Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1)
gp: K = bnfinit(y^36 - 16*y^33 + 470*y^30 + 3624*y^27 + 44003*y^24 + 27532*y^21 + 50596*y^18 - 38116*y^15 + 32635*y^12 - 8692*y^9 + 2129*y^6 + 44*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1)
\( x^{36} - 16 x^{33} + 470 x^{30} + 3624 x^{27} + 44003 x^{24} + 27532 x^{21} + 50596 x^{18} - 38116 x^{15} + \cdots + 1 \)
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: |
\(42497246625894555234552515412349089862939543796539306640625\)
\(\medspace = 3^{54}\cdot 5^{18}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.52\) | 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: | $3^{3/2}5^{1/2}7^{2/3}\approx 42.517290220802025$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| 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$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(4,·)$, $\chi_{315}(86,·)$, $\chi_{315}(134,·)$, $\chi_{315}(11,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(281,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(289,·)$, $\chi_{315}(296,·)$, $\chi_{315}(169,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(176,·)$, $\chi_{315}(179,·)$, $\chi_{315}(184,·)$, $\chi_{315}(191,·)$, $\chi_{315}(64,·)$, $\chi_{315}(71,·)$, $\chi_{315}(74,·)$, $\chi_{315}(79,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(221,·)$, $\chi_{315}(226,·)$, $\chi_{315}(106,·)$, $\chi_{315}(109,·)$, $\chi_{315}(239,·)$, $\chi_{315}(116,·)$, $\chi_{315}(121,·)$, $\chi_{315}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{17}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{26}a^{24}+\frac{3}{26}a^{21}-\frac{1}{13}a^{18}+\frac{3}{26}a^{15}+\frac{6}{13}a^{12}+\frac{1}{13}a^{9}-\frac{11}{26}a^{6}-\frac{1}{13}a^{3}+\frac{1}{26}$, $\frac{1}{26}a^{25}+\frac{3}{26}a^{22}-\frac{1}{13}a^{19}+\frac{3}{26}a^{16}+\frac{6}{13}a^{13}+\frac{1}{13}a^{10}-\frac{11}{26}a^{7}-\frac{1}{13}a^{4}+\frac{1}{26}a$, $\frac{1}{26}a^{26}+\frac{3}{26}a^{23}-\frac{1}{13}a^{20}+\frac{3}{26}a^{17}+\frac{6}{13}a^{14}+\frac{1}{13}a^{11}-\frac{11}{26}a^{8}-\frac{1}{13}a^{5}+\frac{1}{26}a^{2}$, $\frac{1}{442}a^{27}-\frac{3}{13}a^{21}-\frac{95}{442}a^{18}+\frac{2}{13}a^{15}-\frac{1}{13}a^{12}+\frac{50}{221}a^{9}-\frac{1}{13}a^{6}-\frac{4}{13}a^{3}-\frac{81}{442}$, $\frac{1}{442}a^{28}-\frac{3}{13}a^{22}-\frac{95}{442}a^{19}+\frac{2}{13}a^{16}-\frac{1}{13}a^{13}+\frac{50}{221}a^{10}-\frac{1}{13}a^{7}-\frac{4}{13}a^{4}-\frac{81}{442}a$, $\frac{1}{442}a^{29}-\frac{3}{13}a^{23}-\frac{95}{442}a^{20}+\frac{2}{13}a^{17}-\frac{1}{13}a^{14}+\frac{50}{221}a^{11}-\frac{1}{13}a^{8}-\frac{4}{13}a^{5}-\frac{81}{442}a^{2}$, $\frac{1}{7360037698}a^{30}-\frac{1683834}{3680018849}a^{27}+\frac{6441575}{432