Normalized defining polynomial

\( x^{36} - 36 x^{34} + 594 x^{32} - 5952 x^{30} + 40455 x^{28} - x^{27} - 197316 x^{26} + 27 x^{25} + \cdots + 1 \) x^36 - 36*x^34 + 594*x^32 - 5952*x^30 + 40455*x^28 - x^27 - 197316*x^26 + 27*x^25 + 712530*x^24 - 324*x^23 - 1937520*x^22 + 2277*x^21 + 3996135*x^20 - 10395*x^19 - 6249100*x^18 + 32319*x^17 + 7354710*x^16 - 69768*x^15 - 6418656*x^14 + 104652*x^13 + 4056234*x^12 - 107406*x^11 - 1790712*x^10 + 72931*x^9 + 523260*x^8 - 30897*x^7 - 93024*x^6 + 7398*x^5 + 8721*x^4 - 849*x^3 - 324*x^2 + 36*x + 1

Invariants

Degree:		$36$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[36, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(65028396011052373244549315269863064140390224754810333251953125\) 65028396011052373244549315269863064140390224754810333251953125 \(\medspace = 3^{90}\cdot 5^{27}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(52.12\)	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:		$3^{5/2}5^{3/4}\approx 52.123148337972104$
Ramified primes:		\(3\), \(5\) 3, 5	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$36$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(135=3^{3}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{135}(128,·)$, $\chi_{135}(1,·)$, $\chi_{135}(2,·)$, $\chi_{135}(4,·)$, $\chi_{135}(8,·)$, $\chi_{135}(16,·)$, $\chi_{135}(17,·)$, $\chi_{135}(19,·)$, $\chi_{135}(23,·)$, $\chi_{135}(31,·)$, $\chi_{135}(32,·)$, $\chi_{135}(34,·)$, $\chi_{135}(38,·)$, $\chi_{135}(46,·)$, $\chi_{135}(47,·)$, $\chi_{135}(49,·)$, $\chi_{135}(53,·)$, $\chi_{135}(61,·)$, $\chi_{135}(62,·)$, $\chi_{135}(64,·)$, $\chi_{135}(68,·)$, $\chi_{135}(76,·)$, $\chi_{135}(77,·)$, $\chi_{135}(79,·)$, $\chi_{135}(83,·)$, $\chi_{135}(91,·)$, $\chi_{135}(92,·)$, $\chi_{135}(94,·)$, $\chi_{135}(98,·)$, $\chi_{135}(106,·)$, $\chi_{135}(107,·)$, $\chi_{135}(109,·)$, $\chi_{135}(113,·)$, $\chi_{135}(121,·)$, $\chi_{135}(122,·)$, $\chi_{135}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$ 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, a^22, a^23, a^24, a^25, a^26, a^27, a^28, a^29, a^30, a^31, a^32, a^33, a^34, a^35

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$35$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	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:		$a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{33}-33a^{31}-a^{30}+495a^{29}+30a^{28}-4467a^{27}-405a^{26}+27054a^{25}+3249a^{24}-116154a^{23}-17226a^{22}+364067a^{21}+63504a^{20}-845295a^{19}-166725a^{18}+1459998a^{17}+313956a^{16}-1867586a^{15}-421362a^{14}+1746138a^{13}+395149a^{12}-1165554a^{11}-249690a^{10}+534820a^{9}+100089a^{8}-158796a^{7}-23044a^{6}+27567a^{5}+2589a^{4}-2316a^{3}-117a^{2}+63a+1$, $a^{33}-33a^{31}-a^{30}+495a^{29}+30a^{28}-4466a^{27}-405a^{26}+27027a^{25}+3249a^{24}-115830a^{23}-17226a^{22}+361790a^{21