LMFDB - Number field 36.0.32387780912664470931540162651878867184371649203720968942991179776.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827)

gp: K = bnfinit(y^36 - 42*y^34 - 12*y^33 + 798*y^32 + 444*y^31 - 8930*y^30 - 7224*y^29 + 64293*y^28 + 66808*y^27 - 308040*y^26 - 380412*y^25 + 1018573*y^24 + 1390296*y^23 - 2660256*y^22 - 3730568*y^21 + 6607119*y^20 + 9624456*y^19 - 7505542*y^18 - 2964660*y^17 + 28706040*y^16 + 6977356*y^15 - 34628742*y^14 + 28138392*y^13 + 70034750*y^12 - 59344320*y^11 - 97239042*y^10 + 110557824*y^9 + 103706163*y^8 - 146720820*y^7 + 54606518*y^6 - 96816432*y^5 + 280912497*y^4 - 164862264*y^3 + 36478572*y^2 - 7070100*y + 9128827, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827)

$ x^{36} - 42 x^{34} - 12 x^{33} + 798 x^{32} + 444 x^{31} - 8930 x^{30} - 7224 x^{29} + 64293 x^{28} + \cdots + 9128827 $ x^36 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827

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:	$32387780912664470931540162651878867184371649203720968942991179776$ 32387780912664470931540162651878867184371649203720968942991179776 $\medspace = 2^{54}\cdot 3^{48}\cdot 7^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$61.94$	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:	$2^{3/2}3^{4/3}7^{5/6}\approx 61.937694144633795$
Ramified primes:	$2$, $3$, $7$ 2, 3, 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:	$504=2^{3}\cdot 3^{2}\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(115,·)$, $\chi_{504}(265,·)$, $\chi_{504}(139,·)$, $\chi_{504}(145,·)$, $\chi_{504}(403,·)$, $\chi_{504}(409,·)$, $\chi_{504}(25,·)$, $\chi_{504}(283,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(169,·)$, $\chi_{504}(43,·)$, $\chi_{504}(433,·)$, $\chi_{504}(307,·)$, $\chi_{504}(73,·)$, $\chi_{504}(313,·)$, $\chi_{504}(187,·)$, $\chi_{504}(19,·)$, $\chi_{504}(193,·)$, $\chi_{504}(67,·)$, $\chi_{504}(97,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(337,·)$, $\chi_{504}(211,·)$, $\chi_{504}(475,·)$, $\chi_{504}(481,·)$, $\chi_{504}(355,·)$, $\chi_{504}(361,·)$, $\chi_{504}(235,·)$, $\chi_{504}(451,·)$, $\chi_{504}(241,·)$, $\chi_{504}(499,·)$, $\chi_{504}(121,·)$, $\chi_{504}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{8}+\frac{3}{8}a^{4}+\frac{3}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{9}+\frac{3}{8}a^{5}+\frac{3}{8}a^{3}+\frac{1}{8}a$, $\frac{1}{8}a^{20}-\frac{1}{8}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{8}a^{6}+\frac{3}{8}a^{4}-\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{17}-\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{4}a^{11}-\frac{1}{8}a^{7}+\frac{3}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{22}-\frac{1}{8}a^{16}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{1}{8}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{23}-\frac{1}{8}a^{17}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{4}a^{5}-\frac{3}{8}a^{3}-\frac{1}{8}a$, $\frac{1}{16}a^{24}-\frac{1}{8}a^{16}-\frac{1}{8}a^{12}+\frac{1}{16}a^{8}-\frac{1}{8}a^{4}+\frac{1}{16}$, $\frac{1}{16}a^{25}-\frac{1}{8}a^{17}-\frac{1}{8}a^{13}+\frac{1}{16}a^{9}-\frac{1}{8}a^{5}+\frac{1}{16}a$, $\frac{1}{16}a^{26}-\frac{1}{8}a^{12}-\frac{1}{16}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{5}{16}a^{2}-\frac{3}{8}$, $\frac{1}{16}a^{27}-\frac{1}{8}a^{13}-\frac{1}{16}a^{11}-\frac{1}{4}a^{9}+\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{5}{16}a^{3}-\frac{3}{8}a$, $\frac{1}{16}a^{28}-\frac{1}{8}a^{14}-\frac{1}{16}a^{12}-\frac{1}{4}a^{10}+\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{5}{16}a^{4}-\frac{3}{8}a^{2}$, $\frac{1}{16}a^{29}-\frac{1}{8}a^{15}-\frac{1}{16}a^{13}-\frac{1}{4}a^{11}+\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{5}{16}a^{5}-\frac{3}{8}a^{3}$, $\frac{1}{32}a^{30}-\frac{1}{32}a^{26}-\frac{1}{32}a^{24}-\frac{1}{16}a^{22}+\frac{1}{16}a^{16}+\frac{3}{32}a^{14}+\frac{1}{16}a^{12}-\frac{3}{32}a^{10}-\frac{1}{32}a^{8}+\frac{3}{32}a^{6}+\frac{1}{16}a^{4}-\frac{1}{32}a^{2}-\frac{1}{32}$, $\frac{1}{32}a^{31}-\frac{1}{32}a^{27}-\frac{1}{32}a^{25}-\frac{1}{16}a^{23}+\frac{1}{16}a^{17}+\frac{3}{32}a^{15}+\frac{1}{16}a^{13}-\frac{3}{32}a^{11}-\frac{1}