943394}a^{24}-\frac{716190659}{3680018849}a^{21}+\frac{518929651}{3680018849}a^{18}+\frac{13028009}{33303338}a^{15}-\frac{1803481330}{3680018849}a^{12}-\frac{1661750423}{3680018849}a^{9}-\frac{35361279}{432943394}a^{6}-\frac{2282011469}{7360037698}a^{3}+\frac{241393413}{7360037698}$, $\frac{1}{7360037698}a^{31}-\frac{1683834}{3680018849}a^{28}+\frac{6441575}{432943394}a^{25}-\frac{716190659}{3680018849}a^{22}+\frac{518929651}{3680018849}a^{19}+\frac{13028009}{33303338}a^{16}-\frac{1803481330}{3680018849}a^{13}-\frac{1661750423}{3680018849}a^{10}-\frac{35361279}{432943394}a^{7}-\frac{2282011469}{7360037698}a^{4}+\frac{241393413}{7360037698}a$, $\frac{1}{7360037698}a^{32}-\frac{1683834}{3680018849}a^{29}+\frac{6441575}{432943394}a^{26}-\frac{716190659}{3680018849}a^{23}+\frac{518929651}{3680018849}a^{20}+\frac{13028009}{33303338}a^{17}-\frac{1803481330}{3680018849}a^{14}-\frac{1661750423}{3680018849}a^{11}-\frac{35361279}{432943394}a^{8}-\frac{2282011469}{7360037698}a^{5}+\frac{241393413}{7360037698}a^{2}$, $\frac{1}{12\!\cdots\!18}a^{33}-\frac{833324575}{12\!\cdots\!18}a^{30}+\frac{15\!\cdots\!43}{12\!\cdots\!18}a^{27}-\frac{51\!\cdots\!31}{12\!\cdots\!18}a^{24}-\frac{68\!\cdots\!39}{12\!\cdots\!18}a^{21}-\frac{76\!\cdots\!13}{61\!\cdots\!09}a^{18}+\frac{19\!\cdots\!39}{61\!\cdots\!09}a^{15}+\frac{44\!\cdots\!37}{12\!\cdots\!18}a^{12}-\frac{13\!\cdots\!75}{61\!\cdots\!09}a^{9}-\frac{17\!\cdots\!56}{61\!\cdots\!09}a^{6}-\frac{69\!\cdots\!60}{61\!\cdots\!09}a^{3}+\frac{17\!\cdots\!48}{47\!\cdots\!93}$, $\frac{1}{12\!\cdots\!18}a^{34}-\frac{833324575}{12\!\cdots\!18}a^{31}+\frac{15\!\cdots\!43}{12\!\cdots\!18}a^{28}-\frac{51\!\cdots\!31}{12\!\cdots\!18}a^{25}-\frac{68\!\cdots\!39}{12\!\cdots\!18}a^{22}-\frac{76\!\cdots\!13}{61\!\cdots\!09}a^{19}+\frac{19\!\cdots\!39}{61\!\cdots\!09}a^{16}+\frac{44\!\cdots\!37}{12\!\cdots\!18}a^{13}-\frac{13\!\cdots\!75}{61\!\cdots\!09}a^{10}-\frac{17\!\cdots\!56}{61\!\cdots\!09}a^{7}-\frac{69\!\cdots\!60}{61\!\cdots\!09}a^{4}+\frac{17\!\cdots\!48}{47\!\cdots\!93}a$, $\frac{1}{12\!\cdots\!18}a^{35}-\frac{833324575}{12\!\cdots\!18}a^{32}+\frac{15\!\cdots\!43}{12\!\cdots\!18}a^{29}-\frac{51\!\cdots\!31}{12\!\cdots\!18}a^{26}-\frac{68\!\cdots\!39}{12\!\cdots\!18}a^{23}-\frac{76\!\cdots\!13}{61\!\cdots\!09}a^{20}+\frac{19\!\cdots\!39}{61\!\cdots\!09}a^{17}+\frac{44\!\cdots\!37}{12\!\cdots\!18}a^{14}-\frac{13\!\cdots\!75}{61\!\cdots\!09}a^{11}-\frac{17\!\cdots\!56}{61\!\cdots\!09}a^{8}-\frac{69\!\cdots\!60}{61\!\cdots\!09}a^{5}+\frac{17\!\cdots\!48}{47\!\cdots\!93}a^{2}$
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
$C_{2}\times C_{74}$, which has order $148$ (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: |