}+63504a^{20}-834900a^{19}-166725a^{18}+1427679a^{17}+313956a^{16}-1797818a^{15}-421362a^{14}+1641486a^{13}+395149a^{12}-1058148a^{11}-249690a^{10}+461890a^{9}+100089a^{8}-127908a^{7}-23043a^{6}+20196a^{5}+2583a^{4}-1497a^{3}-108a^{2}+36a$, $a^{33}-33a^{31}-a^{30}+495a^{29}+30a^{28}-4466a^{27}-405a^{26}+27027a^{25}+3250a^{24}-115830a^{23}-17250a^{22}+361791a^{21}+63756a^{20}-834921a^{19}-168245a^{18}+1427868a^{17}+319770a^{16}-1798771a^{15}-436050a^{14}+1644441a^{13}+419900a^{12}-1063971a^{11}-277134a^{10}+469172a^{9}+119340a^{8}-133506a^{7}-30939a^{6}+22653a^{5}+4194a^{4}-2022a^{3}-216a^{2}+72a+1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{33}-33a^{31}-a^{30}+495a^{29}+30a^{28}-4466a^{27}-405a^{26}+27027a^{25}+3250a^{24}-115830a^{23}-17250a^{22}+361791a^{21}+63756a^{20}-834921a^{19}-168245a^{18}+1427868a^{17}+319770a^{16}-1798771a^{15}-436050a^{14}+1644441a^{13}+419900a^{12}-1063971a^{11}-277134a^{10}+469172a^{9}+119340a^{8}-133506a^{7}-30939a^{6}+22653a^{5}+4194a^{4}-2021a^{3}-216a^{2}+69a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72931a^{9}+30897a^{7}-7398a^{5}+849a^{3}-36a$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}+a^{12}-1058148a^{11}-12a^{10}+461890a^{9}+54a^{8}-127908a^{7}-112a^{6}+20196a^{5}+105a^{4}-1496a^{3}-36a^{2}+33a+2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+3$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+a^{19}+8645a^{18}-19a^{17}-25194a^{16}+152a^{15}+50388a^{14}-665a^{13}-68952a^{12}+1729a^{11}+63206a^{10}-2717a^{9}-37180a^{8}+2508a^{7}+13013a^{6}-1254a^{5}-2366a^{4}+285a^{3}+169a^{2}-19a-3$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-1$, $a^{7}-7a^{5}+14a^{3}-7a+1$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}+a^{14}-700910a^{13}-14a^{12}+520676a^{11}+77a^{10}-260338a^{9}-210a^{8}+82212a^{7}+294a^{6}-14756a^{5}-196a^{4}+1240a^{3}+49a^{2}-31a-3$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a+1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+a^{17}+94962a^{16}-17a^{15}-155040a^{14}+119a^{13}+176358a^{12}-442a^{11}-136136a^{10}+935a^{9}+68068a^{8}-1122a^{7}-20384a^{6}+714a^{5}+3185a^{4}-204a^{3}-196a^{2}+17a+1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+3$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}+a^{16}-224808a^{15}-16a^{14}+281010a^{13}+104a^{12}-243542a^{11}-352a^{10}+140998a^{9}+660a^{8}-51272a^{7}-672a^{6}+10556a^{5}+336a^{4}-1015a^{3}-64a^{2}+29a+1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}-a^{28}+35525a^{27}+27a^{26}-166257a^{25}-324a^{24}+573300a^{23}+2277a^{22}-1480050a^{21}-10395a^{20}+2877875a^{19}+32319a^{18}-4206125a^{17}-69768a^{16}+4576264a^{15}+104652a^{14}-3640210a^{13}-107406a^{12}+2057510a^{11}+72931a^{10}-791350a^{9}-30897a^{8}+193800a^{7}+7398a^{6}-27132a^{5}-849a^{4}+1785a^{3}+36a^{2}-36a-1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