419329}a^{31}+\frac{128841577}{1257987}a^{22}-\frac{4077245}{1257987}a^{18}-\frac{763494239}{419329}a^{9}+\frac{72056165}{1257987}a^{5}-1$, $\frac{65870}{3773961}a^{34}+\frac{23503}{3773961}a^{32}+\frac{2543}{3773961}a^{30}+\frac{137931491}{3773961}a^{21}+\frac{49215007}{3773961}a^{19}+\frac{5324477}{3773961}a^{17}-\frac{2452391452}{3773961}a^{8}-\frac{875248376}{3773961}a^{6}-\frac{96204034}{3773961}a^{4}$, $\frac{3359122}{3773961}a^{35}-\frac{510500}{1257987}a^{34}+\frac{8397805}{3773961}a^{33}-\frac{8391964}{3773961}a^{32}+\frac{23506672}{3773961}a^{31}-\frac{10636819}{1257987}a^{30}+\frac{23513407}{1257987}a^{29}-\frac{110850718}{3773961}a^{28}+\frac{24446928}{419329}a^{27}-\frac{123727604}{1257987}a^{26}+\frac{233458979}{1257987}a^{25}-\frac{407573413}{1257987}a^{24}+\frac{250254589}{419329}a^{23}+\frac{3036646288}{3773961}a^{22}+\frac{452472958}{419329}a^{21}+\frac{4563367237}{3773961}a^{20}+\frac{6009944066}{3773961}a^{19}+\frac{6876200776}{3773961}a^{18}+\frac{1074452369}{419329}a^{17}+\frac{1123314501}{419329}a^{16}+\frac{16114355591}{3773961}a^{15}+\frac{1530981153}{419329}a^{14}+\frac{9520431365}{1257987}a^{13}+\frac{5037003439}{1257987}a^{12}+\frac{18596779009}{1257987}a^{11}+\frac{332553078}{419329}a^{10}-\frac{1331891873}{3773961}a^{9}-\frac{1639196179}{1257987}a^{8}-\frac{263691077}{3773961}a^{7}-\frac{98599225}{3773961}a^{6}+\frac{217491319}{3773961}a^{5}-\frac{6433655}{1257987}a^{4}+\frac{14437283}{1257987}a^{3}-\frac{3441313}{3773961}a^{2}+\frac{988315}{419329}a$, $\frac{65870}{3773961}a^{35}-\frac{932074}{3773961}a^{34}+\frac{1330592}{3773961}a^{33}-\frac{332648}{419329}a^{32}+\frac{4657072}{3773961}a^{31}-\frac{9314144}{3773961}a^{30}+\frac{15634456}{3773961}a^{29}-\frac{29605672}{3773961}a^{28}+\frac{51560440}{3773961}a^{27}-\frac{95137328}{3773961}a^{26}+\frac{56217512}{1257987}a^{25}-\frac{102455584}{1257987}a^{24}+\frac{183178099}{1257987}a^{23}-\frac{857683774}{3773961}a^{22}-\frac{164445541}{3773961}a^{21}-\frac{442754488}{3773961}a^{20}-\frac{51227792}{419329}a^{19}-\frac{726835880}{3773961}a^{18}-\frac{638018864}{3773961}a^{17}-\frac{1276703024}{3773961}a^{16}-\frac{726170584}{3773961}a^{15}-\frac{2465254328}{3773961}a^{14}-\frac{263789864}{3773961}a^{13}-\frac{1797629792}{1257987}a^{12}+\frac{800018440}{1257987}a^{11}-\frac{4483351483}{1257987}a^{10}+\frac{10404884261}{3773961}a^{9}-\frac{2663955580}{3773961}a^{8}+\frac{94139384}{3773961}a^{7}-\frac{4657072}{419329}a^{6}+\frac{18628288}{3773961}a^{5}-\frac{8316200}{3773961}a^{4}+\frac{3659128}{3773961}a^{3}-\frac{1663240}{3773961}a^{2}+\frac{665296}{3773961}a-\frac{332648}{3773961}$, $\frac{29344}{3773961}a^{33}-\frac{3298}{3773961}a^{30}+\frac{308}{3773961}a^{27}+\frac{61446742}{3773961}a^{20}-\frac{6907258}{3773961}a^{17}+\frac{647357}{3773