\( -\frac{54750183643159558}{6143810019380517809} a^{34} + \frac{863351337584628236}{6143810019380517809} a^{31} - \frac{51059596888771568813}{12287620038761035618} a^{28} - \frac{204366554761057937494}{6143810019380517809} a^{25} - \frac{2454854100752371198598}{6143810019380517809} a^{22} - \frac{4125402857242667211415}{12287620038761035618} a^{19} - \frac{3101986311157413131518}{6143810019380517809} a^{16} + \frac{1463441623126818098554}{6143810019380517809} a^{13} - \frac{1285209272019518159592}{6143810019380517809} a^{10} + \frac{57740314069469615414}{6143810019380517809} a^{7} + \frac{1187719518152058308}{6143810019380517809} a^{4} - \frac{44600832139091345309}{12287620038761035618} a \)
(order $18$)
| 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: |
$\frac{89\!\cdots\!23}{61\!\cdots\!09}a^{35}-\frac{14\!\cdots\!28}{61\!\cdots\!09}a^{32}+\frac{83\!\cdots\!15}{12\!\cdots\!18}a^{29}+\frac{24\!\cdots\!98}{47\!\cdots\!93}a^{26}+\frac{78\!\cdots\!97}{12\!\cdots\!18}a^{23}+\frac{23\!\cdots\!30}{61\!\cdots\!09}a^{20}+\frac{90\!\cdots\!35}{12\!\cdots\!18}a^{17}-\frac{52\!\cdots\!09}{94\!\cdots\!86}a^{14}+\frac{60\!\cdots\!65}{12\!\cdots\!18}a^{11}-\frac{85\!\cdots\!38}{61\!\cdots\!09}a^{8}+\frac{22\!\cdots\!72}{61\!\cdots\!09}a^{5}-\frac{39\!\cdots\!64}{61\!\cdots\!09}a^{2}$, $\frac{17\!\cdots\!04}{47\!\cdots\!93}a^{33}-\frac{35\!\cdots\!61}{61\!\cdots\!09}a^{30}+\frac{10\!\cdots\!52}{61\!\cdots\!09}a^{27}+\frac{79\!\cdots\!81}{61\!\cdots\!09}a^{24}+\frac{97\!\cdots\!32}{61\!\cdots\!09}a^{21}+\frac{55\!\cdots\!99}{61\!\cdots\!09}a^{18}+\frac{10\!\cdots\!88}{61\!\cdots\!09}a^{15}-\frac{92\!\cdots\!81}{61\!\cdots\!09}a^{12}+\frac{76\!\cdots\!28}{61\!\cdots\!09}a^{9}-\frac{21\!\cdots\!71}{61\!\cdots\!09}a^{6}+\frac{48\!\cdots\!68}{61\!\cdots\!09}a^{3}+\frac{10\!\cdots\!26}{61\!\cdots\!09}$, $\frac{38\!\cdots\!48}{47\!\cdots\!93}a^{33}-\frac{81\!\cdots\!07}{61\!\cdots\!09}a^{30}+\frac{23\!\cdots\!84}{61\!\cdots\!09}a^{27}+\frac{17\!\cdots\!97}{61\!\cdots\!09}a^{24}+\frac{21\!\cdots\!04}{61\!\cdots\!09}a^{21}+\frac{90\!\cdots\!73}{61\!\cdots\!09}a^{18}+\frac{21\!\cdots\!88}{61\!\cdots\!09}a^{15}-\frac{25\!\cdots\!47}{61\!\cdots\!09}a^{12}+\frac{19\!\cdots\!76}{61\!\cdots\!09}a^{9}-\frac{59\!\cdots\!42}{61\!\cdots\!09}a^{6}+\frac{11\!\cdots\!36}{61\!\cdots\!09}a^{3}+\frac{24\!\cdots\!22}{61\!\cdots\!09}$, $\frac{21\!\cdots\!20}{47\!\cdots\!93}a^{33}-\frac{88\!\cdots\!95}{12\!\cdots\!18}a^{30}+\frac{13\!\cdots\!01}{61\!\cdots\!09}a^{27}+\frac{10\!\cdots\!79}{61\!\cdots\!09}a^{24}+\frac{24\!\cdots\!41}{12\!\cdots\!18}a^{21}+\frac{14\!\cdots\!55}{12\!\cdots\!18}a^{18}+\frac{10\!\cdots\!39}{47\!\cdots\!93}a^{15}-\frac{21\!\cdots\!37}{12\!