{32}a^{9}+\frac{3}{32}a^{7}+\frac{1}{16}a^{5}-\frac{1}{32}a^{3}-\frac{1}{32}a$, $\frac{1}{12128}a^{32}-\frac{35}{3032}a^{31}-\frac{147}{12128}a^{30}-\frac{29}{3032}a^{29}+\frac{249}{12128}a^{28}+\frac{31}{3032}a^{27}-\frac{19}{3032}a^{26}+\frac{33}{1516}a^{25}+\frac{317}{12128}a^{24}+\frac{15}{758}a^{23}-\frac{361}{6064}a^{22}-\frac{33}{1516}a^{21}-\frac{20}{379}a^{20}+\frac{163}{3032}a^{19}+\frac{307}{6064}a^{18}-\frac{67}{1516}a^{17}-\frac{619}{12128}a^{16}+\frac{39}{1516}a^{15}+\frac{417}{12128}a^{14}-\frac{29}{1516}a^{13}-\frac{403}{12128}a^{12}+\frac{91}{758}a^{11}+\frac{1129}{6064}a^{10}+\frac{19}{1516}a^{9}-\frac{1437}{6064}a^{8}-\frac{377}{3032}a^{7}-\frac{1823}{12128}a^{6}-\frac{551}{1516}a^{5}-\frac{2105}{12128}a^{4}+\frac{65}{758}a^{3}+\frac{143}{379}a^{2}+\frac{875}{3032}a-\frac{5241}{12128}$, $\frac{1}{12128}a^{33}-\frac{39}{12128}a^{31}+\frac{149}{12128}a^{30}-\frac{73}{12128}a^{29}+\frac{29}{3032}a^{28}-\frac{75}{6064}a^{27}-\frac{143}{12128}a^{26}+\frac{135}{12128}a^{25}+\frac{277}{12128}a^{24}-\frac{237}{6064}a^{23}-\frac{265}{6064}a^{22}+\frac{75}{3032}a^{21}+\frac{31}{758}a^{20}-\frac{291}{6064}a^{19}+\frac{33}{758}a^{18}+\frac{141}{12128}a^{17}+\frac{411}{6064}a^{16}-\frac{1383}{12128}a^{15}+\frac{919}{12128}a^{14}-\frac{289}{12128}a^{13}-\frac{573}{6064}a^{12}+\frac{85}{1516}a^{11}-\frac{1709}{12128}a^{10}-\frac{515}{3032}a^{9}-\frac{2507}{12128}a^{8}-\frac{703}{12128}a^{7}+\frac{3019}{12128}a^{6}-\frac{4487}{12128}a^{5}+\frac{2875}{6064}a^{4}-\frac{1849}{6064}a^{3}+\frac{2493}{12128}a^{2}-\frac{1119}{12128}a+\frac{2657}{12128}$, $\frac{1}{14\!\cdots\!64}a^{34}-\frac{12\!\cdots\!67}{35\!\cdots\!16}a^{33}-\frac{29\!\cdots\!89}{17\!\cdots\!08}a^{32}+\frac{29\!\cdots\!07}{71\!\cdots\!32}a^{31}-\frac{46\!\cdots\!51}{71\!\cdots\!32}a^{30}-\frac{31\!\cdots\!93}{71\!\cdots\!32}a^{29}-\frac{25\!\cdots\!69}{14\!\cdots\!64}a^{28}-\frac{10\!\cdots\!59}{35\!\cdots\!16}a^{27}-\frac{40\!\cdots\!25}{14\!\cdots\!64}a^{26}-\frac{20\!\cdots\!01}{17\!\cdots\!08}a^{25}+\frac{11\!\cdots\!05}{14\!\cdots\!64}a^{24}+\frac{21\!\cdots\!85}{35\!\cdots\!16}a^{23}+\frac{22\!\cdots\!91}{71\!\cdots\!32}a^{22}+\frac{69\!\cdots\!35}{35\!\cdots\!16}a^{21}+\frac{44\!\cdots\!41}{71\!\cdots\!32}a^{20}-\frac{64\!\cdots\!74}{44\!\cdots\!27}a^{19}-\frac{33\!\cdots\!77}{14\!\cdots\!64}a^{18}-\frac{20\!\cdots\!81}{17\!\cdots\!08}a^{17}+\frac{37\!\cdots\!29}{35\!\cdots\!16}a^{16}-\frac{37\!\cdots\!39}{71\!\cdots\!32}a^{15}+\frac{27\!\cdots\!35}{71\!\cdots\!32}a^{14}+\frac{63\!\cdots\!49}{71\!\cdots\!32}a^{13}-\frac{10\!\cdots\!07}{14\!\cdots\!64}a^{12}+\frac{58\!\cdots\!57}{35\!\cdots\!16}a^{11}-\frac{81\!\cdots\!45}{71\!\cdots\!32}a^{10}-\frac{87\!\cdots\!49}{89\!\cdots\!54}a^{9}-\frac{34\!\cdots\!37}{14\!\cdots\!64}a^{8}-\frac{15\!\cdots\!33}{71\!\cdots\!32}a^{7}+\frac{31\!\cdots\!31}{35\!\cdots\!16}a^{6}-\frac{26\!\cdots\!23}{71\!\cdots\!32}a^{5}+\frac{42\!\cdots\!53}{14\!\cdots\!64}a^{4}+\frac{19\!\cdots\!77}{44\!\cdots\!27}a^{3}-\frac{31\!\cdots\!11}{14\!\cdots\!64}a^{2}-\frac{13\!\cdots\!21}{35\!\cdots\!16}a-\frac{66\!\cdots\!79}{14\!\cdots\!64}$, $\frac{1}{84\!\cdots\!96}a^{35}-\frac{19\!\cdots\!99}{84\!\cdots\!96}a^{34}+\frac{25\!\cdots\!43}{52\!\cdots\!06}a^{33}+\frac{66\!\cdots\!19}{42\!\cdots\!48}a^{32}+\frac{26\!\cdots\!45}{21\!\cdots\!24}a^{31}+\frac{92\!\cdots\!61}{42\!\cdots\!48}a^{30}-\frac{14\!\cdots\!95}{84\!\cdots\!96}a^{29}+\frac{19\!\cdots\!85}{84\!\cdots\!96}a^{28}+\frac{97\!\cdots\!65}{84\!\cdots\!96}a^{27}-\frac{62\!\cdots\!37}{84\!\cdots\!96}a^{26}+\frac{17\!\cdots\!27}{84\!\cdots\!96}a^{25}-\frac{16\!\cdots\!99}{84\!\cdots\!96}a^{24}+\frac{25\!\cdots\!29}{42\!\cdots\!48}a^{23}-\frac{25\!\cdots\!97}{42\!\cdots\!48}a^{22}+\frac{23\!\cdots\!69}{42\!\cdots\!48}a^{21}+\frac{16\!\cdots\!67}{42\!\cdots\!48}a^{20}-\frac{13\!\cdots\!41}{84\!\cdots\!96}a^{19}+\frac{28\!\cdots\!87}{84\!\cdots\!96}a^{18}-\frac{63\!\cdots\!51}{52\!\cdots\!06}a^{17}-\frac{31\!\cdots\!63}{42\!\cdots\!48}a^{16}-\frac{21\!\cdots\!95}{21\!\cdots\!24}a^{15}-\frac{17\!\cdots\!71}{42\!\cdots\!48}a^{14}-\frac{84\!\cdots\!41}{84\!\cdots\!96}a^{13}-\frac{47\!\cdots\!97}{84\!\cdots\!96}a^{12}+\frac{21\!