}-a^{28}+35525a^{27}+27a^{26}-166257a^{25}-324a^{24}+573300a^{23}+2277a^{22}-1480050a^{21}-10395a^{20}+2877875a^{19}+32319a^{18}-4206125a^{17}-69768a^{16}+4576264a^{15}+104652a^{14}-3640210a^{13}-107406a^{12}+2057510a^{11}+72931a^{10}-791350a^{9}-30898a^{8}+193800a^{7}+7406a^{6}-27132a^{5}-869a^{4}+1785a^{3}+52a^{2}-36a-1$, $a^{32}-a^{31}-32a^{30}+31a^{29}+464a^{28}-434a^{27}-4032a^{26}+3627a^{25}+23400a^{24}-20150a^{23}-95679a^{22}+78430a^{21}+283338a^{20}-219604a^{19}-615087a^{18}+447051a^{17}+979506a^{16}-660858a^{15}-1133221a^{14}+700910a^{13}+932582a^{12}-520676a^{11}-526538a^{10}+260338a^{9}+192324a^{8}-82212a^{7}-41271a^{6}+14757a^{5}+4426a^{4}-1245a^{3}-184a^{2}+36a+1$, $a^{4}-4a^{2}+3$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}-a^{28}+35525a^{27}+28a^{26}-166257a^{25}-350a^{24}+573300a^{23}+2576a^{22}-1480050a^{21}-12397a^{20}+2877876a^{19}+40964a^{18}-4206145a^{17}-94962a^{16}+4576433a^{15}+155040a^{14}-3640994a^{13}-176358a^{12}+2059681a^{11}+136136a^{10}-795002a^{9}-68067a^{8}+197430a^{7}+20376a^{6}-29100a^{5}-3165a^{4}+2274a^{3}+180a^{2}-72a$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}-a^{28}+35525a^{27}+28a^{26}-166257a^{25}-350a^{24}+573300a^{23}+2576a^{22}-1480050a^{21}-12397a^{20}+2877876a^{19}+40964a^{18}-4206145a^{17}-94962a^{16}+4576433a^{15}+155040a^{14}-3640994a^{13}-176358a^{12}+2059681a^{11}+136136a^{10}-795002a^{9}-68067a^{8}+197430a^{7}+20376a^{6}-29100a^{5}-3165a^{4}+2274a^{3}+180a^{2}-72a+1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}-a^{28}+35525a^{27}+28a^{26}-166257a^{25}-350a^{24}+573300a^{23}+2576a^{22}-1480050a^{21}-12397a^{20}+2877876a^{19}+40964a^{18}-4206145a^{17}-94962a^{16}+4576433a^{15}+155040a^{14}-3640994a^{13}-176358a^{12}+2059681a^{11}+136136a^{10}-795002a^{9}-68067a^{8}+197430a^{7}+20376a^{6}-29100a^{5}-3165a^{4}+2274a^{3}+180a^{2}-71a-1$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}-2778446a^{16}+2778446a^{14}-1998724a^{12}+999362a^{10}-329460a^{8}+65892a^{6}-6936a^{4}+289a^{2}-1$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a$, $a^{2}-1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}+a^{10}-791350a^{9}-10a^{8}+193800a^{7}+35a^{6}-27132a^{5}-50a^{4}+1785a^{3}+25a^{2}-35a-2$, $a^{32}-a^{31}-32a^{30}+31a^{29}+464a^{28}-435a^{27}-4032a^{26}+3654a^{25}+23400a^{24}-20475a^{23}-95680a^{22}+80730a^{21}+283360a^{20}-230229a^{19}-615296a^{18}+480681a^{17}+980628a^{16}-735318a^{15}-1136960a^{14}+816511a^{13}+940576a^{12}-644839a^{11}-537472a^{10}+349778a^{9}+201552a^{8}-123123a^{7}-45696a^{6}+25598a^{5}+5439a^{4}-2656a^{3}-252a^{2}+94a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a+1$ a^27 - 27*a^25 + 324*a^23 - 2277*a^21 + 10395*a^19 - 32319*a^17 + 69768*a^15 - 104652*a^13 + 107406*a^11 - 72930*a^9 + 30888*a^7 - 7371*a^5 + 819*a^3 - 27*a, a^30 - 30*a^28 + 405*a^26 - 3250*a^24 + 17250*a^22 - 63756*a^20 + 168245*a^18 - 319770*a^16 + 436050*a^14 - 419900*a^12 + 277134*a^10 - 119340*a^8 + 30940*a^6 - 4200*a^4 + 225*a^2 - 2, a^15 - 15*a^13 + 90*a^11 - 275*a^9 + 450*a^7 - 378*a^5 + 140*a^3 - 15*a, a^33 - 33*a^31 - a^30 + 495*a^29 + 30*a^28 - 4467*a^27 - 405*a^26 + 27054*a^25 + 3249*a^24 - 116154*a^23 - 17226*a^22 + 364067*a^21 + 63504*a^20 - 845295*a^19 - 166725*a^18 + 1459998*a^17 + 313956*a^16 - 1867586*a^15 - 421362*a^14 + 1746138*a^13 + 395149*a^12 - 1165554*a^11 - 249690*a^10 + 534820*a^9 + 100089*a^8 - 158796*a^7 - 23044*a^6 + 27567*a^5 + 2589*a^4 - 2316*a^3 - 117*a^2 + 63*a + 1, a^33 - 33*a^31 - a^30 + 495*a^29 + 30*a^28 - 4466*a^27 - 405*a^26 + 27027*a^25 + 3249*a^24 - 115830*a^23 - 17226*a^22 + 361790*a^21 + 63504*a^20 - 834900*a^19 - 166725*a^18 + 1427679*a^17 + 313956*a^16 - 1797818*a^15 - 421362*a^14 + 1641486*a^13 + 395149*a^12 - 1058148*a^11 - 249690*a^10 + 461890*a^9 + 100089*a^8 - 127908*a^7 - 23043*a^6 + 20196*a^5 + 2583*a^4 - 1497*a^3 - 108*a^2 + 36*a, a^33 - 33*a^31 - a^30 + 495*a^29 + 30*a^28 - 4466*a^27 - 405*a^26 + 27027*a^25 + 3250*a^24 - 115830*a^23 - 17250*a^22 + 361791*a^21 + 63756*a^20 - 834921*a^19 - 168245*a^18 + 1427868*a^17 + 319770*a^16 - 1798771*a^15 - 436050*a^14 + 1644441*a^13 + 419900*a^12 - 1063971*a^11 - 277134*a^10 + 469172*a^9 + 119340*a^8 - 133506*a^7 - 30939*a^6 + 22653*a^5 + 4194*a^4 - 2022*a^3 - 216*a^2 + 72*a + 1, a^33 - 33*a^31 + 495*a^29 - 4466*a^27 + 27027*a^25 - 115830*a^23 + 361790*a^21 - 834900*a^19 + 1427679*a^17 - 1797818*a^15 + 1641486*a^13 - 1058148*a^11 + 461890*a^9 - 127908*a^7 + 20196*a^5 - 1496*a^3 + 33*a, a^12 - 12*a^10 + 54*a^8 - 112*a^6 + 105*a^4 - 36*a^2 + 2, a^33 - 33*a^31 - a^30 + 495*a^29 + 30*a^28 - 4466*a^27 - 405*a^26 + 27027*a^25 + 3250*a^24 - 115830*a^23 - 17250*a^22 + 361791*a^21 + 63756*a^20 - 834921*a^19 - 168245*a^18 + 1427868*a^17 + 319770*a^16 - 1798771*a^15 - 436050*a^14 + 1644441*a^13 + 419900*a^12 - 1063971*a^11 - 277134*a^10 + 469172*a^9 + 119340*a^8 - 133506*a^7 - 30939*a^6 + 22653*a^5 + 4194*a^4 - 2021*a^3 - 216*a^2 + 69*a, a^27 - 27*a^25 + 324*a^23 - 2277*a^21 + 10395*a^19 - 32319*a^17 + 69768*a^15 - 104652*a^13 + 107406*a^11 - 72931*a^9 + 30897*a^7 - 7398*a^5 + 849*a^3 - 36*a, a^33 - 33*a^31 + 495*a^29 - 4466*a^27 + 27027*a^25 - 115830*a^23 + 361790*a^21 - 834900*a^19 + 1427679*a^17 - 1797818*a^15 + 1641486*a^13 + a^12 - 1058148*a^11 - 12*a^10 + 461890*a^9 + 54*a^8 - 127908*a^7 - 112*a^6 + 20196*a^5 + 105*a^4 - 1496*a^3 - 36*a^2 + 33*a + 2, a^32 - 32*a^30 + 464*a^28 - 4032*a^26 + 23400*a^24 - 95680*a^22 + 283360*a^20 - 615296*a^18 + 980628*a^16 - 1136960*a^14 + 940576*a^12 - 537472*a^10 + 201552*a^8 - 45696*a^6 + 5440*a^4 - 256*a^2 + 3, a^26 - 26*a^24 + 299*a^22 - 2002*a^20 + a^19 + 8645*a^18 - 19*a^17 - 25194*a^16 + 152*a^15 + 50388*a^14 - 665*a^13 - 68952*a^12 + 1729*a^11 + 63206*a^10 - 2717*a^9 - 37180*a^8 + 2508*a^7 + 13013*a^6 - 1254*a^5 - 2366*a^4 + 285*a^3 + 169*a^2 - 19*a - 3, a^26 - 26*a^24 + 299*a^22 - 2002*a^20 + 8645*a^18 - 25194*a^16 + 50388*a^14 - 68952*a^12 + 63206*a^10 - 37180*a^8 + 13013*a^6 - 2366*a^4 + 169*a^2 - 1, a^7 - 7*a^5 + 14*a^3 - 7*a + 1, a^14 - 14*a^12 + 77*a^10 - 210*a^8 + 294*a^6 - 196*a^4 + 49*a^2 - 1, a^31 - 31*a^29 + 434*a^27 - 3627*a^25 + 20150*a^23 - 78430*a^21 + 219604*a^19 - 447051*a^17 + 660858*a^15 + a^14 - 700910*a^13 - 14*a^12 + 520676*a^11 + 77*a^10 - 260338*a^9 - 210*a^8 + 82212*a^7 + 294*a^6 - 14756*a^5 - 196*a^4 + 1240*a^3 + 49*a^2 - 31*a - 3, a^17 - 17*a^15 + 119*a^13 - 442*a^11 + 935*a^9 - 1122*a^7 + 714*a^5 - 204*a^3 + 17*a + 1, a^28 - 28*a^26 + 350*a^24 - 2576*a^22 + 12397*a^20 - 40964*a^18 + a^17 + 94962*a^16 - 17*a^15 - 155040*a^14 + 119*a^13 + 176358*a^12 - 442*a^11 - 136136*a^10 + 935*a^9 + 68068*a^8 - 1122*a^7 - 20384*a^6 + 714*a^5 + 3185*a^4 - 204*a^3 - 196*a^2 + 17*a + 1, a^16 - 16*a^14 + 104*a^12 - 352*a^10 + 660*a^8 - 672*a^6 + 336*a^4 - 64*a^2 + 3, a^29 - 29*a^27 + 377*a^25 - 2900*a^23 + 14674*a^21 - 51359*a^19 + 127281*a^17 + a^16 - 224808*a^15 - 16*a^14 + 281010*a^13 + 104*a^12 - 243542*a^11 - 352*a^10 + 140998*a^9 + 660*a^8 - 51272*a^7 - 672*a^6 + 10556*a^5 + 336*a^4 - 1015*a^3 - 64*a^2 + 29*a + 1, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 - a^28 + 35525*a^27 + 27*a^26 - 166257*a^25 - 324*a^24 + 573300*a^23 + 2277*a^22 - 1480050*a^21 - 10395*a^20 + 2877875*a^19 + 32319*a^18 - 4206125*a^17 - 69768*a^16 + 4576264*a^15 + 104652*a^14 - 3640210*a^13 - 107406*a^12 + 2057510*a^11 + 72931*a^10 - 791350*a^9 - 30897*a^8 + 193800*a^7 + 7398*a^6 - 27132*a^5 - 849*a^4 + 1785*a^3 + 36*a^2 - 36*a - 1, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 - a^28 + 35525*a^27 + 27*a^26 - 166257*a^25 - 324*a^24 + 573300*a^23 + 2277*a^22 - 1480050*a^21 - 10395*a^20 + 2877875*a^19 + 32319*a^18 - 4206125*a^17 - 69768*a^16 + 4576264*a^15 + 104652*a^14 - 3640210*a^13 - 107406*a^12 + 2057510*a^11 + 72931*a^10 - 791350*a^9 - 30898*a^8 + 193800*a^7 + 7406*a^6 - 27132*a^5 - 869*a^4 + 1785*a^3 + 52*a^2 - 36*a - 1, a^32 - a^31 - 32*a^30 + 31*a^29 + 464*a^28 - 434*a^27 - 4032*a^26 + 3627*a^25 + 23400*a^24 - 20150*a^23 - 95679*a^22 + 78430*a^21 + 283338*a^20 - 219604*a^19 - 615087*a^18 + 447051*a^17 + 979506*a^16 - 660858*a^15 - 1133221*a^14 + 700910*a^13 + 932582*a^12 - 520676*a^11 - 526538*a^10 + 260338*a^9 + 192324*a^8 - 82212*a^7 - 41271*a^6 + 14757*a^5 + 4426*a^4 - 1245*a^3 - 184*a^2 + 36*a + 1, a^4 - 4*a^2 + 3, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 - a^28 + 35525*a^27 + 28*a^26 - 166257*a^25 - 350*a^24 + 573300*a^23 + 2576*a^22 - 1480050*a^21 - 12397*a^20 + 2877876*a^19 + 40964*a^18 - 4206145*a^17 - 94962*a^16 + 4576433*a^15 + 155040*a^14 - 