\cdots\!18}a^{12}+\frac{93\!\cdots\!11}{61\!\cdots\!09}a^{9}-\frac{52\!\cdots\!53}{12\!\cdots\!18}a^{6}+\frac{65\!\cdots\!15}{61\!\cdots\!09}a^{3}-\frac{24\!\cdots\!67}{12\!\cdots\!18}$, $\frac{27\!\cdots\!35}{61\!\cdots\!09}a^{35}-\frac{42\!\cdots\!24}{61\!\cdots\!09}a^{32}+\frac{25\!\cdots\!81}{12\!\cdots\!18}a^{29}+\frac{10\!\cdots\!62}{61\!\cdots\!09}a^{26}+\frac{12\!\cdots\!91}{61\!\cdots\!09}a^{23}+\frac{21\!\cdots\!63}{12\!\cdots\!18}a^{20}+\frac{15\!\cdots\!75}{61\!\cdots\!09}a^{17}-\frac{72\!\cdots\!26}{61\!\cdots\!09}a^{14}+\frac{55\!\cdots\!37}{61\!\cdots\!09}a^{11}-\frac{28\!\cdots\!78}{61\!\cdots\!09}a^{8}-\frac{58\!\cdots\!54}{61\!\cdots\!09}a^{5}+\frac{19\!\cdots\!05}{12\!\cdots\!18}a^{2}$, $\frac{33\!\cdots\!67}{12\!\cdots\!18}a^{35}-\frac{41\!\cdots\!67}{61\!\cdots\!09}a^{34}-\frac{20\!\cdots\!37}{47\!\cdots\!93}a^{32}+\frac{66\!\cdots\!66}{61\!\cdots\!09}a^{31}+\frac{79\!\cdots\!74}{61\!\cdots\!09}a^{29}-\frac{38\!\cdots\!29}{12\!\cdots\!18}a^{28}+\frac{12\!\cdots\!01}{12\!\cdots\!18}a^{26}-\frac{11\!\cdots\!23}{47\!\cdots\!93}a^{25}+\frac{74\!\cdots\!91}{61\!\cdots\!09}a^{23}-\frac{36\!\cdots\!15}{12\!\cdots\!18}a^{22}+\frac{45\!\cdots\!31}{61\!\cdots\!09}a^{20}-\frac{21\!\cdots\!25}{12\!\cdots\!18}a^{19}+\frac{84\!\cdots\!36}{61\!\cdots\!09}a^{17}-\frac{41\!\cdots\!73}{12\!\cdots\!18}a^{16}-\frac{64\!\cdots\!21}{61\!\cdots\!09}a^{14}+\frac{12\!\cdots\!66}{47\!\cdots\!93}a^{13}+\frac{55\!\cdots\!31}{61\!\cdots\!09}a^{11}-\frac{27\!\cdots\!19}{12\!\cdots\!18}a^{10}-\frac{29\!\cdots\!17}{12\!\cdots\!18}a^{8}+\frac{78\!\cdots\!55}{12\!\cdots\!18}a^{7}+\frac{35\!\cdots\!37}{61\!\cdots\!09}a^{5}-\frac{19\!\cdots\!61}{12\!\cdots\!18}a^{4}+\frac{74\!\cdots\!98}{61\!\cdots\!09}a^{2}+\frac{36\!\cdots\!41}{12\!\cdots\!18}a$, $\frac{33\!\cdots\!67}{12\!\cdots\!18}a^{35}-\frac{20\!\cdots\!37}{47\!\cdots\!93}a^{32}+\frac{79\!\cdots\!74}{61\!\cdots\!09}a^{29}+\frac{12\!\cdots\!01}{12\!\cdots\!18}a^{26}+\frac{74\!\cdots\!91}{61\!\cdots\!09}a^{23}+\frac{45\!\cdots\!31}{61\!\cdots\!09}a^{20}+\frac{84\!\cdots\!36}{61\!\cdots\!09}a^{17}-\frac{64\!\cdots\!21}{61\!\cdots\!09}a^{14}+\frac{55\!\cdots\!31}{61\!\cdots\!09}a^{11}-\frac{29\!\cdots\!17}{12\!\cdots\!18}a^{8}+\frac{35\!\cdots\!37}{61\!\cdots\!09}a^{5}+\frac{74\!\cdots\!98}{61\!\cdots\!09}a^{2}-1$, $\frac{23\!\cdots\!69}{12\!\cdots\!18}a^{35}-\frac{73\!\cdots\!01}{61\!\cdots\!09}a^{34}+\frac{1933004242484}{94539061956707}a^{33}-\frac{18\!\cdots\!33}{61\!\cdots\!09}a^{32}+\frac{91\!\cdots\!90}{47\!\cdots\!93}a^{31}-\frac{30976861095595}{94539061956707}a^{30}+\frac{10\!\cdots\!69}{12\!\cdots\!18}a^{29}-\frac{34\!\cdots\!07}{61\!\cdots\!09}a^{28}+\frac{909282693942500}{94539061956707}a^{27}+\frac{87\!