961}a^{14}-\frac{1091416871}{3773961}a^{7}+\frac{119964461}{3773961}a^{4}-\frac{8479996}{3773961}a$, $\frac{299314}{419329}a^{35}-\frac{149657}{419329}a^{34}+\frac{748285}{419329}a^{33}-\frac{746838}{419329}a^{32}+\frac{2095198}{419329}a^{31}-\frac{2843483}{419329}a^{30}+\frac{56569144}{3773961}a^{29}-\frac{9877362}{419329}a^{28}+\frac{19605067}{419329}a^{27}-\frac{33074197}{419329}a^{26}+\frac{62406969}{419329}a^{25}-\frac{326850898}{1257987}a^{24}+\frac{200690037}{419329}a^{23}+\frac{270579856}{419329}a^{22}+\frac{335231680}{419329}a^{21}+\frac{406618069}{419329}a^{20}+\frac{1612365355}{1257987}a^{19}+\frac{614042671}{419329}a^{18}+\frac{861425692}{419329}a^{17}+\frac{8107244968}{3773961}a^{16}+\frac{1435809258}{419329}a^{15}+\frac{1227786028}{419329}a^{14}+\frac{2544917285}{419329}a^{13}+\frac{1346464029}{419329}a^{12}+\frac{14913445966}{1257987}a^{11}+\frac{266688774}{419329}a^{10}-\frac{118678001}{419329}a^{9}+\frac{52828921}{419329}a^{8}-\frac{23496149}{419329}a^{7}-\frac{130480765}{1257987}a^{6}-\frac{4639367}{419329}a^{5}+\frac{2095198}{419329}a^{4}+\frac{34733341}{3773961}a^{3}+\frac{448971}{419329}a^{2}-\frac{149657}{419329}a+\frac{149657}{419329}$, $\frac{1597370}{3773961}a^{35}-\frac{421574}{1257987}a^{34}+\frac{4446439}{3773961}a^{33}-\frac{5398718}{3773961}a^{32}+\frac{13038556}{3773961}a^{31}-\frac{19369348}{3773961}a^{30}+\frac{40068086}{3773961}a^{29}-\frac{65761799}{3773961}a^{28}+\frac{42177158}{1257987}a^{27}-\frac{217986370}{3773961}a^{26}+\frac{135095896}{1257987}a^{25}-\frac{238243021}{1257987}a^{24}+\frac{435772448}{1257987}a^{23}+\frac{1013404292}{3773961}a^{22}+\frac{527688682}{1257987}a^{21}+\frac{1723790140}{3773961}a^{20}+\frac{2413438495}{3773961}a^{19}+\frac{2641671301}{3773961}a^{18}+\frac{3951303578}{3773961}a^{17}+\frac{3788073335}{3773961}a^{16}+\frac{6769660348}{3773961}a^{15}+\frac{1587920578}{1257987}a^{14}+\frac{12564927011}{3773961}a^{13}+\frac{1244205365}{1257987}a^{12}+\frac{8720260939}{1257987}a^{11}-\frac{2044045058}{1257987}a^{10}+\frac{2729099750}{3773961}a^{9}-\frac{404836334}{1257987}a^{8}+\frac{1026299533}{3773961}a^{7}+\frac{29066128}{3773961}a^{6}+\frac{376686946}{3773961}a^{5}-\frac{47539072}{3773961}a^{4}+\frac{64120781}{3773961}a^{3}-\frac{28309262}{3773961}a^{2}+\frac{480858}{419329}a-\frac{6237535}{3773961}$ 65870/3773961*a^34 - 1202/3773961*a^29 + 137931491/3773961*a^21 - 2518205/3773961*a^16 - 2452391452/3773961*a^8 + 42814819/3773961*a^3, 1494974/3773961*a^35 - 747487/3773961*a^34 + 3737435/3773961*a^33 - 3737435/3773961*a^32 + 10464818/3773961*a^31 - 14202253/3773961*a^30 + 10464818/1257987*a^29 - 16444714/1257987*a^28 + 32640163/1257987*a^27 - 165194627/3773961*a^26 + 103900693/1257987*a^25 - 181390225/1257987*a^24 + 111375563/419329*a^23 + 1351456496/3773961*a^22 + 1674370880/3773961*a^21 + 2030922179/3773961*a^20 + 2669276077/3773961*a^19 + 3066939161/3773961*a^18 + 4302535172/3773961*a^17 + 500068803/419329*a^16 + 796821142/419329*a^15 + 681303999/419329*a^14 + 12711016435/3773961*a^13 + 2241713513/1257987*a^12 + 8276329204/1257987*a^11 + 148002426/419329*a^10 - 592757191/3773961*a^9 + 263862911/3773961*a^8 - 117355459/3773961*a^7 + 52324090/3773961*a^6 - 23172097/3773961*a^5 + 10464818/3773961*a^4 - 1494974/1257987*a^3 + 747487/1257987*a^2 + 2577503/1257987*a + 747487/3773961, 29344/3773961*a^33 - 1447/419329*a^32 + 649/419329*a^31 + 61446742/3773961*a^20 - 9089914/1257987*a^19 + 4077245/1257987*a^18 - 1091416871/3773961*a^7 + 161908735/1257987*a^6 - 72056165/1257987*a^5, 148061/3773961*a^35 + 649/419329*a^31 + 310039982/3773961*a^22 + 4077245/1257987*a^18 - 5510473570/3773961*a^9 - 72056165/1257987*a^5, 148061/3773961*a^35 - 1447/419329*a^32 + 308/3773961*a^27 + 310039982/3773961*a^22 - 9089914/1257987*a^19 + 647357/3773961*a^14 - 5510473570/3773961*a^9 + 161908735/1257987*a^6 - 8479996/3773961*a, 27397/1257987*a^35 + 57369497/1257987*a^22 - 339786902/419329*a^9, 139/3773961*a^26 + 288067/3773961*a^13 - 3026474/3773961, 332648/3773961*a^35 - 332648/1257987*a^34 + 1330592/3773961*a^33 - 332648/419329*a^32 + 4657072/3773961*a^31 - 9314144/3773961*a^30 + 15633254/3773961*a^29 - 29605672/3773961*a^28 + 51560440/3773961*a^27 - 95137159/3773961*a^26 + 56217512/1257987*a^25 - 102455584/1257987*a^24 + 183178099/1257987*a^23 - 299050552/3773961*a^22 - 33597448/419329*a^21 - 442754488/3773961*a^20 - 51227792/419329*a^19 - 726835880/3773961*a^18 - 638018864/3773961*a^17 - 1279221229/3773961*a^16 - 726170584/3773961*a^15 - 2465254328/3773961*a^14 - 263430574/3773961*a^13 - 1797629792/1257987*a^12 + 800018440/1257987*a^11 - 4483351483/1257987*a^10 + 475353992/3773961*a^9 - 70521376/1257987*a^8 + 94139384/3773961*a^7 - 4657072/419329*a^6 + 18628288/3773961*a^5 - 8316200/3773961*a^4 + 46473947/3773961*a^3 - 1663240/3773961*a^2 + 665296/3773961*a - 2012209/3773961, 414839/3773961*a^35 - 414839/1257987*a^34 + 1659356/3773961*a^33 - 414839/419329*a^32 + 5813587/3773961*a^31 - 11615492/3773961*a^30 + 19497433/3773961*a^29 - 36921118/3773961*a^28 + 64300045/3773961*a^27 - 118643954/3773961*a^26 + 70107791/1257987*a^25 - 127770412/1257987*a^24 + 228438122/1257987*a^23 - 372940261/3773961*a^22 - 41898739/419329*a^21 - 552150709/3773961*a^20 - 63885206/419329*a^19 - 894191480/3773961*a^18 - 795661202/3773961*a^17 - 1592152082/3773961*a^16 - 906528961/3773961*a^15 - 3074371829/3773961*a^14 - 328967327/3773961*a^13 - 2241789956/1257987*a^12 + 997687795/1257987*a^11 - 5590683956/1257987*a^10 + 592804931/3773961*a^9 - 87945868/1257987*a^8 + 117399437/3773961*a^7 - 5807746/419329*a^6 - 