LMFDB - Number field 36.0.4999758568289868528789868885747458073284974388119052886962890625.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241)

gp: K = bnfinit(y^36 + 36*y^34 - 4*y^33 + 594*y^32 - 132*y^31 + 5951*y^30 - 1980*y^29 + 40425*y^28 - 17788*y^27 + 196911*y^26 - 106056*y^25 + 708976*y^24 - 438696*y^23 + 1912974*y^22 - 1274184*y^21 + 3855771*y^20 - 2551176*y^19 + 5624552*y^18 - 3268836*y^17 + 5371470*y^16 - 1959888*y^15 + 2297349*y^14 + 1184688*y^13 - 837713*y^12 + 3617676*y^11 - 642219*y^10 + 3953440*y^9 - 768960*y^8 + 5792076*y^7 - 5874034*y^6 + 12034116*y^5 - 7662084*y^4 + 4778248*y^3 - 2375280*y^2 - 19886784*y + 31530241, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241)

$ x^{36} + 36 x^{34} - 4 x^{33} + 594 x^{32} - 132 x^{31} + 5951 x^{30} - 1980 x^{29} + 40425 x^{28} + \cdots + 31530241 $ x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

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:	$4999758568289868528789868885747458073284974388119052886962890625$ 4999758568289868528789868885747458073284974388119052886962890625 $\medspace = 3^{54}\cdot 5^{18}\cdot 7^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$58.81$	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:	$3^{3/2}5^{1/2}7^{5/6}\approx 58.8051349456126$
Ramified primes:	$3$, $5$, $7$ 3, 5, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$36$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$315=3^{2}\cdot 5\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(131,·)$, $\chi_{315}(134,·)$, $\chi_{315}(139,·)$, $\chi_{315}(16,·)$, $\chi_{315}(146,·)$, $\chi_{315}(19,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(26,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(34,·)$, $\chi_{315}(41,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(304,·)$, $\chi_{315}(179,·)$, $\chi_{315}(311,·)$, $\chi_{315}(199,·)$, $\chi_{315}(74,·)$, $\chi_{315}(206,·)$, $\chi_{315}(211,·)$, $\chi_{315}(94,·)$, $\chi_{315}(101,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(106,·)$, $\chi_{315}(236,·)$, $\chi_{315}(239,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(251,·)$, $\chi_{315}(124,·)$, $\chi_{315}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{10946}a^{21}+\frac{21}{10946}a^{19}+\frac{189}{10946}a^{17}+\frac{476}{5473}a^{15}+\frac{1470}{5473}a^{13}-\frac{401}{842}a^{11}-\frac{303}{842}a^{9}+\frac{198}{421}a^{7}+\frac{2079}{10946}a^{5}+\frac{385}{10946}a^{3}+\frac{21}{10946}a-\frac{4181}{10946}$, $\frac{1}{10946}a^{22}+\frac{21}{10946}a^{20}+\frac{189}{10946}a^{18}+\frac{476}{5473}a^{16}+\frac{1470}{5473}a^{14}-\frac{401}{842}a^{12}-\frac{303}{842}a^{10}+\frac{198}{421}a^{8}+\frac{2079}{10946}a^{6}+\frac{385}{10946}a^{4}+\frac{21}{10946}a^{2}-\frac{4181}{10946}a$, $\frac{1}{10946}a^{23}-\frac{126}{5473}a^{19}-\frac{3017}{10946}a^{17}+\frac{2420}{5473}a^{15}-\frac{1277}{10946}a^{13}-\frac{151}{421}a^{11}+\frac{23}{842}a^{9}+\frac{3431}{10946}a^{7}+\frac{255}{5473}a^{5}+\frac{1441}{5473}a^{3}-\frac{4181}{10946}a^{2}-\frac{441}{10946}a+\frac{233}{10946}$, $\frac{1}{10946}a^{24}-\frac{126}{5473}a^{20}+\frac{1228}{5473}a^{18}+\frac{2420}{5473}a^{16}-\frac{1}{2}a^{15}+\frac{2098}{5473}a^{14}-\frac{1}{2}a^{13}-\frac{151}{421}a^{12}-\frac{199}{421}a^{10}+\frac{3431}{10946}a^{8}-\frac{1}{2}a^{7}-\frac{4963}{10946}a^{6}-\frac{1}{2}a^{5}+\frac{1441}{5473}a^{4}-\frac{4181}{10946}a^{3}-\frac{441}{10946}a^{2}+\frac{233}{10946}a-\frac{1}{2}$, $\frac{1}{10946}a^{25}+\frac{175}{842}a^{19}-\frac{87}{421}a^{17}-\frac{2185}{10946}a^{15}+\frac{1786}{5473}a^{13}+\frac{11}{842}a^{11}-\frac{4057}{10946}a^{9}-\frac{2384}{5473}a^{7}+\frac{691}{5473}a^{5}-\frac{4181}{10946}a^{4}-\frac{1935}{10946}a^{3}+\frac{233}{10946}a^{2}+\frac{2646}{5473}a-\frac{1398}{5473}$, $\frac{1}{10946}a^{26}+\frac{175}{842}a^{20}-\frac{87}{421}a^{18}-\frac{2185}{10946}a^{16}+\frac{1786}{5473}a^{14}+\frac{11}{842}a^{12}-\frac{4057}{10946}a^{10}-\frac{2384}{5473}a^{8}+\frac{691}{5473}a^{6}-\frac{4181}{10946}a^{5}-\frac{1935}{10946}a^{4}+\frac{233}{10946}a^{3}+\frac{2646}{5473}a^{2}-\frac{1398}{5473}a$, $\frac{1}{186082}a^{27}-\frac{7}{186082}a^{25}+\frac{1}{186082}a^{23}-\frac{1}{186082}a^{21}-\frac{2569}{14314}a^{19}-\frac{2}{17}a^{18}-\frac{45576}{93041}a^{17}-\frac{2}{17}a^{16}+\frac{2715}{10946}a^{15}+\frac{2}{17}a^{14}+\frac{418}{5473}a^{13}-\frac{4}{17}a^{12}-\frac{928}{5473}a^{11}-\frac{7}{17}a^{10}-\frac{4259}{93041}a^{9}+\frac{6}{17}a^{8}+\frac{11669}{186082}a^{7}+\frac{83387}{186082}a^{6}-\frac{1481}{10946}a^{5}+\frac{25696}{93041}a^{4}-\frac{66429}{186082}a^{3}+\frac{23061}{93041}a^{2}+\frac{46071}{186082}a+\frac{34633}{186082}$, $\frac{1}{186082}a^{28}-\frac{7}{186082}a^{26}+\frac{1}{186082}a^{24}-\frac{1}{186082}a^{22}-\frac{2569}{14314}a^{20}-\frac{2}{17}a^{19}+\frac{1889}{186082}a^{18}-\frac{2}{17}a^{17}+\frac{2715}{10946}a^{16}-\frac{13}{34}a^{15}-\frac{4637}{10946}a^{14}+\frac{9}{34}a^{13}-\frac{928}{5473}a^{12}-\frac{7}{17}a^{11}+\frac{84523}{186082}a^{10}+\frac{6}{17}a^{9}+\frac{11669}{186082}a^{8}-\frac{4827}{93041}a^{7}+\frac{1996}{5473}a^{6}-\frac{41649}{186082}a^{5}-\frac{66429}{186082}a^{4}+\frac{23061}{93041}a^{3}+\frac{46071}{186082}a^{2}+\frac{34633}{186082}a-\frac{1}{2}$, $\frac{1}{6557470319842}a^{29}+\frac{12238}{3278735159921}a^{28}+\frac{29}{6557470319842}a^{27}+\frac{71053826}{3278735159921}a^{26}-\frac{103305}{6557470319842}a^{25}+\frac{45549781}{3278735159921}a^{24}+\frac{30796113}{6557470319842}a^{23}-\frac{185995755}{6557470319842}a^{22}+\frac{70144298}{3278735159921}a^{21}-\frac{478134856475}{6557470319842}a^{20}-\frac{93600027659}{3278735159921}a^{19}-\frac{808825282751}{3278735159921}a^{18}+\frac{116838667338}{252210396917}a^{17}+\frac{133279503293}{504420793834}a^{16}+\frac{796955029327}{3278735159921}a^{15}+\frac{29853957775}{192866774113}a^{14}-\frac{3000699778711}{6557470319842}a^{13}+\frac{396644725233}{6557470319842}a^{12}-\frac{88890202373}{252