\cdots\!61}{12\!\cdots\!18}a^{26}-\frac{26\!\cdots\!71}{61\!\cdots\!09}a^{25}+\frac{69\!\cdots\!02}{94539061956707}a^{24}+\frac{10\!\cdots\!41}{12\!\cdots\!18}a^{23}-\frac{64\!\cdots\!83}{12\!\cdots\!18}a^{22}+\frac{84\!\cdots\!60}{94539061956707}a^{21}+\frac{33\!\cdots\!97}{47\!\cdots\!93}a^{20}-\frac{19\!\cdots\!92}{61\!\cdots\!09}a^{19}+\frac{51\!\cdots\!65}{94539061956707}a^{18}+\frac{13\!\cdots\!29}{12\!\cdots\!18}a^{17}-\frac{73\!\cdots\!61}{12\!\cdots\!18}a^{16}+\frac{96\!\cdots\!20}{94539061956707}a^{15}-\frac{62\!\cdots\!79}{12\!\cdots\!18}a^{14}+\frac{29\!\cdots\!40}{61\!\cdots\!09}a^{13}-\frac{76\!\cdots\!95}{94539061956707}a^{12}+\frac{56\!\cdots\!93}{12\!\cdots\!18}a^{11}-\frac{49\!\cdots\!11}{12\!\cdots\!18}a^{10}+\frac{64\!\cdots\!00}{94539061956707}a^{9}-\frac{19\!\cdots\!61}{94\!\cdots\!86}a^{8}+\frac{13\!\cdots\!09}{12\!\cdots\!18}a^{7}-\frac{18\!\cdots\!39}{94539061956707}a^{6}-\frac{25\!\cdots\!71}{61\!\cdots\!09}a^{5}-\frac{31\!\cdots\!53}{12\!\cdots\!18}a^{4}+\frac{41\!\cdots\!80}{94539061956707}a^{3}+\frac{81\!\cdots\!02}{61\!\cdots\!09}a^{2}-\frac{32\!\cdots\!82}{61\!\cdots\!09}a-\frac{8430169977877}{94539061956707}$, $\frac{15\!\cdots\!99}{61\!\cdots\!09}a^{35}+\frac{20\!\cdots\!52}{47\!\cdots\!93}a^{34}-\frac{25\!\cdots\!48}{61\!\cdots\!09}a^{32}-\frac{43\!\cdots\!68}{61\!\cdots\!09}a^{31}+\frac{14\!\cdots\!79}{12\!\cdots\!18}a^{29}+\frac{12\!\cdots\!36}{61\!\cdots\!09}a^{28}+\frac{56\!\cdots\!70}{61\!\cdots\!09}a^{26}+\frac{97\!\cdots\!78}{61\!\cdots\!09}a^{25}+\frac{13\!\cdots\!41}{12\!\cdots\!18}a^{23}+\frac{11\!\cdots\!36}{61\!\cdots\!09}a^{22}+\frac{41\!\cdots\!70}{61\!\cdots\!09}a^{20}+\frac{64\!\cdots\!72}{61\!\cdots\!09}a^{19}+\frac{15\!\cdots\!43}{12\!\cdots\!18}a^{17}+\frac{12\!\cdots\!76}{61\!\cdots\!09}a^{16}-\frac{12\!\cdots\!63}{12\!\cdots\!18}a^{14}-\frac{11\!\cdots\!28}{61\!\cdots\!09}a^{13}+\frac{10\!\cdots\!69}{12\!\cdots\!18}a^{11}+\frac{95\!\cdots\!04}{61\!\cdots\!09}a^{10}-\frac{14\!\cdots\!62}{61\!\cdots\!09}a^{8}-\frac{27\!\cdots\!13}{61\!\cdots\!09}a^{7}+\frac{35\!\cdots\!08}{61\!\cdots\!09}a^{5}+\frac{60\!\cdots\!04}{61\!\cdots\!09}a^{4}-\frac{68\!\cdots\!04}{61\!\cdots\!09}a^{2}+\frac{12\!\cdots\!48}{61\!\cdots\!09}a+1$, $\frac{15\!\cdots\!99}{61\!\cdots\!09}a^{35}-\frac{2355747673117}{283759093798606}a^{34}-\frac{1933004242484}{94539061956707}a^{33}-\frac{25\!\cdots\!48}{61\!\cdots\!09}a^{32}+\frac{18508000823902}{141879546899303}a^{31}+\frac{30976861095595}{94539061956707}a^{30}+\frac{14\!\cdots\!79}{12\!\cdots\!18}a^{29}-\frac{548154992075615}{141879546899303}a^{28}-\frac{909282693942500}{94539061956707}a^{27}+\frac{56\!\cdots\!70}{61\!\cdots\!09}a^{26}-\frac{44\!\cdots\!38}{141879546899303}a^{25}-\frac{69\!