\cdots\!31}{21\!\cdots\!24}a^{11}+\frac{85\!\cdots\!29}{42\!\cdots\!48}a^{10}+\frac{34\!\cdots\!77}{84\!\cdots\!96}a^{9}-\frac{15\!\cdots\!67}{84\!\cdots\!96}a^{8}+\frac{93\!\cdots\!23}{42\!\cdots\!48}a^{7}+\frac{14\!\cdots\!61}{10\!\cdots\!12}a^{6}-\frac{28\!\cdots\!13}{84\!\cdots\!96}a^{5}+\frac{45\!\cdots\!15}{84\!\cdots\!96}a^{4}-\frac{39\!\cdots\!61}{84\!\cdots\!96}a^{3}+\frac{61\!\cdots\!93}{84\!\cdots\!96}a^{2}-\frac{17\!\cdots\!57}{84\!\cdots\!96}a+\frac{41\!\cdots\!57}{84\!\cdots\!96}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^2 - 1/2, 1/2*a^7 - 1/2*a^3 - 1/2*a, 1/2*a^8 - 1/2*a^4 - 1/2*a^2, 1/2*a^9 - 1/2*a^5 - 1/2*a^3, 1/2*a^10 - 1/2*a^4 - 1/2*a^2 - 1/2, 1/2*a^11 - 1/2*a^5 - 1/2*a^3 - 1/2*a, 1/4*a^12 - 1/4*a^4 - 1/2*a^2 - 1/4, 1/4*a^13 - 1/4*a^5 - 1/2*a^3 - 1/4*a, 1/4*a^14 - 1/4*a^6 - 1/2*a^4 - 1/4*a^2, 1/4*a^15 - 1/4*a^7 - 1/2*a^5 - 1/4*a^3, 1/4*a^16 - 1/4*a^8 - 1/4*a^4 - 1/2*a^2 - 1/2, 1/4*a^17 - 1/4*a^9 - 1/4*a^5 - 1/2*a^3 - 1/2*a, 1/8*a^18 - 1/8*a^14 - 1/8*a^12 - 1/8*a^10 - 1/4*a^8 + 3/8*a^4 + 3/8*a^2 + 1/8, 1/8*a^19 - 1/8*a^15 - 1/8*a^13 - 1/8*a^11 - 1/4*a^9 + 3/8*a^5 + 3/8*a^3 + 1/8*a, 1/8*a^20 - 1/8*a^16 - 1/8*a^14 - 1/8*a^12 - 1/4*a^10 - 1/8*a^6 + 3/8*a^4 - 3/8*a^2 - 1/2, 1/8*a^21 - 1/8*a^17 - 1/8*a^15 - 1/8*a^13 - 1/4*a^11 - 1/8*a^7 + 3/8*a^5 - 3/8*a^3 - 1/2*a, 1/8*a^22 - 1/8*a^16 - 1/8*a^12 - 1/8*a^10 + 1/8*a^8 + 1/8*a^6 - 1/4*a^4 - 3/8*a^2 - 1/8, 1/8*a^23 - 1/8*a^17 - 1/8*a^13 - 1/8*a^11 + 1/8*a^9 + 1/8*a^7 - 1/4*a^5 - 3/8*a^3 - 1/8*a, 1/16*a^24 - 1/8*a^16 - 1/8*a^12 + 1/16*a^8 - 1/8*a^4 + 1/16, 1/16*a^25 - 1/8*a^17 - 1/8*a^13 + 1/16*a^9 - 1/8*a^5 + 1/16*a, 1/16*a^26 - 1/8*a^12 - 1/16*a^10 - 1/4*a^8 + 1/8*a^6 - 1/8*a^4 - 5/16*a^2 - 3/8, 1/16*a^27 - 1/8*a^13 - 1/16*a^11 - 1/4*a^9 + 1/8*a^7 - 1/8*a^5 - 5/16*a^3 - 3/8*a, 1/16*a^28 - 1/8*a^14 - 1/16*a^12 - 1/4*a^10 + 1/8*a^8 - 1/8*a^6 - 5/16*a^4 - 3/8*a^2, 1/16*a^29 - 1/8*a^15 - 1/16*a^13 - 1/4*a^11 + 1/8*a^9 - 1/8*a^7 - 5/16*a^5 - 3/8*a^3, 1/32*a^30 - 1/32*a^26 - 1/32*a^24 - 1/16*a^22 + 1/16*a^16 + 3/32*a^14 + 1/16*a^12 - 3/32*a^10 - 1/32*a^8 + 3/32*a^6 + 1/16*a^4 - 1/32*a^2 - 1/32, 1/32*a^31 - 1/32*a^27 - 1/32*a^25 - 1/16*a^23 + 1/16*a^17 + 3/32*a^15 + 1/16*a^13 - 3/32*a^11 - 1/32*a^9 + 3/32*a^7 + 1/16*a^5 - 1/32*a^3 - 1/32*a, 1/12128*a^32 - 35/3032*a^31 - 147/12128*a^30 - 29/3032*a^29 + 249/12128*a^28 + 31/3032*a^27 - 19/3032*a^26 + 33/1516*a^25 + 317/12128*a^24 + 15/758*a^23 - 361/6064*a^22 - 33/1516*a^21 - 20/379*a^20 + 163/3032*a^19 + 307/6064*a^18 - 67/1516*a^17 - 619/12128*a^16 + 39/1516*a^15 + 417/12128*a^14 - 29/1516*a^13 - 403/12128*a^12 + 91/758*a^11 + 1129/6064*a^10 + 19/1516*a^9 - 1437/6064*a^8 - 377/3032*a^7 - 1823/12128*a^6 - 551/1516*a^5 - 2105/12128*a^4 + 65/758*a^3 + 143/379*a^2 + 875/3032*a - 5241/12128, 1/12128*a^33 - 39/12128*a^31 + 149/12128*a^30 - 73/12128*a^29 + 29/3032*a^28 - 75/6064*a^27 - 143/12128*a^26 + 135/12128*a^25 + 277/12128*a^24 - 237/6064*a^23 - 265/6064*a^22 + 75/3032*a^21 + 31/758*a^20 - 291/6064*a^19 + 33/758*a^18 + 141/12128*a^17 + 411/6064*a^16 - 1383/12128*a^15 + 919/12128*a^14 - 289/12128*a^13 - 573/6064*a^12 + 85/1516*a^11 - 1709/12128*a^10 - 515/3032*a^9 - 2507/12128*a^8 - 703/12128*a^7 + 3019/12128*a^6 - 4487/12128*a^5 + 2875/6064*a^4 - 1849/6064*a^3 + 2493/12128*a^2 - 1119/12128*a + 2657/12128, 1/1432658498741408708332691891198406841348257179217673030979430458464*a^34 - 12262935239310317472165616543745051603490761360000426623929567/358164624685352177083172972799601710337064294804418257744857614616*a^33 - 2911099187463437700745291096881969367672968880943380616203289/179082312342676088541586486399800855168532147402209128872428807308*a^32 + 2927519237738541987623311317513304623218177520499396735432037707/716329249370704354166345945599203420674128589608836515489715229232*a^31 - 4642244694325973365662881799722724045918931927914733466932080551/716329249370704354166345945599203420674128589608836515489715229232*a^30 - 3121426689501134006247113544711607254806591787045166649620293693/716329249370704354166345945599203420674128589608836515489715229232*a^29 - 25311559341246323060865760056261477617637879101266507239584736969/1432658498741408708332691891198406841348257179217673030979430458464*a^28 - 10863715642578861090313116687990948083652308693079187697770588059/358164624685352177083172972799601710337064294804418257744857614616*a^27 - 40261706465633503518183630525426199143057207914517368751878365925/1432658498741408708332691891198406841348257179217673030979430458464*a^26 - 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33035200005924258482000094595200490744571191985463127081185572477/1432658498741408708332691891198406841348257179217673030979430458464*a^18 - 20583104361609401137174702838245870243463276898800244043844605781/179082312342676088541586486399800855168532147402209128872428807308*a^17 + 37995466128498883921814970251064437593586149471019472941264483329/358164624685352177083172972799601710337064294804418257744857614616*a^16 - 37637591347373236496039698726612438683190140406744791711297323339/716329249370704354166345945599203420674128589608836515489715229232*a^15 + 27860727704906534498006273788085121961595755289635120791398704635/716329249370704354166345945599203420674128589608836515489715229232*a^14 + 63490299185206937338645358715051461133634390052438940342458899049/716329249370704354166345945599203420674128589608836515489715229232*a^13 - 10648975839773942366184302629880164344551916365937154112959664607/1432658498741408708332691891198406841348257179217673030979430458464*a^12 + 5835493726822230426577342490199601012709505453191354824268455557/358164624685352177083172972799601710337064294804418257744857614616*a^11 - 81451758215509339444096851864463745334625106855555957423375970045/716329249370704354166345945599203420674128589608836515489715229232*a^10 - 8757210864249478935558127658758293651033203629901257206191645049/89541156171338044270793243199900427584266073701104564436214403654*a^9 - 342004062377566398301873661853717332847453834829561931252866046337/1432658498741408708332691891198406841348257179217673030979430458464*a^8 - 153916763226340493124721337234664842486819389273480365082823407833/716329249370704354166345945599203420674128589608836515489715229232*a^7 + 31699668541364439584077333573621454777837875149300776330592560931/358164624685352177083172972799601710337064294804418257744857614616*a^6 - 264931431707274993523447142401506456170194247152401918287308418723/716329249370704354166345945599203420674128589608836515489715229232*a^5 + 428175185722214056967465667483192708444515085057904934773203638853/1432658498741408708332691891198406841348257179217673030979430458464*a^4 + 19665874847707245944346882455201477784593574549436015417267933077/44770578085669022135396621599950213792133036850552282218107201827*a^3 - 317233842331281797372620565689001627899827161593508744012277502111/1432658498741408708332691891198406841348257179217673030979430458464*a^2 - 130548135050014040200582055962633461070768771241779104088629254321/358164624685352177083172972799601710337064294804418257744857614616*a - 666161374971431442399525585378585107502097024141627262985369728179/1432658498741408708332691891198406841348257179217673030979430458464, 1/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^35 - 19838082629499399/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^34 + 25462104834077261049170252639467373902505990442925522212674907324277115451943/5283565909856084363519521680752963046297639004865229075588661925056751224451650006*a^33 + 662346721585283465829355558983027397902249738369117411822382406079070168308719/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^32 + 262606107634083518839717437162731684221531313683109932932767017756600361414387745/21134263639424337454078086723011852185190556019460916302354647700227004897806600024*a^31 + 92579440014564398534720133050099083347296426305436860384498121427072159957277861/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^30 - 142566733275057504827198383413840142976726928370285801110074395605741855079243695/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^29 + 1922494413297639891479001670760372693469025374379067646202131124786846938073763385/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^28 + 97318634970018151075172216016929040928979021876455591195485143594061299181643265/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^27 - 626621289766035856377245720343275665221051576305528854741727282345563870749091337/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^26 + 1776831936676075554452420461164065049256023103742098994026305368136925536094799827/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^25 - 1623960968809823601244038923946997801586865345077137315942512696851805755153598899/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^24 + 2532317207028482665264589163171276182726186701122738416298697375164308275792856429/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^23 - 256208467196528282933793467187765814736767099486860294555490467966737018498476397/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^22 + 2319677750588217959190892938564945884033641528825325354655127333143519197243458569/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^21 + 1649485471433265501558210463593054155684545299720041776136839662316692637949888867/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^20 - 1346931956513427927819767500364227367840624996994925208616532466341774239826751641/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^19 + 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8431142224749123469723822292470713827055844995747966274791759404177069909403671841/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^13 - 4705366459039878323654340644948231753288356385538115138445750897202284957797867197/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^12 + 2147380613131880195224271281597172076400217339527303222988101597343270609210768731/21134263639424337454078086723011852185190556019460916302354647700227004897806600024*a^11 + 852298008317053184986198573694188193958067268575839418115261013775989659843451429/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^10 + 3472604473799770347481165365818882640313141854240573136710171773048197120209094477/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^9 - 15460958774005409097276582440065530049962037422592675319776573613644645196967830567/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^8 + 9302825926550583307023169636128746623574077100348243472947868227107357561182033623/42268527278848674908156173446023704370381112038921832604709295400454009795613200048*a^7 + 1483532992689165395001321810928199789992639173436755266607117418316061778779598861/10567131819712168727039043361505926092595278009730458151177323850113502448903300012*a^6 - 28895716086790476899033665411166433607449791111923852338945867433160231890940305413/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^5 + 4594889183196993186702842037671937059282115579816545889760406802236005063292923515/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^4 - 39284042283602941530125722594924707239314277011346755176195560294862275363563761161/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^3 + 6168110829043916688862579795341147884759041049426988407848460326769518312232586193/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a^2 - 177633102272286173490100981864487251944703884308993454466017818662078741514422557/84537054557697349816312346892047408740762224077843665209418590800908019591226400096*a + 41250355846507397198257613005542937002671202775980701567482522131165541260706785557/84537054557697349816312346892047408740762224077843665209418590800908019591226400096

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

not computed

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:	\( -\frac{9184140999306974922247097181328673990348288143853597837307}{358164624685352177083172972799601710337064294804418257744857614616} a^{35} - \frac{47510439334793455254206855397865580954033502996129263893125}{1432658498741408708332691891198406841348257179217673030979430458464} a^{34} + \frac{768840675402678397575197476630144746807019429604458161156333}{716329249370704354166345945599203420674128589608836515489715229232} a^{33} + \frac{2458819936818752120245203501918467895832585239773346798832263}{1432658498741408708332691891198406841348257179217673030979430458464} a^{32} - \frac{7122592969890571040074306709305083078595040702