210396917}a^{11}+\frac{1574529199077}{3278735159921}a^{10}+\frac{1350791062839}{3278735159921}a^{9}-\frac{1494789215638}{3278735159921}a^{8}-\frac{1619857863435}{6557470319842}a^{7}+\frac{779742212257}{6557470319842}a^{6}-\frac{1825125509731}{6557470319842}a^{5}-\frac{2102702539823}{6557470319842}a^{4}-\frac{1636459959438}{3278735159921}a^{3}-\frac{1444761878466}{3278735159921}a^{2}+\frac{2135469084367}{6557470319842}a+\frac{387135740986}{3278735159921}$, $\frac{1}{6557470319842}a^{30}+\frac{15}{3278735159921}a^{28}+\frac{219562}{3278735159921}a^{27}+\frac{405}{6557470319842}a^{26}+\frac{5928174}{3278735159921}a^{25}+\frac{33492195}{6557470319842}a^{24}+\frac{71138088}{3278735159921}a^{23}+\frac{204677353}{6557470319842}a^{22}+\frac{19954951}{504420793834}a^{21}-\frac{4141288221}{6557470319842}a^{20}-\frac{10985196705}{6557470319842}a^{19}-\frac{31161850173}{3278735159921}a^{18}-\frac{9673612893}{504420793834}a^{17}-\frac{14448054238}{252210396917}a^{16}+\frac{1015846644265}{6557470319842}a^{15}-\frac{634829568900}{3278735159921}a^{14}-\frac{231449037625}{504420793834}a^{13}-\frac{1405149745175}{3278735159921}a^{12}-\frac{2907333273257}{6557470319842}a^{11}+\frac{832914510039}{6557470319842}a^{10}+\frac{712810668212}{3278735159921}a^{9}-\frac{323101112917}{6557470319842}a^{8}-\frac{1553656964778}{3278735159921}a^{7}+\frac{1322754874990}{3278735159921}a^{6}-\frac{2909769944891}{6557470319842}a^{5}-\frac{329700782287}{3278735159921}a^{4}+\frac{526443664912}{3278735159921}a^{3}+\frac{62143740323}{385733548226}a^{2}-\frac{122533724124}{252210396917}a-\frac{37161884305}{252210396917}$, $\frac{1}{6557470319842}a^{31}-\frac{147578}{3278735159921}a^{28}-\frac{465}{6557470319842}a^{27}-\frac{57851173}{6557470319842}a^{26}+\frac{36591345}{6557470319842}a^{25}-\frac{97206188}{3278735159921}a^{24}-\frac{60065730}{3278735159921}a^{23}-\frac{151458757}{6557470319842}a^{22}+\frac{37097977}{6557470319842}a^{21}+\frac{192504182037}{6557470319842}a^{20}-\frac{489315614207}{3278735159921}a^{19}+\frac{523756880089}{3278735159921}a^{18}-\frac{1603108642014}{3278735159921}a^{17}+\frac{1507386261117}{6557470319842}a^{16}+\frac{570923968284}{3278735159921}a^{15}-\frac{1543696220151}{3278735159921}a^{14}+\frac{85450733767}{192866774113}a^{13}-\frac{2221316316631}{6557470319842}a^{12}-\frac{2136236486305}{6557470319842}a^{11}+\frac{1390320289785}{3278735159921}a^{10}+\frac{40594029943}{504420793834}a^{9}+\frac{1343109118587}{6557470319842}a^{8}-\frac{1888881357369}{6557470319842}a^{7}-\frac{1115310367082}{3278735159921}a^{6}-\frac{717528481964}{3278735159921}a^{5}+\frac{1601960102677}{6557470319842}a^{4}-\frac{1602150667788}{3278735159921}a^{3}+\frac{10701780357}{192866774113}a^{2}+\frac{71880883651}{385733548226}a-\frac{1208146827090}{3278735159921}$, $\frac{1}{6557470319842}a^{32}+\frac{51593}{3278735159921}a^{28}-\frac{7025984}{3278735159921}a^{27}+\frac{2894169}{6557470319842}a^{26}+\frac{219697281}{6557470319842}a^{25}+\frac{81355520}{3278735159921}a^{24}-\frac{217247269}{6557470319842}a^{23}-\frac{293712573}{6557470319842}a^{22}-\frac{124421774}{3278735159921}a^{21}+\frac{139787780345}{6557470319842}a^{20}+\frac{16873788735}{6557470319842}a^{19}-\frac{60031627725}{3278735159921}a^{18}-\frac{710141652439}{3278735159921}a^{17}+\frac{921921718513}{3278735159921}a^{16}-\frac{1393886782843}{6557470319842}a^{15}+\frac{203266366027}{504420793834}a^{14}-\frac{1131768051116}{3278735159921}a^{13}+\frac{2575267333209}{6557470319842}a^{12}-\frac{28542334011}{252210396917}a^{11}-\frac{283214456467}{3278735159921}a^{10}-\frac{1511349968328}{3278735159921}a^{9}+\frac{1929276729561}{6557470319842}a^{8}+\frac{1088048171747}{6557470319842}a^{7}+\frac{972677600603}{6557470319842}a^{6}-\frac{2907147115055}{6557470319842}a^{5}+\frac{3153895105367}{6557470319842}a^{4}+\frac{195484891}{1198149154}a^{3}+\frac{1504802598831}{6557470319842}a^{2}-\frac{71950513075}{385733548226}a+\frac{356705315256}{3278735159921}$, $\frac{1}{6557470319842}a^{33}-\frac{1187736}{3278735159921}a^{28}-\frac{98225}{6557470319842}a^{27}-\frac{139697253}{6557470319842}a^{26}+\frac{19499192}{3278735159921}a^{25}-\frac{98252685}{3278735159921}a^{24}+\frac{34601197}{3278735159921}a^{23}-\frac{92487807}{3278735159921}a^{22}-\frac{176658901}{6557470319842}a^{21}+\frac{89260645158}{3278735159921}a^{20}-\frac{1570220861921}{6557470319842}a^{19}-\frac{744787523645}{3278735159921}a^{18}+\frac{687754268211}{3278735159921}a^{17}+\frac{1181177070001}{6557470319842}a^{16}-\frac{868874492288}{3278735159921}a^{15}+\frac{2236877975201}{6557470319842}a^{14}-\frac{3271426145159}{6557470319842}a^{13}-\frac{221784248575}{504420793834}a^{12}-\frac{173949192245}{385733548226}a^{11}-\frac{318249567853}{3278735159921}a^{10}+\frac{1782869432423}{6557470319842}a^{9}+\frac{326693637761}{3278735159921}a^{8}-\frac{2716141651135}{6557470319842}a^{7}+\frac{556609336635}{6557470319842}a^{6}-\frac{298584471688}{3278735159921}a^{5}+\frac{1523353189387}{6557470319842}a^{4}-\frac{193364395447}{3278735159921}a^{3}+\frac{816303705630}{3278735159921}a^{2}-\frac{952782060709}{6557470319842}a-\frac{272828370034}{3278735159921}$, $\frac{1}{6557470319842}a^{34}-\frac{3519203}{6557470319842}a^{28}-\frac{329203}{6557470319842}a^{27}-\frac{49240206}{3278735159921}a^{26}+\frac{131655283}{6557470319842}a^{25}+\frac{92090845}{3278735159921}a^{24}-\frac{5977991}{192866774113}a^{23}-\frac{164735271}{6557470319842}a^{22}+\frac{149834491}{6557470319842}a^{21}+\frac{333131900251}{6557470319842}a^{20}-\frac{323794103151}{6557470319842}a^{19}+\frac{250734694266}{3278735159921}a^{18}+\frac{458878117981}{3278735159921}a^{17}-\frac{2602471167481}{6557470319842}a^{16}+\frac{635135407997}{6557470319842}a^{15}+\frac{1568675489171}{3278735159921}a^{14}-\frac{2182562871395}{6557470319842}a^{13}-\frac{187337756661}{385733548226}a^{12}+\frac{1783520188583}{6557470319842}a^{11}-\frac{934669355859