\cdots\!02}{94539061956707}a^{24}+\frac{13\!\cdots\!41}{12\!\cdots\!18}a^{23}-\frac{10\!\cdots\!37}{283759093798606}a^{22}-\frac{84\!\cdots\!60}{94539061956707}a^{21}+\frac{41\!\cdots\!70}{61\!\cdots\!09}a^{20}-\frac{47\!\cdots\!66}{141879546899303}a^{19}-\frac{51\!\cdots\!65}{94539061956707}a^{18}+\frac{15\!\cdots\!43}{12\!\cdots\!18}a^{17}-\frac{13\!\cdots\!57}{283759093798606}a^{16}-\frac{96\!\cdots\!20}{94539061956707}a^{15}-\frac{12\!\cdots\!63}{12\!\cdots\!18}a^{14}+\frac{31\!\cdots\!78}{141879546899303}a^{13}+\frac{76\!\cdots\!95}{94539061956707}a^{12}+\frac{10\!\cdots\!69}{12\!\cdots\!18}a^{11}-\frac{21\!\cdots\!49}{141879546899303}a^{10}-\frac{64\!\cdots\!00}{94539061956707}a^{9}-\frac{14\!\cdots\!62}{61\!\cdots\!09}a^{8}+\frac{12\!\cdots\!08}{141879546899303}a^{7}+\frac{18\!\cdots\!39}{94539061956707}a^{6}+\frac{35\!\cdots\!08}{61\!\cdots\!09}a^{5}+\frac{25643739303041}{141879546899303}a^{4}-\frac{41\!\cdots\!80}{94539061956707}a^{3}-\frac{68\!\cdots\!04}{61\!\cdots\!09}a^{2}-\frac{457671131688289}{283759093798606}a-\frac{86108891978830}{94539061956707}$, $\frac{66\!\cdots\!59}{61\!\cdots\!09}a^{35}-\frac{72\!\cdots\!63}{61\!\cdots\!09}a^{34}+\frac{74405397285436}{27\!\cdots\!29}a^{33}-\frac{10\!\cdots\!27}{61\!\cdots\!09}a^{32}+\frac{23\!\cdots\!69}{12\!\cdots\!18}a^{31}-\frac{15\!\cdots\!74}{36\!\cdots\!77}a^{30}+\frac{62\!\cdots\!79}{12\!\cdots\!18}a^{29}-\frac{68\!\cdots\!37}{12\!\cdots\!18}a^{28}+\frac{44\!\cdots\!80}{36\!\cdots\!77}a^{27}+\frac{49\!\cdots\!33}{12\!\cdots\!18}a^{26}-\frac{20\!\cdots\!41}{47\!\cdots\!93}a^{25}+\frac{36\!\cdots\!43}{36\!\cdots\!77}a^{24}+\frac{29\!\cdots\!65}{61\!\cdots\!09}a^{23}-\frac{31\!\cdots\!60}{61\!\cdots\!09}a^{22}+\frac{43\!\cdots\!60}{36\!\cdots\!77}a^{21}+\frac{18\!\cdots\!51}{47\!\cdots\!93}a^{20}-\frac{38\!\cdots\!15}{12\!\cdots\!18}a^{19}+\frac{40\!\cdots\!83}{36\!\cdots\!77}a^{18}+\frac{37\!\cdots\!29}{61\!\cdots\!09}a^{17}-\frac{73\!\cdots\!89}{12\!\cdots\!18}a^{16}+\frac{55\!\cdots\!08}{36\!\cdots\!77}a^{15}-\frac{35\!\cdots\!11}{12\!\cdots\!18}a^{14}+\frac{21\!\cdots\!31}{47\!\cdots\!93}a^{13}-\frac{26\!\cdots\!41}{36\!\cdots\!77}a^{12}+\frac{38\!\cdots\!99}{12\!\cdots\!18}a^{11}-\frac{49\!\cdots\!07}{12\!\cdots\!18}a^{10}+\frac{12\!\cdots\!12}{36\!\cdots\!77}a^{9}-\frac{10\!\cdots\!65}{94\!\cdots\!86}a^{8}+\frac{13\!\cdots\!69}{12\!\cdots\!18}a^{7}-\frac{10\!\cdots\!35}{36\!\cdots\!77}a^{6}-\frac{14\!\cdots\!48}{61\!\cdots\!09}a^{5}-\frac{17\!\cdots\!93}{61\!\cdots\!09}a^{4}-\frac{21\!\cdots\!16}{36\!\cdots\!77}a^{3}+\frac{13\!\cdots\!19}{12\!\cdots\!18}a^{2}+\frac{64\!\cdots\!95}{12\!\cdots\!18}a-\frac{65\!\cdots\!76}{36\!\cdots\!77}$, $\frac{33\!\cdots\!67}{12\!