880528115152585}{358164624685352177083172972799601710337064294804418257744857614616} a^{31} - \frac{13771457315911402162150735904959126118169836634402326842158261}{358164624685352177083172972799601710337064294804418257744857614616} a^{30} + \frac{150510815316246211097495029762368507082190831473625527817383137}{716329249370704354166345945599203420674128589608836515489715229232} a^{29} + \frac{351945105961642826283901367874759762493147355427818375300678601}{716329249370704354166345945599203420674128589608836515489715229232} a^{28} - \frac{483357595347797955192372159943834048380810030263063279062839009}{358164624685352177083172972799601710337064294804418257744857614616} a^{27} - \frac{2817960103240782314771750177721575417653110036256346877460403115}{716329249370704354166345945599203420674128589608836515489715229232} a^{26} + \frac{228930726162206110746101674091757272754809810621344728005618370}{44770578085669022135396621599950213792133036850552282218107201827} a^{25} + \frac{29194453854320240288606388420611348164359467535975457861203456701}{1432658498741408708332691891198406841348257179217673030979430458464} a^{24} - \frac{875551105034287682104040962050897391849879215511623284251373601}{89541156171338044270793243199900427584266073701104564436214403654} a^{23} - \frac{49951534465721728351328603295797018694566186037985064984439084171}{716329249370704354166345945599203420674128589608836515489715229232} a^{22} + \frac{248902842583982292521613251291040977400908633959636174391168263}{44770578085669022135396621599950213792133036850552282218107201827} a^{21} + \frac{126736692346680854956030003172628807101597328513980727992537971701}{716329249370704354166345945599203420674128589608836515489715229232} a^{20} + \frac{1522043440431059143360920546804308711419236250147154249840970477}{358164624685352177083172972799601710337064294804418257744857614616} a^{19} - \frac{611412624049512549854449809667171250468081148493014950943685972441}{1432658498741408708332691891198406841348257179217673030979430458464} a^{18} - \frac{171714675040716311403796241881284319880253109753665875369744192817}{716329249370704354166345945599203420674128589608836515489715229232} a^{17} + \frac{303687958149263753427904893243226294416964269260256363760891793097}{1432658498741408708332691891198406841348257179217673030979430458464} a^{16} - \frac{148423461741786656203160503822358348561890723786380159848996244379}{358164624685352177083172972799601710337064294804418257744857614616} a^{15} - \frac{579563349467052862210909331723704152405320110965417642809578386273}{716329249370704354166345945599203420674128589608836515489715229232} a^{14} + \frac{436148984949054136496238131000773229467135535774549997700120278819}{716329249370704354166345945599203420674128589608836515489715229232} a^{13} + \frac{341276001058789189524365934791140343823597873724428467530514532031}{716329249370704354166345945599203420674128589608836515489715229232} a^{12} - \frac{102628217041981101973821980875365678242287008566863115590220135961}{44770578085669022135396621599950213792133036850552282218107201827} a^{11} - \frac{1628571495841942386975268757135420581197230489780250376220941081473}{1432658498741408708332691891198406841348257179217673030979430458464} a^{10} + \frac{2556535350256257975349552517396749386015139542342444033546915617065}{716329249370704354166345945599203420674128589608836515489715229232} a^{9} + \frac{979534418624631172335253390877686954837879212590371239885122093847}{716329249370704354166345945599203420674128589608836515489715229232} a^{8} - \frac{1600272902095787881742361980208072876619077707703929058025355630613}{358164624685352177083172972799601710337064294804418257744857614616} a^{7} - \frac{99049463161543613270754798946611582366203545509124903030012649269}{89541156171338044270793243199900427584266073701104564436214403654} a^{6} + \frac{784461246538332010627819647012525008366030216711259846588527022345}{716329249370704354166345945599203420674128589608836515489715229232} a^{5} + \frac{942997609475975669860550463037234444662425880261567460997057286477}{358164624685352177083172972799601710337064294804418257744857614616} a^{4} - \frac{675505911511869284750732752124635158137278795893578237505466204369}{358164624685352177083172972799601710337064294804418257744857614616} a^{3} - \frac{630984043852746660236580170141695690867200083707270709027334484161}{179082312342676088541586486399800855168532147402209128872428807308} a^{2} + \frac{446815178726753994023770301284671690244872016915230130803773648283}{358164624685352177083172972799601710337064294804418257744857614616} a - \frac{864557963503414620632377737738297991474716817261661658332203991905}{1432658498741408708332691891198406841348257179217673030979430458464} \) -(9184140999306974922247097181328673990348288143853597837307)/(358164624685352177083172972799601710337064294804418257744857614616)a^(35) - (47510439334793455254206855397865580954033502996129263893125)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(34) + (768840675402678397575197476630144746807019429604458161156333)/(716329249370704354166345945599203420674128589608836515489715229232)a^(33) + (2458819936818752120245203501918467895832585239773346798832263)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(32) - (7122592969890571040074306709305083078595040702880528115152585)/(358164624685352177083172972799601710337064294804418257744857614616)a^(31) - (13771457315911402162150735904959126118169836634402326842158261)/(358164624685352177083172972799601710337064294804418257744857614616)a^(30) + (150510815316246211097495029762368507082190831473625527817383137)/(716329249370704354166345945599203420674128589608836515489715229232)a^(29) + (351945105961642826283901367874759762493147355427818375300678601)/(716329249370704354166345945599203420674128589608836515489715229232)a^(28) - (483357595347797955192372159943834048380810030263063279062839009)/(358164624685352177083172972799601710337064294804418257744857614616)a^(27) - (2817960103240782314771750177721575417653110036256346877460403115)/(716329249370704354166345945599203420674128589608836515489715229232)a^(26) + (228930726162206110746101674091757272754809810621344728005618370)/(44770578085669022135396621599950213792133036850552282218107201827)a^(25) + (29194453854320240288606388420611348164359467535975457861203456701)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(24) - (875551105034287682104040962050897391849879215511623284251373601)/(89541156171338044270793243199900427584266073701104564436214403654)a^(23) - (49951534465721728351328603295797018694566186037985064984439084171)/(716329249370704354166345945599203420674128589608836515489715229232)a^(22) + (248902842583982292521613251291040977400908633959636174391168263)/(44770578085669022135396621599950213792133036850552282218107201827)a^(21) + (126736692346680854956030003172628807101597328513980727992537971701)/(716329249370704354166345945599203420674128589608836515489715229232)a^(20) + (1522043440431059143360920546804308711419236250147154249840970477)/(358164624685352177083172972799601710337064294804418257744857614616)a^(19) - (611412624049512549854449809667171250468081148493014950943685972441)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(18) - (171714675040716311403796241881284319880253109753665875369744192817)/(716329249370704354166345945599203420674128589608836515489715229232)a^(17) + (303687958149263753427904893243226294416964269260256363760891793097)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(16) - (148423461741786656203160503822358348561890723786380159848996244379)/(358164624685352177083172972799601710337064294804418257744857614616)a^(15) - (579563349467052862210909331723704152405320110965417642809578386273)/(716329249370704354166345945599203420674128589608836515489715229232)a^(14) + (436148984949054136496238131000773229467135535774549997700120278819)/(716329249370704354166345945599203420674128589608836515489715229232)a^(13) + (341276001058789189524365934791140343823597873724428467530514532031)/(716329249370704354166345945599203420674128589608836515489715229232)a^(12) - (102628217041981101973821980875365678242287008566863115590220135961)/(44770578085669022135396621599950213792133036850552282218107201827)a^(11) - (1628571495841942386975268757135420581197230489780250376220