}{6557470319842}a^{10}+\frac{529240215956}{3278735159921}a^{9}+\frac{122856819591}{252210396917}a^{8}+\frac{919744795172}{3278735159921}a^{7}+\frac{3022825680787}{6557470319842}a^{6}+\frac{351355817751}{3278735159921}a^{5}-\frac{95268052803}{504420793834}a^{4}+\frac{1292009126149}{3278735159921}a^{3}-\frac{596441757126}{3278735159921}a^{2}+\frac{13394954433}{192866774113}a-\frac{257043476305}{6557470319842}$, $\frac{1}{6557470319842}a^{35}+\frac{582533}{385733548226}a^{28}+\frac{3576475}{6557470319842}a^{27}-\frac{240395721}{6557470319842}a^{26}+\frac{135592007}{3278735159921}a^{25}+\frac{34341681}{6557470319842}a^{24}+\frac{112473376}{3278735159921}a^{23}-\frac{139307797}{6557470319842}a^{22}+\frac{72809190}{3278735159921}a^{21}-\frac{592091457184}{3278735159921}a^{20}+\frac{105126979397}{504420793834}a^{19}+\frac{19055207393}{192866774113}a^{18}+\frac{2810006014441}{6557470319842}a^{17}+\frac{1199835981784}{3278735159921}a^{16}+\frac{123837091023}{385733548226}a^{15}-\frac{2616871093929}{6557470319842}a^{14}-\frac{1112018722739}{3278735159921}a^{13}-\frac{9705183267}{252210396917}a^{12}-\frac{456346362494}{3278735159921}a^{11}-\frac{669535128852}{3278735159921}a^{10}-\frac{2294137359039}{6557470319842}a^{9}-\frac{3086772664205}{6557470319842}a^{8}+\frac{126678951870}{3278735159921}a^{7}-\frac{2650357741815}{6557470319842}a^{6}-\frac{2173859726253}{6557470319842}a^{5}+\frac{2670018880951}{6557470319842}a^{4}+\frac{3213598992477}{6557470319842}a^{3}+\frac{1415480584970}{3278735159921}a^{2}+\frac{56110566346}{252210396917}a-\frac{1522679865033}{6557470319842}$ 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, 1/2*a^18 - 1/2*a^15 - 1/2*a^14 - 1/2*a^13 - 1/2*a^10 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2, 1/2*a^19 - 1/2*a^16 - 1/2*a^15 - 1/2*a^14 - 1/2*a^11 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a, 1/2*a^20 - 1/2*a^17 - 1/2*a^16 - 1/2*a^15 - 1/2*a^12 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^2, 1/10946*a^21 + 21/10946*a^19 + 189/10946*a^17 + 476/5473*a^15 + 1470/5473*a^13 - 401/842*a^11 - 303/842*a^9 + 198/421*a^7 + 2079/10946*a^5 + 385/10946*a^3 + 21/10946*a - 4181/10946, 1/10946*a^22 + 21/10946*a^20 + 189/10946*a^18 + 476/5473*a^16 + 1470/5473*a^14 - 401/842*a^12 - 303/842*a^10 + 198/421*a^8 + 2079/10946*a^6 + 385/10946*a^4 + 21/10946*a^2 - 4181/10946*a, 1/10946*a^23 - 126/5473*a^19 - 3017/10946*a^17 + 2420/5473*a^15 - 1277/10946*a^13 - 151/421*a^11 + 23/842*a^9 + 3431/10946*a^7 + 255/5473*a^5 + 1441/5473*a^3 - 4181/10946*a^2 - 441/10946*a + 233/10946, 1/10946*a^24 - 126/5473*a^20 + 1228/5473*a^18 + 2420/5473*a^16 - 1/2*a^15 + 2098/5473*a^14 - 1/2*a^13 - 151/421*a^12 - 199/421*a^10 + 3431/10946*a^8 - 1/2*a^7 - 4963/10946*a^6 - 1/2*a^5 + 1441/5473*a^4 - 4181/10946*a^3 - 441/10946*a^2 + 233/10946*a - 1/2, 1/10946*a^25 + 175/842*a^19 - 87/421*a^17 - 2185/10946*a^15 + 1786/5473*a^13 + 11/842*a^11 - 4057/10946*a^9 - 2384/5473*a^7 + 691/5473*a^5 - 4181/10946*a^4 - 1935/10946*a^3 + 233/10946*a^2 + 2646/5473*a - 1398/5473, 1/10946*a^26 + 175/842*a^20 - 87/421*a^18 - 2185/10946*a^16 + 1786/5473*a^14 + 11/842*a^12 - 4057/10946*a^10 - 2384/5473*a^8 + 691/5473*a^6 - 4181/10946*a^5 - 1935/10946*a^4 + 233/10946*a^3 + 2646/5473*a^2 - 1398/5473*a, 1/186082*a^27 - 7/186082*a^25 + 1/186082*a^23 - 1/186082*a^21 - 2569/14314*a^19 - 2/17*a^18 - 45576/93041*a^17 - 2/17*a^16 + 2715/10946*a^15 + 2/17*a^14 + 418/5473*a^13 - 4/17*a^12 - 928/5473*a^11 - 7/17*a^10 - 4259/93041*a^9 + 6/17*a^8 + 11669/186082*a^7 + 83387/186082*a^6 - 1481/10946*a^5 + 25696/93041*a^4 - 66429/186082*a^3 + 23061/93041*a^2 + 46071/186082*a + 34633/186082, 1/186082*a^28 - 7/186082*a^26 + 1/186082*a^24 - 1/186082*a^22 - 2569/14314*a^20 - 2/17*a^19 + 1889/186082*a^18 - 2/17*a^17 + 2715/10946*a^16 - 13/34*a^15 - 4637/10946*a^14 + 9/34*a^13 - 928/5473*a^12 - 7/17*a^11 + 84523/186082*a^10 + 6/17*a^9 + 11669/186082*a^8 - 4827/93041*a^7 + 1996/5473*a^6 - 41649/186082*a^5 - 66429/186082*a^4 + 23061/93041*a^3 + 46071/186082*a^2 + 34633/186082*a - 1/2, 1/6557470319842*a^29 + 12238/3278735159921*a^28 + 29/6557470319842*a^27 + 71053826/3278735159921*a^26 - 103305/6557470319842*a^25 + 45549781/3278735159921*a^24 + 30796113/6557470319842*a^23 - 185995755/6557470319842*a^22 + 70144298/3278735159921*a^21 - 478134856475/6557470319842*a^20 - 93600027659/3278735159921*a^19 - 808825282751/3278735159921*a^18 + 116838667338/252210396917*a^17 + 133279503293/504420793834*a^16 + 796955029327/3278735159921*a^15 + 29853957775/192866774113*a^14 - 3000699778711/6557470319842*a^13 + 396644725233/6557470319842*a^12 - 88890202373/252210396917*a^11 + 1574529199077/3278735159921*a^10 + 1350791062839/3278735159921*a^9 - 1494789215638/3278735159921*a^8 - 1619857863435/6557470319842*a^7 + 779742212257/6557470319842*a^6 - 1825125509731/6557470319842*a^5 - 2102702539823/6557470319842*a^4 - 1636459959438/3278735159921*a^3 - 1444761878466/3278735159921*a^2 + 2135469084367/6557470319842*a + 387135740986/3278735159921, 1/6557470319842*a^30 + 15/3278735159921*a^28 + 219562/3278735159921*a^27 + 405/6557470319842*a^26 + 5928174/3278735159921*a^25 + 33492195/6557470319842*a^24 + 71138088/3278735159921*a^23 + 204677353/6557470319842*a^22 + 19954951/504420793834*a^21 - 4141288221/6557470319842*a^20 - 10985196705/6557470319842*a^19 - 31161850173/3278735159921*a^18 - 9673612893/504420793834*a^17 - 14448054238/252210396917*a^16 + 1015846644265/6557470319842*a^15 - 634829568900/3278735159921*a^14 - 231449037625/504420793834*a^13 - 1405149745175/3278735159921*a^12 - 2907333273257/6557470319842*a^11 + 832914510039/6557470319842*a^10 + 712810668212/3278735159921*a^9 - 323101112917/6557470319842*a^8 - 1553656964778/3278735159921*a^7 + 1322754874990/3278735159921*a^6 - 2909769944891/6557470319842*a^5 - 329700782287/3278735159921*a^4 + 526443664912/3278735159921*a^3 + 62143740323/385733548226*a^2 - 122533724124/252210396917*a - 37161884305/252210396917, 1/6557470319842*a^31 - 147578/3278735159921*a^28 - 465/6557470319842*a^27 - 57851173/6557470319842*a^26 + 36591345/6557470319842*a^25 - 97206188/3278735159921*a^24 - 60065730/3278735159921*a^23 - 151458757/6557470319842*a^22 + 37097977/6557470319842*a^21 + 192504182037/6557470319842*a^20 - 489315614207/3278735159921*a^19 + 523756880089/3278735159921*a^18 - 1603108642014/3278735159921*a^17 + 1507386261117/6557470319842*a^16 + 570923968284/3278735159921*a^15 - 1543696220151/3278735159921*a^14 + 85450733767/192866774113*a^13 - 2221316316631/6557470319842*a^12 - 2136236486305/6557470319842*a^11 + 1390320289785/3278735159921*a^10 + 40594029943/504420793834*a^9 + 1343109118587/6557470319842*a^8 - 1888881357369/6557470319842*a^7 - 1115310367082/3278735159921*a^6 - 717528481964/3278735159921*a^5 + 1601960102677/6557470319842*a^4 - 1602150667788/3278735159921*a^3 + 10701780357/192866774113*a^2 + 71880883651/385733548226*a - 1208146827090/3278735159921, 1/6557470319842*a^32 + 51593/3278735159921*a^28 - 7025984/3278735159921*a^27 + 2894169/6557470319842*a^26 + 219697281/6557470319842*a^25 + 81355520/3278735159921*a^24 - 217247269/6557470319842*a^23 - 293712573/6557470319842*a^22 - 124421774/3278735159921*a^21 + 139787780345/6557470319842*a^20 + 16873788735/6557470319842*a^19 - 60031627725/3278735159921*a^18 - 710141652439/3278735159921*a^17 + 921921718513/3278735159921*a^16 - 1393886782843/6557470319842*a^15 + 203266366027/504420793834*a^14 - 1131768051116/3278735159921*a^13 + 2575267333209/6557470319842*a^12 - 28542334011/252210396917*a^11 - 283214456467/3278735159921*a^10 - 1511349968328/3278735159921*a^9 + 1929276729561/6557470319842*a^8 + 1088048171747/6557470319842*a^7 + 972677600603/6557470319842*a^6 - 2907147115055/6557470319842*a^5 + 3153895105367/6557470319842*a^4 + 195484891/1198149154*a^3 + 1504802598831/6557470319842*a^2 - 71950513075/385733548226*a + 356705315256/3278735159921, 1/6557470319842*a^33 - 1187736/3278735159921*a^28 - 98225/6557470319842*a^27 - 139697253/6557470319842*a^26 + 19499192/3278735159921*a^25 - 98252685/3278735159921*a^24 + 34601197/3278735159921*a^23 - 92487807/3278735159921*a^22 - 176658901/6557470319842*a^21 + 89260645158/3278735159921*a^20 - 1570220861921/6557470319842*a^19 - 744787523645/3278735159921*a^18 + 687754268211/3278735159921*a^17 + 1181177070001/6557470319842*a^16 - 868874492288/3278735159921*a^15 + 2236877975201/6557470319842*a^14 - 3271426145159/6557470319842*a^13 - 221784248575/504420793834*a^12 - 173949192245/385733548226*a^11 - 318249567853/3278735159921*a^10 + 1782869432423/6557470319842*a^9 + 326693637761/3278735159921*a^8 - 2716141651135/6557470319842*a^7 + 556609336635/6557470319842*a^6 - 298584471688/3278735159921*a^5 + 1523353189387/6557470319842*a^4 - 193364395447/3278735159921*a^3 + 816303705630/3278735159921*a^2 - 952782060709/6557470319842*a - 272828370034/3278735159921, 1/6557470319842*a^34 - 3519203/6557470319842*a^28 - 329203/6557470319842*a^27 - 49240206/3278735159921*a^26 + 131655283/6557470319842*a^25 + 92090845/3278735159921*a^24 - 5977991/192866774113*a^23 - 164735271/6557470319842*a^22 + 149834491/6557470319842*a^21 + 333131900251/6557470319842*a^20 - 323794103151/6557470319842*a^19 + 250734694266/3278735159921*a^18 + 458878117981/3278735159921*a^17 - 2602471167481/6557470319842*a^16 + 635135407997/6557470319842*a^15 + 1568675489171/3278735159921*a^14 - 2182562871395/6557470319842*a^13 - 187337756661/385733548226*a^12 + 1783520188583/6557470319842*a^11 - 934669355859/6557470319842*a^10 + 529240215956/3278735159921*a^9 + 122856819591/252210396917*a^8 + 919744795172/3278735159921*a^7 + 3022825680787/6557470319842*a^6 + 351355817751/3278735159921*a^5 - 95268052803/504420793834*a^4 + 1292009126149/3278735159921*a^3 - 596441757126/3278735159921*a^2 + 13394954433/192866774113*a - 257043476305/6557470319842, 1/6557470319842*a^35 + 582533/385733548226*a^28 + 3576475/6557470319842*a^27 - 240395721/6557470319842*a^26 + 135592007/3278735159921*a^25 + 34341681/6557470319842*a^24 + 112473376/3278735159921*a^23 - 139307797/6557470319842*a^22 + 72809190/3278735159921*a^21 - 592091457184/3278735159921*a^20 + 105126979397/504420793834*a^19 + 19055207393/192866774113*a^18 + 2810006014441/6557470319842*a^17 + 1199835981784/3278735159921*a^16 + 123837091023/385733548226*a^15 - 2616871093929/6557470319842*a^14 - 1112018722739/3278735159921*a^13 - 9705183267/252210396917*a^12 - 456346362494/3278735159921*a^11 - 669535128852/3278735159921*a^10 - 2294137359039/6557470319842*a^9 - 3086772664205/6557470319842*a^8 + 126678951870/3278735159921*a^7 - 2650357741815/6557470319842*a^6 - 2173859726253/6557470319842*a^5 + 2670018880951/6557470319842*a^4 + 3213598992477/6557470319842*a^3 + 1415480584970/3278735159921*a^2 + 56110566346/252210396917*a - 1522679865033/6557470319842

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{666}$, which has order $95904$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$17$	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:	