\cdots\!18}a^{35}-\frac{41\!\cdots\!67}{61\!\cdots\!09}a^{34}+\frac{17\!\cdots\!04}{47\!\cdots\!93}a^{33}-\frac{20\!\cdots\!37}{47\!\cdots\!93}a^{32}+\frac{66\!\cdots\!66}{61\!\cdots\!09}a^{31}-\frac{35\!\cdots\!61}{61\!\cdots\!09}a^{30}+\frac{79\!\cdots\!74}{61\!\cdots\!09}a^{29}-\frac{38\!\cdots\!29}{12\!\cdots\!18}a^{28}+\frac{10\!\cdots\!52}{61\!\cdots\!09}a^{27}+\frac{12\!\cdots\!01}{12\!\cdots\!18}a^{26}-\frac{11\!\cdots\!23}{47\!\cdots\!93}a^{25}+\frac{79\!\cdots\!81}{61\!\cdots\!09}a^{24}+\frac{74\!\cdots\!91}{61\!\cdots\!09}a^{23}-\frac{36\!\cdots\!15}{12\!\cdots\!18}a^{22}+\frac{97\!\cdots\!32}{61\!\cdots\!09}a^{21}+\frac{45\!\cdots\!31}{61\!\cdots\!09}a^{20}-\frac{21\!\cdots\!25}{12\!\cdots\!18}a^{19}+\frac{55\!\cdots\!99}{61\!\cdots\!09}a^{18}+\frac{84\!\cdots\!36}{61\!\cdots\!09}a^{17}-\frac{41\!\cdots\!73}{12\!\cdots\!18}a^{16}+\frac{10\!\cdots\!88}{61\!\cdots\!09}a^{15}-\frac{64\!\cdots\!21}{61\!\cdots\!09}a^{14}+\frac{12\!\cdots\!66}{47\!\cdots\!93}a^{13}-\frac{92\!\cdots\!81}{61\!\cdots\!09}a^{12}+\frac{55\!\cdots\!31}{61\!\cdots\!09}a^{11}-\frac{27\!\cdots\!19}{12\!\cdots\!18}a^{10}+\frac{76\!\cdots\!28}{61\!\cdots\!09}a^{9}-\frac{29\!\cdots\!17}{12\!\cdots\!18}a^{8}+\frac{78\!\cdots\!55}{12\!\cdots\!18}a^{7}-\frac{21\!\cdots\!71}{61\!\cdots\!09}a^{6}+\frac{35\!\cdots\!37}{61\!\cdots\!09}a^{5}-\frac{19\!\cdots\!61}{12\!\cdots\!18}a^{4}+\frac{48\!\cdots\!68}{61\!\cdots\!09}a^{3}+\frac{74\!\cdots\!98}{61\!\cdots\!09}a^{2}+\frac{36\!\cdots\!41}{12\!\cdots\!18}a+\frac{10\!\cdots\!26}{61\!\cdots\!09}$, $\frac{23\!\cdots\!69}{12\!\cdots\!18}a^{35}-\frac{46\!\cdots\!25}{61\!\cdots\!09}a^{34}-\frac{92989721159168}{27\!\cdots\!29}a^{33}-\frac{18\!\cdots\!33}{61\!\cdots\!09}a^{32}+\frac{74\!\cdots\!02}{61\!\cdots\!09}a^{31}+\frac{18\!\cdots\!30}{36\!\cdots\!77}a^{30}+\frac{10\!\cdots\!69}{12\!\cdots\!18}a^{29}-\frac{16\!\cdots\!67}{47\!\cdots\!93}a^{28}-\frac{56\!\cdots\!72}{36\!\cdots\!77}a^{27}+\frac{87\!\cdots\!61}{12\!\cdots\!18}a^{26}-\frac{16\!\cdots\!93}{61\!\cdots\!09}a^{25}-\frac{45\!\cdots\!53}{36\!\cdots\!77}a^{24}+\frac{10\!\cdots\!41}{12\!\cdots\!18}a^{23}-\frac{41\!\cdots\!11}{12\!\cdots\!18}a^{22}-\frac{54\!\cdots\!32}{36\!\cdots\!77}a^{21}+\frac{33\!\cdots\!97}{47\!\cdots\!93}a^{20}-\frac{13\!\cdots\!20}{61\!\cdots\!09}a^{19}-\frac{50\!\cdots\!76}{36\!\cdots\!77}a^{18}+\frac{13\!\cdots\!29}{12\!\cdots\!18}a^{17}-\frac{47\!\cdots\!09}{12\!\cdots\!18}a^{16}-\frac{68\!\cdots\!84}{36\!\cdots\!77}a^{15}-\frac{62\!\cdots\!79}{12\!\cdots\!18}a^{14}+\frac{17\!\cdots\!12}{61\!\cdots\!09}a^{13}+\frac{32\!\cdots\!15}{36\!\cdots\!77}a^{12}+\frac{56\!\cdots\!93}{12\!\cdots\!18}a^{11}-\frac{30\!\cdots\!03}{12\!\cdots\!18}a^{10}-\frac{15\!\cdots\!36}{36\!