941081473)/(1432658498741408708332691891198406841348257179217673030979430458464)a^(10) + (2556535350256257975349552517396749386015139542342444033546915617065)/(716329249370704354166345945599203420674128589608836515489715229232)a^(9) + (979534418624631172335253390877686954837879212590371239885122093847)/(716329249370704354166345945599203420674128589608836515489715229232)a^(8) - (1600272902095787881742361980208072876619077707703929058025355630613)/(358164624685352177083172972799601710337064294804418257744857614616)a^(7) - (99049463161543613270754798946611582366203545509124903030012649269)/(89541156171338044270793243199900427584266073701104564436214403654)a^(6) + (784461246538332010627819647012525008366030216711259846588527022345)/(716329249370704354166345945599203420674128589608836515489715229232)a^(5) + (942997609475975669860550463037234444662425880261567460997057286477)/(358164624685352177083172972799601710337064294804418257744857614616)a^(4) - (675505911511869284750732752124635158137278795893578237505466204369)/(358164624685352177083172972799601710337064294804418257744857614616)a^(3) - (630984043852746660236580170141695690867200083707270709027334484161)/(179082312342676088541586486399800855168532147402209128872428807308)a^(2) + (446815178726753994023770301284671690244872016915230130803773648283)/(358164624685352177083172972799601710337064294804418257744857614616)*a - (864557963503414620632377737738297991474716817261661658332203991905)/(1432658498741408708332691891198406841348257179217673030979430458464) (order $14$)	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:	not computed	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:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^36 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827)

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 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827, 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 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827);

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 - 42*x^34 - 12*x^33 + 798*x^32 + 444*x^31 - 8930*x^30 - 7224*x^29 + 64293*x^28 + 66808*x^27 - 308040*x^26 - 380412*x^25 + 1018573*x^24 + 1390296*x^23 - 2660256*x^22 - 3730568*x^21 + 6607119*x^20 + 9624456*x^19 - 7505542*x^18 - 2964660*x^17 + 28706040*x^16 + 6977356*x^15 - 34628742*x^14 + 28138392*x^13 + 70034750*x^12 - 59344320*x^11 - 97239042*x^10 + 110557824*x^9 + 103706163*x^8 - 146720820*x^7 + 54606518*x^6 - 96816432*x^5 + 280912497*x^4 - 164862264*x^3 + 36478572*x^2 - 7070100*x + 9128827);

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 $C_6^2$

Intermediate fields

$\Q(\sqrt{-2}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{14}) $, $\Q(\zeta_{9})^+$, $\Q(\zeta_{7})^+$, 3.3.3969.1, 3.3.3969.2, $\Q(\sqrt{-2}, \sqrt{-7})$, 6.0.3359232.1, 6.0.1229312.1, 6.0.8065516032.1, 6.0.8065516032.2, 6.0.2250423.1, 6.6.1152216576.1, $\Q(\zeta_{7})$, 6.6.8605184.1, 6.0.110270727.2, 6.6.56458612224.1, 6.0.110270727.1, 6.6.56458612224.2, 9.9.62523502209.1, 12.0.1327603038009163776.1, 12.0.74049191673856.1, 12.0.3187574894260002226176.2, 12.0.3187574894260002226176.3, 18.0.524682375772545974113841184768.4, 18.0.1340851596668237962730583.1, 18.18.179966054889983269121047526375424.1

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	R	R	${\href{/padicField/5.6.0.1}{6} }^{6}$	R	${\href{/padicField/11.3.0.1}{3} }^{12}$	${\href{/padicField/13.6.0.1}{6} }^{6}$	${\href{/padicField/17.6.0.1}{6} }^{6}$	${\href{/padicField/19.6.0.1}{6} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{6}$	${\href{/padicField/29.6.0.1}{6} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{6}$	${\href{/padicField/43.3.0.1}{3} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{6}$	${\href{/padicField/53.6.0.1}{6} }^{6}$	${\href{/padicField/59.6.0.1}{6} }^{6}$

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
$2$ 2	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.5	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
$3$ 3		Deg $18$	$3$	$6$	$24$
$3$ 3		Deg $18$	$3$	$6$	$24$
$7$ 7		Deg $36$	$6$	$6$	$30$

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