$\frac{1368}{192866774113}a^{33}+\frac{45144}{192866774113}a^{31}-\frac{5776}{192866774113}a^{30}+\frac{677160}{192866774113}a^{29}-\frac{173280}{192866774113}a^{28}+\frac{6109412}{192866774113}a^{27}-\frac{2339280}{192866774113}a^{26}+\frac{36970884}{192866774113}a^{25}-\frac{20528208}{192866774113}a^{24}+\frac{158430816}{192866774113}a^{23}-\frac{141784992}{192866774113}a^{22}+\frac{494316616}{192866774113}a^{21}-\frac{810819072}{192866774113}a^{20}+\frac{1132133088}{192866774113}a^{19}-\frac{3607845551}{192866774113}a^{18}+\frac{1867627800}{192866774113}a^{17}-\frac{11456857710}{192866774113}a^{16}+\frac{2044038088}{192866774113}a^{15}-\frac{23808354489}{192866774113}a^{14}+\frac{1065330456}{192866774113}a^{13}-\frac{27672837342}{192866774113}a^{12}-\frac{366437268}{192866774113}a^{11}-\frac{6865870329}{192866774113}a^{10}-\frac{32201504}{14835905701}a^{9}+\frac{24884991558}{192866774113}a^{8}+\frac{192400308}{192866774113}a^{7}+\frac{32624689498}{192866774113}a^{6}-\frac{1727346924}{192866774113}a^{5}+\frac{18588594300}{192866774113}a^{4}-\frac{48836802000}{192866774113}a^{3}+\frac{7842820689}{192866774113}a^{2}-\frac{137732313204}{192866774113}a+\frac{183348339859}{192866774113}$, $\frac{1368}{192866774113}a^{33}+\frac{45144}{192866774113}a^{31}-\frac{5776}{192866774113}a^{30}+\frac{677160}{192866774113}a^{29}-\frac{173280}{192866774113}a^{28}+\frac{6109412}{192866774113}a^{27}-\frac{2339280}{192866774113}a^{26}+\frac{36970884}{192866774113}a^{25}-\frac{20528208}{192866774113}a^{24}+\frac{158430816}{192866774113}a^{23}-\frac{141784992}{192866774113}a^{22}+\frac{494316616}{192866774113}a^{21}-\frac{810819072}{192866774113}a^{20}+\frac{1132133088}{192866774113}a^{19}-\frac{3607845551}{192866774113}a^{18}+\frac{1867627800}{192866774113}a^{17}-\frac{11456857710}{192866774113}a^{16}+\frac{2044038088}{192866774113}a^{15}-\frac{23808354489}{192866774113}a^{14}+\frac{1065330456}{192866774113}a^{13}-\frac{27672837342}{192866774113}a^{12}-\frac{366437268}{192866774113}a^{11}-\frac{6865870329}{192866774113}a^{10}-\frac{32201504}{14835905701}a^{9}+\frac{24884991558}{192866774113}a^{8}+\frac{192400308}{192866774113}a^{7}+\frac{32624689498}{192866774113}a^{6}-\frac{1727346924}{192866774113}a^{5}+\frac{18588594300}{192866774113}a^{4}-\frac{48836802000}{192866774113}a^{3}+\frac{7842820689}{192866774113}a^{2}-\frac{137732313204}{192866774113}a-\frac{9518434254}{192866774113}$, $\frac{416020}{3278735159921}a^{33}+\frac{13728660}{3278735159921}a^{31}+\frac{1762289}{3278735159921}a^{30}+\frac{205929900}{3278735159921}a^{29}+\frac{52868670}{3278735159921}a^{28}+\frac{1857945320}{3278735159921}a^{27}+\frac{713727045}{3278735159921}a^{26}+\frac{864905580}{252210396917}a^{25}+\frac{440572250}{252210396917}a^{24}+\frac{3706738200}{252210396917}a^{23}+\frac{30399485250}{3278735159921}a^{22}+\frac{11577836600}{252210396917}a^{21}+\frac{112356497484}{3278735159921}a^{20}+\frac{347335098000}{3278735159921}a^{19}+\frac{296496312805}{3278735159921}a^{18}+\frac{593943017580}{3278735159921}a^{17}+\frac{33148656090}{192866774113}a^{16}+\frac{43995779080}{192866774113}a^{15}+\frac{45202712850}{192866774113}a^{14}+\frac{40170059160}{192866774113}a^{13}+\frac{3348349100}{14835905701}a^{12}+\frac{1991903760}{14835905701}a^{11}+\frac{2209910406}{14835905701}a^{10}+\frac{869481800}{14835905701}a^{9}+\frac{951636060}{14835905701}a^{8}+\frac{3130134480}{192866774113}a^{7}+\frac{246720460}{14835905701}a^{6}+\frac{494231760}{192866774113}a^{5}+\frac{7401613800}{3278735159921}a^{4}+\frac{36609760}{192866774113}a^{3}+\frac{396515025}{3278735159921}a^{2}+\frac{13728660}{3278735159921}a-\frac{3278731635343}{3278735159921}$, $\frac{417312}{3278735159921}a^{33}+\frac{13771296}{3278735159921}a^{31}-\frac{1761984}{3278735159921}a^{30}+\frac{206569440}{3278735159921}a^{29}-\frac{52859520}{3278735159921}a^{28}+\frac{1863692208}{3278735159921}a^{27}-\frac{713603520}{3278735159921}a^{26}+\frac{11278065456}{3278735159921}a^{25}-\frac{5663109295}{3278735159921}a^{24}+\frac{48329737344}{3278735159921}a^{23}-\frac{28874095080}{3278735159921}a^{22}+\frac{8870152032}{192866774113}a^{21}-\frac{96375698244}{3278735159921}a^{20}+\frac{20315304576}{192866774113}a^{19}-\frac{200739380048}{3278735159921}a^{18}+\frac{569724775200}{3278735159921}a^{17}-\frac{205424237034}{3278735159921}a^{16}+\frac{623539197792}{3278735159921}a^{15}+\frac{85161944160}{3278735159921}a^{14}+\frac{324981859104}{3278735159921}a^{13}+\frac{517145321392}{3278735159921}a^{12}-\frac{111782652912}{3278735159921}a^{11}+\frac{518544568128}{3278735159921}a^{10}-\frac{9823153536}{252210396917}a^{9}-\frac{57260871}{3278735159921}a^{8}+\frac{58692220272}{3278735159921}a^{7}-\frac{149804392696}{3278735159921}a^{6}-\frac{526931724816}{3278735159921}a^{5}+\frac{893534573172}{3278735159921}a^{4}-\frac{1008849974832}{3278735159921}a^{3}+\frac{1607993277840}{3278735159921}a^{2}-\frac{26828432064}{252210396917}a-\frac{6202660343311}{3278735159921}$, $\frac{391616}{3278735159921}a^{33}+\frac{12923328}{3278735159921}a^{31}+\frac{97904}{3278735159921}a^{30}+\frac{193849920}{3278735159921}a^{29}+\frac{2937120}{3278735159921}a^{28}+\frac{133962796}{252210396917}a^{27}+\frac{39651120}{3278735159921}a^{26}+\frac{10383306516}{3278735159921}a^{25}+\frac{347950816}{3278735159921}a^{24}+\frac{42950091888}{3278735159921}a^{23}+\frac{2403151584}{3278735159921}a^{22}+\frac{124747701228}{3278735159921}a^{21}+\frac{13742197056}{3278735159921}a^{20}+\frac{249770293524}{3278735159921}a^{19}+\frac{61145747392}{3278735159921}a^{18}+\frac{18825411204}{192866774113}a^{17}+\frac{194167128960}{3278735159921}a^{16}+\frac{194276893988}{3278735159921}a^{15}+\frac{403492440528}{3278735159921}a^{14}-\frac{80041637940}{3278735159921}a^{13}+\frac{479136987088}{3278735159921}a^{12}-\frac{138518533008}{3278735159921}a^{11}+\frac{238193676624}{3278735159921}a^{10}+\frac{271924153220}{3278735159921}a^{9}+\frac{126513506880}{3278735159921}a^{8}+\frac{511266706344}{3278735159921}a^{7}+\frac{401481100447}{3278735159921}a^{6}-\frac{272387361924}{3278735159921}a^{5}-\frac{345303983190}{3278735159921}a^{4}-\frac{132327832348}{3278735159921}a^{3}-\frac{1411878579849}{3278735159921}a^{2}+\frac{1982940175248}{3278735159921}a-\frac{173636851105}{3278735159921}$, $\frac{121393}{6557470319842}a^{35}+\frac{242786}{3278735159921}a^{34}+\frac{4248755}{6557470319842}a^{33}+\frac{8011938}{3278735159921}a^{32}+\frac{4005969}{385733548226}a^{31}+\frac{120179070}{3278735159921}a^{30}+\frac{662198815}{6557470319842}a^{29}+\frac{1079669342}{3278735159921}a^{28}+\frac{2180825245}{3278735159921}a^{27}+\frac{12898613825}{6557470319842}a^{26}+\frac{20613866723}{6557470319842}a^{25}+\frac{26941363385}{3278735159921}a^{24}+\frac{36287159739}{3278735159921}a^{