3640994*a^13 - 176358*a^12 + 2059681*a^11 + 136136*a^10 - 795002*a^9 - 68067*a^8 + 197430*a^7 + 20376*a^6 - 29100*a^5 - 3165*a^4 + 2274*a^3 + 180*a^2 - 72*a, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 - a^28 + 35525*a^27 + 28*a^26 - 166257*a^25 - 350*a^24 + 573300*a^23 + 2576*a^22 - 1480050*a^21 - 12397*a^20 + 2877876*a^19 + 40964*a^18 - 4206145*a^17 - 94962*a^16 + 4576433*a^15 + 155040*a^14 - 3640994*a^13 - 176358*a^12 + 2059681*a^11 + 136136*a^10 - 795002*a^9 - 68067*a^8 + 197430*a^7 + 20376*a^6 - 29100*a^5 - 3165*a^4 + 2274*a^3 + 180*a^2 - 72*a + 1, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 - a^28 + 35525*a^27 + 28*a^26 - 166257*a^25 - 350*a^24 + 573300*a^23 + 2576*a^22 - 1480050*a^21 - 12397*a^20 + 2877876*a^19 + 40964*a^18 - 4206145*a^17 - 94962*a^16 + 4576433*a^15 + 155040*a^14 - 3640994*a^13 - 176358*a^12 + 2059681*a^11 + 136136*a^10 - 795002*a^9 - 68067*a^8 + 197430*a^7 + 20376*a^6 - 29100*a^5 - 3165*a^4 + 2274*a^3 + 180*a^2 - 71*a - 1, a^34 - 34*a^32 + 527*a^30 - 4930*a^28 + 31059*a^26 - 139230*a^24 + 457470*a^22 - 1118260*a^20 + 2042975*a^18 - 2778446*a^16 + 2778446*a^14 - 1998724*a^12 + 999362*a^10 - 329460*a^8 + 65892*a^6 - 6936*a^4 + 289*a^2 - 1, a^22 - 22*a^20 + 209*a^18 - 1122*a^16 + 3740*a^14 - 8008*a^12 + 11011*a^10 - 9438*a^8 + 4719*a^6 - 1210*a^4 + 121*a^2 - 1, a^31 - 31*a^29 + 434*a^27 - 3627*a^25 + 20150*a^23 - 78430*a^21 + 219604*a^19 - 447051*a^17 + 660858*a^15 - 700910*a^13 + 520676*a^11 - 260338*a^9 + 82212*a^7 - 14756*a^5 + 1240*a^3 - 31*a, a^2 - 1, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 + 35525*a^27 - 166257*a^25 + 573300*a^23 - 1480050*a^21 + 2877875*a^19 - 4206125*a^17 + 4576264*a^15 - 3640210*a^13 + 2057510*a^11 + a^10 - 791350*a^9 - 10*a^8 + 193800*a^7 + 35*a^6 - 27132*a^5 - 50*a^4 + 1785*a^3 + 25*a^2 - 35*a - 2, a^32 - a^31 - 32*a^30 + 31*a^29 + 464*a^28 - 435*a^27 - 4032*a^26 + 3654*a^25 + 23400*a^24 - 20475*a^23 - 95680*a^22 + 80730*a^21 + 283360*a^20 - 230229*a^19 - 615296*a^18 + 480681*a^17 + 980628*a^16 - 735318*a^15 - 1136960*a^14 + 816511*a^13 + 940576*a^12 - 644839*a^11 - 537472*a^10 + 349778*a^9 + 201552*a^8 - 123123*a^7 - 45696*a^6 + 25598*a^5 + 5439*a^4 - 2656*a^3 - 252*a^2 + 94*a, a^23 - 23*a^21 + 230*a^19 - 1311*a^17 + 4692*a^15 - 10948*a^13 + 16744*a^11 - 16445*a^9 + 9867*a^7 - 3289*a^5 + 506*a^3 - 23*a + 1 (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:		\( 31157843880432600000 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{36}\cdot(2\pi)^{0}\cdot 31157843880432600000 \cdot 1}{2\cdot\sqrt{65028396011052373244549315269863064140390224754810333251953125}}\cr\approx \mathstrut & 0.132759535499642 \end{aligned}\] (assuming GRH)

Galois group

$C_{36}$ (as 36T1):

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
\(3\) 3		Deg $36$	$18$	$2$	$90$
\(5\) 5		Deg $36$	$4$	$9$	$27$