192937511/3773961*a^5 - 10370975/3773961*a^4 + 4563229/3773961*a^3 + 16980197/3773961*a^2 + 829678/3773961*a - 414839/3773961, 1679561/3773961*a^35 - 1679561/3773961*a^34 + 5068027/3773961*a^33 - 6718244/3773961*a^32 + 1679561/419329*a^31 - 23513854/3773961*a^30 + 47029804/3773961*a^29 - 78939367/3773961*a^28 + 149480929/3773961*a^27 - 260331955/3773961*a^26 + 53372735/419329*a^25 - 283845809/1257987*a^24 + 517304788/1257987*a^23 + 742365962/3773961*a^22 + 1509925339/3773961*a^21 + 1588167691/3773961*a^20 + 2235495691/3773961*a^19 + 258652394/419329*a^18 + 3669840785/3773961*a^17 + 3225787051/3773961*a^16 + 6446155118/3773961*a^15 + 3666481663/3773961*a^14 + 12447226571/3773961*a^13 + 148027907/419329*a^12 + 9076347644/1257987*a^11 - 4039344205/1257987*a^10 + 5393070371/3773961*a^9 - 2400092669/3773961*a^8 - 23216075/3773961*a^7 - 475315763/3773961*a^6 + 23513854/419329*a^5 - 94055416/3773961*a^4 - 35160617/3773961*a^3 - 18475171/3773961*a^2 + 8397805/3773961*a - 3359122/3773961, 61529/1257987*a^35 - 649/419329*a^31 + 128841577/1257987*a^22 - 4077245/1257987*a^18 - 763494239/419329*a^9 + 72056165/1257987*a^5 - 1, 65870/3773961*a^34 + 23503/3773961*a^32 + 2543/3773961*a^30 + 137931491/3773961*a^21 + 49215007/3773961*a^19 + 5324477/3773961*a^17 - 2452391452/3773961*a^8 - 875248376/3773961*a^6 - 96204034/3773961*a^4, 3359122/3773961*a^35 - 510500/1257987*a^34 + 8397805/3773961*a^33 - 8391964/3773961*a^32 + 23506672/3773961*a^31 - 10636819/1257987*a^30 + 23513407/1257987*a^29 - 110850718/3773961*a^28 + 24446928/419329*a^27 - 123727604/1257987*a^26 + 233458979/1257987*a^25 - 407573413/1257987*a^24 + 250254589/419329*a^23 + 3036646288/3773961*a^22 + 452472958/419329*a^21 + 4563367237/3773961*a^20 + 6009944066/3773961*a^19 + 6876200776/3773961*a^18 + 1074452369/419329*a^17 + 1123314501/419329*a^16 + 16114355591/3773961*a^15 + 1530981153/419329*a^14 + 9520431365/1257987*a^13 + 5037003439/1257987*a^12 + 18596779009/1257987*a^11 + 332553078/419329*a^10 - 1331891873/3773961*a^9 - 1639196179/1257987*a^8 - 263691077/3773961*a^7 - 98599225/3773961*a^6 + 217491319/3773961*a^5 - 6433655/1257987*a^4 + 14437283/1257987*a^3 - 3441313/3773961*a^2 + 988315/419329*a, 65870/3773961*a^35 - 932074/3773961*a^34 + 1330592/3773961*a^33 - 332648/419329*a^32 + 4657072/3773961*a^31 - 9314144/3773961*a^30 + 15634456/3773961*a^29 - 29605672/3773961*a^28 + 51560440/3773961*a^27 - 95137328/3773961*a^26 + 56217512/1257987*a^25 - 102455584/1257987*a^24 + 183178099/1257987*a^23 - 857683774/3773961*a^22 - 164445541/3773961*a^21 - 442754488/3773961*a^20 - 51227792/419329*a^19 - 726835880/3773961*a^18 - 638018864/3773961*a^17 - 1276703024/3773961*a^16 - 726170584/3773961*a^15 - 2465254328/3773961*a^14 - 263789864/3773961*a^13 - 1797629792/1257987*a^12 + 800018440/1257987*a^11 - 4483351483/1257987*a^10 + 10404884261/3773961*a^9 - 2663955580/3773961*a^8 + 94139384/3773961*a^7 - 4657072/419329*a^6 + 18628288/3773961*a^5 - 8316200/3773961*a^4 + 3659128/3773961*a^3 - 1663240/3773961*a^2 + 665296/3773961*a - 332648/3773961, 29344/3773961*a^33 - 3298/3773961*a^30 + 308/3773961*a^27 + 61446742/3773961*a^20 - 6907258/3773961*a^17 + 647357/3773961*a^14 - 1091416871/3773961*a^7 + 119964461/3773961*a^4 - 8479996/3773961*a, 299314/419329*a^35 - 149657/419329*a^34 + 748285/419329*a^33 - 746838/419329*a^32 + 2095198/419329*a^31 - 2843483/419329*a^30 + 56569144/3773961*a^29 - 9877362/419329*a^28 + 19605067/419329*a^27 - 33074197/419329*a^26 + 62406969/419329*a^25 - 326850898/1257987*a^24 + 200690037/419329*a^23 + 270579856/419329*a^22 + 335231680/419329*a^21 + 406618069/419329*a^20 + 1612365355/1257987*a^19 + 614042671/419329*a^18 + 861425692/419329*a^17 + 8107244968/3773961*a^16 + 1435809258/419329*a^15 + 1227786028/419329*a^14 + 2544917285/419329*a^13 + 1346464029/419329*a^12 + 14913445966/1257987*a^11 + 266688774/419329*a^10 - 118678001/419329*a^9 + 52828921/419329*a^8 - 23496149/419329*a^7 - 130480765/1257987*a^6 - 4639367/419329*a^5 + 2095198/419329*a^4 + 34733341/3773961*a^3 + 448971/419329*a^2 - 149657/419329*a + 149657/419329, 1597370/3773961*a^35 - 421574/1257987*a^34 + 4446439/3773961*a^33 - 5398718/3773961*a^32 + 13038556/3773961*a^31 - 19369348/3773961*a^30 + 40068086/3773961*a^29 - 65761799/3773961*a^28 + 42177158/1257987*a^27 - 217986370/3773961*a^26 + 135095896/1257987*a^25 - 238243021/1257987*a^24 + 435772448/1257987*a^23 + 1013404292/3773961*a^22 + 527688682/1257987*a^21 + 1723790140/3773961*a^20 + 2413438495/3773961*a^19 + 2641671301/3773961*a^18 + 3951303578/3773961*a^17 + 3788073335/3773961*a^16 + 6769660348/3773961*a^15 + 1587920578/1257987*a^14 + 12564927011/3773961*a^13 + 1244205365/1257987*a^12 + 8720260939/1257987*a^11 - 2044045058/1257987*a^10 + 2729099750/3773961*a^9 - 404836334/1257987*a^8 + 1026299533/3773961*a^7 + 29066128/3773961*a^6 + 376686946/3773961*a^5 - 47539072/3773961*a^4 + 64120781/3773961*a^3 - 28309262/3773961*a^2 + 480858/419329*a - 6237535/3773961 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 4866030378143.887 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 4866030378143.887 \cdot 148}{26\cdot\sqrt{1102766555593920971763188134004988790509406708671598011853}}\cr\approx \mathstrut & 0.194293834986093 \end{aligned}\] (assuming GRH)

Galois group

$C_3\times C_{12}$ (as 36T3):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(7\) 7		Deg $36$	$3$	$12$	$24$
\(13\) 13	13.12.11.4	$x^{12} + 13$	$12$	$1$	$11$	$C_{12}$	$[\ ]_{12}$
	13.12.11.4	$x^{12} + 13$	$12$	$1$	$11$	$C_{12}$	$[\ ]_{12}$
	13.12.11.4	$x^{12} + 13$	$12$	$1$	$11$	$C_{12}$	$[\ ]_{12}$