23}+\frac{161900205595}{6557470319842}a^{22}+\frac{98353458351}{3278735159921}a^{21}+\frac{179029428901}{3278735159921}a^{20}+\frac{425169635239}{6557470319842}a^{19}+\frac{605658136513}{6557470319842}a^{18}+\frac{57795814265}{504420793834}a^{17}+\frac{423274362741}{3278735159921}a^{16}+\frac{1055824236403}{6557470319842}a^{15}+\frac{536725626306}{3278735159921}a^{14}+\frac{1035986920701}{6557470319842}a^{13}+\frac{613284627558}{3278735159921}a^{12}+\frac{545108104313}{6557470319842}a^{11}+\frac{523499519691}{3278735159921}a^{10}+\frac{126465406505}{3278735159921}a^{9}+\frac{97535077096}{3278735159921}a^{8}+\frac{373942517597}{3278735159921}a^{7}-\frac{1146491770967}{6557470319842}a^{6}+\frac{71882743557}{504420793834}a^{5}-\frac{261444232221}{3278735159921}a^{4}+\frac{288598382877}{6557470319842}a^{3}+\frac{2418199041185}{6557470319842}a^{2}-\frac{1913773027086}{3278735159921}a+\frac{3278759317738}{3278735159921}$, $\frac{61190}{3278735159921}a^{35}+\frac{244760}{3278735159921}a^{34}+\frac{2141650}{3278735159921}a^{33}+\frac{8077080}{3278735159921}a^{32}+\frac{2019270}{192866774113}a^{31}+\frac{121156200}{3278735159921}a^{30}+\frac{333791450}{3278735159921}a^{29}+\frac{1088447720}{3278735159921}a^{28}+\frac{2198556700}{3278735159921}a^{27}+\frac{12979133285}{6557470319842}a^{26}+\frac{10390735090}{3278735159921}a^{25}+\frac{26843808305}{3278735159921}a^{24}+\frac{36582196740}{3278735159921}a^{23}+\frac{155934639415}{6557470319842}a^{22}+\frac{99153132660}{3278735159921}a^{21}+\frac{156106464445}{3278735159921}a^{20}+\frac{214313263370}{3278735159921}a^{19}+\frac{400040192905}{6557470319842}a^{18}+\frac{29132864950}{252210396917}a^{17}+\frac{242846592295}{6557470319842}a^{16}+\frac{532204369490}{3278735159921}a^{15}-\frac{48527949670}{3278735159921}a^{14}+\frac{522205066830}{3278735159921}a^{13}-\frac{65606032040}{3278735159921}a^{12}+\frac{274770084790}{3278735159921}a^{11}+\frac{285274792175}{3278735159921}a^{10}+\frac{127493648300}{3278735159921}a^{9}+\frac{634056014560}{3278735159921}a^{8}+\frac{376982901020}{3278735159921}a^{7}+\frac{540155527705}{6557470319842}a^{6}+\frac{350786922561}{6557470319842}a^{5}+\frac{210841655475}{3278735159921}a^{4}-\frac{2665488171675}{6557470319842}a^{3}+\frac{2625448491405}{6557470319842}a^{2}-\frac{6815099403015}{6557470319842}a-\frac{3275739787031}{3278735159921}$, $\frac{1346269}{6557470319842}a^{32}+\frac{2178309}{6557470319842}a^{31}+\frac{21540304}{3278735159921}a^{30}+\frac{67527579}{6557470319842}a^{29}+\frac{312334408}{3278735159921}a^{28}+\frac{472693053}{3278735159921}a^{27}+\frac{2714078304}{3278735159921}a^{26}+\frac{607748211}{504420793834}a^{25}+\frac{1211642100}{252210396917}a^{24}+\frac{1688189475}{252210396917}a^{23}+\frac{4954269920}{252210396917}a^{22}+\frac{85422387435}{3278735159921}a^{21}+\frac{190739391920}{3278735159921}a^{20}+\frac{239182684818}{3278735159921}a^{19}+\frac{414176965312}{3278735159921}a^{18}+\frac{973815216759}{6557470319842}a^{17}+\frac{38829090498}{192866774113}a^{16}+\frac{42339792033}{192866774113}a^{15}+\frac{45019235360}{192866774113}a^{14}+\frac{44905840035}{192866774113}a^{13}+\frac{2864860432}{14835905701}a^{12}+\frac{2566048002}{14835905701}a^{11}+\frac{1637063104}{14835905701}a^{10}+\frac{1283024001}{14835905701}a^{9}+\frac{613898664}{14835905701}a^{8}+\frac{405165474}{14835905701}a^{7}+\frac{1809385536}{192866774113}a^{6}+\frac{945386106}{192866774113}a^{5}+\frac{215403040}{192866774113}a^{4}+\frac{1350551580}{3278735159921}a^{3}+\frac{172322432}{3278735159921}a^{2}+\frac{67527579}{6557470319842}a+\frac{3278736506190}{3278735159921}$, $\frac{514229}{6557470319842}a^{34}+\frac{514229}{192866774113}a^{32}+\frac{15941099}{385733548226}a^{30}+\frac{5702887}{6557470319842}a^{29}+\frac{74563205}{192866774113}a^{28}+\frac{165383723}{6557470319842}a^{27}+\frac{939496383}{385733548226}a^{26}+\frac{165383723}{504420793834}a^{25}+\frac{161982135}{14835905701}a^{24}+\frac{8269186150}{3278735159921}a^{23}+\frac{532227015}{14835905701}a^{22}+\frac{41842081919}{3278735159921}a^{21}+\frac{1300999370}{14835905701}a^{20}+\frac{292894573433}{6557470319842}a^{19}+\frac{61797470075}{385733548226}a^{18}+\frac{725869160247}{6557470319842}a^{17}+\frac{42022279651}{192866774113}a^{16}+\frac{37707488844}{192866774113}a^{15}+\frac{42022279651}{192866774113}a^{14}+\frac{47134361055}{192866774113}a^{13}+\frac{2325343538}{14835905701}a^{12}+\frac{3142290737}{14835905701}a^{11}+\frac{1162671769}{14835905701}a^{10}+\frac{1819220953}{14835905701}a^{9}+\frac{4982879010}{192866774113}a^{8}+\frac{661534892}{14835905701}a^{7}+\frac{996575802}{192866774113}a^{6}+\frac{2315372122}{252210396917}a^{5}+\frac{104902716}{192866774113}a^{4}+\frac{5788430305}{6557470319842}a^{3}+\frac{8741893}{385733548226}a^{2}+\frac{165383723}{6557470319842}a+\frac{3278735674150}{3278735159921}$, $\frac{233}{6557470319842}a^{34}+\frac{233}{192866774113}a^{32}+\frac{7223}{385733548226}a^{30}+\frac{2851454}{3278735159921}a^{29}+\frac{33785}{192866774113}a^{28}+\frac{82692166}{3278735159921}a^{27}+\frac{425691}{385733548226}a^{26}+\frac{82692166}{252210396917}a^{25}+\frac{73395}{14835905701}a^{24}+\frac{8269216600}{3278735159921}a^{23}+\frac{241155}{14835905701}a^{22}+\frac{41842235996}{3278735159921}a^{21}+\frac{589490}{14835905701}a^{20}+\frac{146447825986}{3278735159921}a^{19}+\frac{28000775}{385733548226}a^{18}+\frac{362935916574}{3278735159921}a^{17}+\frac{19040527}{192866774113}a^{16}+\frac{37707627696}{192866774113}a^{15}+\frac{19040527}{192866774113}a^{14}+\frac{93529035939}{385733548226}a^{13}+\frac{1053626}{14835905701}a^{12}+\frac{5544571315}{29671811402}a^{11}+\frac{526813}{14835905701}a^{10}-\frac{61711201}{29671811402}a^{9}+\frac{8215361213}{385733548226}a^{8}-\frac{3778662478}{14835905701}a^{7}+\frac{32843834246}{192866774113}a^{6}-\frac{85748582171}{252210396917}a^{5}+\frac{82108504262}{192866774113}a^{4}-\frac{1139043065027}{6557470319842}a^{3}+\frac{131373534729}{385733548226}a^{2}-\frac{163381975189}{6557470319842}a-\frac{3139150783247}{3278735159921}$, $\frac{57314}{3278735159921}a^{35}+\frac{28657}{385733548226}a^{34}+\frac{2005990}{3278735159921}a^{33}+\frac{945681}{385733548226}a^{32}+\frac{31064188}{3278735159921}a^{31}+\frac{14185215}{385733548226}a^{30}+\frac{278947238}{3278735159921}a^{29}+\frac{63991081}{192866774113}a^{28}+\frac{1588342882}{3278735159921}a^{27}+\frac{6587728474}{3278735159921}a^{26}+\frac{5805564316}{3278735159921}a^{25}+\frac{2179078280}{252210396917}a^{24}+\frac{25131587211}{6557470319842}a^{23}+\frac{6873438764}{252210396917}a^{22}+\frac{19514184933}{6557470319842}a^{21}+\frac{210392587064}{3278735159921}a^{20}-\frac{22055773159}{3278735159921}a^{19}+\frac{738302069027}{6557470319842}a^{18}-\frac{103502281803}{6557470319842}a^{17}+\frac{54287448259}{385733548226}a^{16}+\frac{44835098862}{3278735159921}a^{15}+\frac{698532170997}{6557470319842}a^{14}+\frac{310249896032}{3278735159921}a^{13}+\frac{77381895303}{3278735159921}a^{12}+\frac{480298723168}{3278735159921}a^{11}-\frac{5006836412}{3278735159921}a^{10}+\frac{535646315755}{6557470319842}a^{9}+\frac{308166516997}{6557470319842}a^{8}-\frac{12714159147}{504420793834}a^{7}-\frac{45696738770}{3278735159921}a^{6}+\frac{547317633467}{6557470319842}a^{5}-\frac{739483711765}{6557470319842}a^{4}+\frac{1164888224375}{3278735159921}a^{3}-\frac{599258964138}{3278735159921}a^{2}-\frac{145417861271}{385733548226}a+\frac{139716063073}{192866774113}$, $\frac{48952}{3278735159921}a^{35}+\frac{12238}{192866774113}a^{34}+\frac{1713320}{3278735159921}a^{33}+\frac{403854}{192866774113}a^{32}+\frac{26531984}{3278735159921}a^{31}+\frac{6057810}{192866774113}a^{30}+\frac{238249384}{3278735159921}a^{29}+\frac{54654908}{192866774113}a^{28}+\frac{1356606776}{3278735159921}a^{27}+\frac{5626591832}{3278735159921}a^{26}+\frac{4958543888}{3278735159921}a^{25}+\frac{1861155040}{252210396917}a^{24}+\frac{21377534207}{6557470319842}a^{23}+\frac{5870617552}{252210396917}a^{22}+\frac{14656816201}{6557470319842}a^{21}+\frac{179696721952}{3278735159921}a^{20}-\frac{57179558677}{6557470319842}a^{19}+\frac{315292623818}{3278735159921}a^{18}-\frac{95802736046}{3278735159921}a^{17}+\frac{23183508106}{192866774113}a^{16}-\frac{121225909918}{3278735159921}a^{15}+\frac{298308849798}{3278735159921}a^{14}-\frac{28542159211}{6557470319842}a^{13}+\frac{66092028804}{3278735159921}a^{12}+\frac{392758818377}{6557470319842}a^{11}-\frac{4276348816}{3278735159921}a^{10}+\frac{323913321152}{3278735159921}a^{9}+\frac{131602813798}{3278735159921}a^{8}+\frac{498896258721}{6557470319842}a^{7}-\frac{39029674360}{3278735159921}a^{6}+\frac{931233474765}{6557470319842}a^{5}-\frac{315797245510}{3278735159921}a^{4}+\frac{162798197823}{504420793834}a^{3}-\frac{3528387323605}{6557470319842}a^{2}-\frac{4755783818}{14835905701}a-\frac{385600007094}{192866774113}$, $\frac{150050}{3278735159921}a^{35}-\frac{75025}{6557470319842}a^{34}+\frac{5251750}{3278735159921}a^{33}-\frac{2475825}{6557470319842}a^{32}+\frac{4951650}{192866774113}a^{31}-\frac{37137375}{6557470319842}a^{30}+\frac{815821850}{3278735159921}a^{29}-\frac{334986625}{6557470319842}a^{28}+\frac{5312970400}{3278735159921}a^{27}-\frac{2002792375}{6557470319842}a^{26}+\frac{48865958369}{6557470319842}a^{25}-\frac{237379100}{192866774113}a^{24}+\frac{162399193625}{6557470319842}a^{23}-\frac{10053875175}{3278735159921}a^{22}+\frac{393342535075}{6557470319842}a^{21}-\frac{7989562300}{3278735159921}a^{20}+\frac{348226337725}{3278735159921}a^{19}+\frac{89231809025}{6557470319842}a^{18}+\frac{53884273125}{385733548226}a^{17}+\frac{398333308525}{6557470319842}a^{16}+\frac{481364346210}{3278735159921}a^{15}+\frac{777117577875}{6557470319842}a^{14}+\frac{473134528500}{3278735159921}a^{13}+\frac{766913127525}{6557470319842}a^{12}+\frac{444128975000}{3278735159921}a^{11}+\frac{193516484000}{3278735159921}a^{10}+\frac{184818933925}{3278735159921}a^{9}+\frac{308429200475}{3278735159921}a^{8}-\frac{1073259535825}{6557470319842}a^{7}+\frac{1465915650725}{6557470319842}a^{6}-\frac{151428528935}{504420793834}a^{5}+\frac{664356803475}{3278735159921}a^{4}+\frac{400218583525}{3278735159921}a^{3}-\frac{1004559241500}{3278735159921}a^{2}+\frac{2984989143425}{6557470319842}a+\frac{53716406113}{192866774113}$, $\frac{48952}{3278735159921}a^{35}+\frac{12238}{192866774113}a^{34}+\frac{1713320}{3278735159921}a^{33}+\frac{403854}{192866774113}a^{32}+\frac{26531984}{3278735159921}a^{31}+\frac{6057810}{192866774113}a^{30}+\frac{238249384}{3278735159921}a^{29}+\frac{54654908}{192866774113}a^{28}+\frac{1356606776}{3278735159921}a^{27}+\frac{5626591832}{3278735159921}a^{26}+\frac{4958543888}{3278735159921}a^{25}+\frac{1861155040}{252210396917}a^{24}+\frac{21377534207}{6557470319842}a^{23}+\frac{5870617552}{252210396917}a^{22}+\frac{14656816201}{6557470319842}a^{21}+\frac{179696721952}{3278735159921}a^{20}-\frac{57179558677}{6557470319842}a^{19}+\frac{315292623818}{3278735159921}a^{18}-\frac{95802736046}{3278735159921}a^{17}+\frac{23183508106}{192866774113}a^{16}-\frac{121225909918}{3278735159921}a^{15}+\frac{298308849798}{3278735159921}a^{14}-\frac{28542159211}{6557470319842}a^{13}+\frac{66092028804}{3278735159921}a^{12}+\frac{392758818377}{6557470319842}a^{11}-\frac{4276348816}{3278735159921}a^{10}+\frac{323913321152}{3278735159921}a^{9}+\frac{131602813798}{3278735159921}a^{8}+\frac{498896258721}{6557470319842}a^{7}-\frac{39029674360}{3278735159921}a^{6}+\frac{931233474765}{6557470319842}a^{5}-\frac{315797245510}{3278735159921}a^{4}+\frac{162798197823}{504420793834}a^{3}-\frac{3528387323605}{6557470319842}a^{2}-\frac{4755783818}{14835905701}a-\frac{192733232981}{192866774113}$, $\frac{366458}{3278735159921}a^{35}+\frac{412377}{6557470319842}a^{34}+\frac{12826030}{3278735159921}a^{33}+\frac{6804337}{3278735159921}a^{32}+\frac{410848065}{6557470319842}a^{31}+\frac{204134071}{6557470319842}a^{30}+\frac{1994469271}{3278735159921}a^{29}+\frac{1841858819}{6557470319842}a^{28}+\frac{26217025093}{6557470319842}a^{27}+\frac{5602029566}{3278735159921}a^{26}+\frac{9491515099}{504420793834}a^{25}+\frac{24622844791}{3278735159921}a^{24}+\frac{429299721431}{6557470319842}a^{23}+\frac{12776693381}{504420793834}a^{22}+\frac{1125216875251}{6557470319842}a^{21}+\frac{457073306835}{6557470319842}a^{20}+\frac{1123682603025}{3278735159921}a^{19}+\frac{535730212728}{3278735159921}a^{18}+\frac{1719863124502}{3278735159921}a^{17}+\frac{1056654462839}{3278735159921}a^{16}+\frac{2011827701561}{3278735159921}a^{15}+\frac{1632650236175}{3278735159921}a^{14}+\frac{273625505739}{504420793834}a^{13}+\frac{1802480331253}{3278735159921}a^{12}+\frac{1198579906532}{3278735159921}a^{11}+\frac{1341108279064}{3278735159921}a^{10}+\frac{582654756253}{3278735159921}a^{9}+\frac{1566554188697}{6557470319842}a^{8}-\frac{578449218429}{6557470319842}a^{7}+\frac{1593003061369}{6557470319842}a^{6}-\frac{1060643113337}{3278735159921}a^{5}+\frac{1320949175003}{3278735159921}a^{4}+\frac{275288636923}{6557470319842}a^{3}-\frac{569941296059}{3278735159921}a^{2}-\frac{519935275760}{3278735159921}a-\frac{240946676941}{252210396917}$, $\frac{61190}{3278735159921}a^{35}+\frac{244760}{3278735159921}a^{34}+\frac{2141650}{3278735159921}a^{33}+\frac{8077080}{3278735159921}a^{32}+\frac{2019270}{192866774113}a^{31}+\frac{121156200}{3278735159921}a^{30}+\frac{333791450}{3278735159921}a^{29}+\frac{1088447720}{3278735159921}a^{28}+\frac{2198556700}{3278735159921}a^{27}+\frac{12979133285}{6557470319842}a^{26}+\frac{10390735090}{3278735159921}a^{25}+\frac{26843808305}{3278735159921}a^{24}+\frac{36582196740}{3278735159921}a^{23}+\frac{155934639415}{6557470319842}a^{22}+\frac{99153132660}{3278735159921}a^{21}+\frac{156106464445}{3278735159921}a^{20}+\frac{214313263370}{3278735159921}a^{19}+\frac{400040192905}{6557470319842}a^{18}+\frac{29132864950}{252210396917}a^{17}+\frac{242846592295}{6557470319842}a^{16}+\frac{532204369490}{3278735159921}a^{15}-\frac{48527949670}{3278735159921}a^{14}+\frac{522205066830}{3278735159921}a^{13}-\frac{65606032040}{3278735159921}a^{12}+\frac{274770084790}{3278735159921}a^{11}+\frac{285274792175}{3278735159921}a^{10}+\frac{127493648300}{3278735159921}a^{9}+\frac{634056014560}{3278735159921}a^{8}+\frac{376982901020}{3278735159921}a^{7}+\frac{540155527705}{6557470319842}a^{6}+\frac{350786922561}{6557470319842}a^{5}+\frac{210841655475}{3278735159921}a^{4}-\frac{2665488171675}{6557470319842}a^{3}+\frac{2625448491405}{6557470319842}a^{2}-\frac{6815099403015}{6557470319842}a+\frac{2995372890}{3278735159921}$, $\frac{89}{6557470319842}a^{32}+\frac{1089182}{3278735159921}a^{31}+\frac{1424}{3278735159921}a^{30}+\frac{33764642}{3278735159921}a^{29}+\frac{20648}{3278735159921}a^{28}+\frac{472704988}{3278735159921}a^{27}+\frac{179424}{3278735159921}a^{26}+\frac{303881778}{252210396917}a^{25}+\frac{80100}{252210396917}a^{24}+\frac{1688232100}{252210396917}a^{23}+\frac{327520}{252210396917}a^{22}+\frac{85424544260}{3278735159921}a^{21}+\frac{12609520}{3278735159921}a^{20}+\frac{239188723928}{3278735159921}a^{19}+\frac{27380672}{3278735159921}a^{18}+\frac{486919902282}{3278735159921}a^{17}+\frac{2566938}{192866774113}a^{16}+\frac{42340861068}{192866774113}a^{15}+\frac{2976160}{192866774113}a^{14}+\frac{44906973860}{192866774113}a^{13}+\frac{189392}{14835905701}a^{12}+\frac{64780750137}{385733548226}a^{11}+\frac{3139145433}{385733548226}a^{10}+\frac{12039459291}{385733548226}a^{9}+\frac{15682185637}{192866774113}a^{8}-\frac{37372729858}{192866774113}a^{7}+\frac{8443988119}{29671811402}a^{6}-\frac{147349229083}{385733548226}a^{5}+\frac{78408304465}{192866774113}a^{4}-\frac{1809499424065}{6557470319842}a^{3}+\frac{1332940956609}{6557470319842}a^{2}-\frac{362372589801}{6557470319842}a-\frac{3225417522479}{3278735159921}$ 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[K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 816369751172.7767 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 816369751172.7767 \cdot 95904}{2\cdot\sqrt{4999758568289868528789868885747458073284974388119052886962890625}}\cr\approx \mathstrut & 0.128960510565241 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_6^2$ (as 36T4):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

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{-35}) $, $\Q(\sqrt{21}) $, $\Q(\sqrt{-15}) $, $\Q(\zeta_{9})^+$, 3.3.3969.2, $\Q(\zeta_{7})^+$, 3.3.3969.1, $\Q(\sqrt{-15}, \sqrt{21})$, 6.0.281302875.3, 6.0.13783840875.2, 6.0.2100875.1, 6.0.13783840875.1, 6.6.6751269.1, 6.0.2460375.1, 6.6.330812181.1, 6.0.5907360375.2, $\Q(\zeta_{21})^+$, 6.0.8103375.1, 6.6.330812181.2, 6.0.5907360375.1, 9.9.62523502209.1, 12.0.712181767349390625.3, 12.0.1709948423405886890625.3, 12.0.3217569633140625.1, 12.0.1709948423405886890625.2, 18.0.2618850774742652270958169921875.4, $\Q(\zeta_{63})^+$, 18.0.206148603259625688967552734375.1

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

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{6}$	R	R	R	${\href{/padicField/11.6.0.1}{6} }^{6}$	${\href{/padicField/13.6.0.1}{6} }^{6}$	${\href{/padicField/17.3.0.1}{3} }^{12}$	${\href{/padicField/19.6.0.1}{6} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{6}$	${\href{/padicField/29.6.0.1}{6} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{6}$	${\href{/padicField/43.6.0.1}{6} }^{6}$	${\href{/padicField/47.3.0.1}{3} }^{12}$	${\href{/padicField/53.6.0.1}{6} }^{6}$	${\href{/padicField/59.6.0.1}{6} }^{6}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3		Deg $18$	$6$	$3$	$27$
$3$ 3		Deg $18$	$6$	$3$	$27$
$5$ 5	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$ 7		Deg $36$	$6$	$6$	$30$

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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