\cdots\!77}a^{9}-\frac{19\!\cdots\!61}{94\!\cdots\!86}a^{8}+\frac{79\!\cdots\!83}{12\!\cdots\!18}a^{7}+\frac{12\!\cdots\!21}{36\!\cdots\!77}a^{6}-\frac{25\!\cdots\!71}{61\!\cdots\!09}a^{5}-\frac{19\!\cdots\!45}{12\!\cdots\!18}a^{4}+\frac{26\!\cdots\!56}{36\!\cdots\!77}a^{3}+\frac{81\!\cdots\!02}{61\!\cdots\!09}a^{2}-\frac{20\!\cdots\!34}{61\!\cdots\!09}a+\frac{16\!\cdots\!43}{36\!\cdots\!77}$, $\frac{84\!\cdots\!79}{94\!\cdots\!86}a^{35}+\frac{77\!\cdots\!85}{12\!\cdots\!18}a^{34}-\frac{36\!\cdots\!01}{94\!\cdots\!86}a^{33}-\frac{86\!\cdots\!96}{61\!\cdots\!09}a^{32}-\frac{96\!\cdots\!23}{94\!\cdots\!86}a^{31}+\frac{29\!\cdots\!40}{47\!\cdots\!93}a^{30}+\frac{25\!\cdots\!64}{61\!\cdots\!09}a^{29}+\frac{18\!\cdots\!67}{61\!\cdots\!09}a^{28}-\frac{11\!\cdots\!51}{61\!\cdots\!09}a^{27}+\frac{20\!\cdots\!25}{61\!\cdots\!09}a^{26}+\frac{27\!\cdots\!49}{12\!\cdots\!18}a^{25}-\frac{84\!\cdots\!95}{61\!\cdots\!09}a^{24}+\frac{49\!\cdots\!99}{12\!\cdots\!18}a^{23}+\frac{33\!\cdots\!29}{12\!\cdots\!18}a^{22}-\frac{20\!\cdots\!01}{12\!\cdots\!18}a^{21}+\frac{20\!\cdots\!04}{61\!\cdots\!09}a^{20}+\frac{77\!\cdots\!90}{61\!\cdots\!09}a^{19}-\frac{50\!\cdots\!85}{61\!\cdots\!09}a^{18}+\frac{62\!\cdots\!47}{12\!\cdots\!18}a^{17}+\frac{17\!\cdots\!89}{61\!\cdots\!09}a^{16}-\frac{21\!\cdots\!53}{12\!\cdots\!18}a^{15}-\frac{14\!\cdots\!59}{61\!\cdots\!09}a^{14}-\frac{18\!\cdots\!94}{61\!\cdots\!09}a^{13}+\frac{83\!\cdots\!07}{47\!\cdots\!93}a^{12}+\frac{12\!\cdots\!73}{61\!\cdots\!09}a^{11}+\frac{14\!\cdots\!84}{61\!\cdots\!09}a^{10}-\frac{88\!\cdots\!61}{61\!\cdots\!09}a^{9}-\frac{58\!\cdots\!31}{61\!\cdots\!09}a^{8}-\frac{98\!\cdots\!91}{12\!\cdots\!18}a^{7}+\frac{27\!\cdots\!84}{61\!\cdots\!09}a^{6}-\frac{11\!\cdots\!95}{61\!\cdots\!09}a^{5}+\frac{17\!\cdots\!69}{94\!\cdots\!86}a^{4}-\frac{60\!\cdots\!55}{61\!\cdots\!09}a^{3}+\frac{50\!\cdots\!17}{12\!\cdots\!18}a^{2}-\frac{52\!\cdots\!57}{61\!\cdots\!09}a+\frac{16\!\cdots\!55}{12\!\cdots\!18}$, $\frac{33\!\cdots\!67}{12\!\cdots\!18}a^{35}+\frac{46\!\cdots\!25}{61\!\cdots\!09}a^{34}-\frac{18\!\cdots\!88}{61\!\cdots\!09}a^{33}-\frac{20\!\cdots\!37}{47\!\cdots\!93}a^{32}-\frac{74\!\cdots\!02}{61\!\cdots\!09}a^{31}+\frac{60\!\cdots\!07}{12\!\cdots\!18}a^{30}+\frac{79\!\cdots\!74}{61\!\cdots\!09}a^{29}+\frac{16\!\cdots\!67}{47\!\cdots\!93}a^{28}-\frac{89\!\cdots\!04}{61\!\cdots\!09}a^{27}+\frac{12\!\cdots\!01}{12\!\cdots\!18}a^{26}+\frac{16\!\cdots\!93}{61\!\cdots\!09}a^{25}-\frac{52\!\cdots\!20}{47\!\cdots\!93}a^{24}+\frac{74\!\cdots\!91}{61\!\cdots\!09}a^{23}+\frac{41\!\cdots\!11}{12\!\cdots\!18}a^{22}-\frac{16\!\cdots\!65}{12\!\cdots\!18}a^{21}+\frac{45\!\cdots\!31}{61\!\cdots\!09}a^{20}+\frac{13\!\cdots\!20}{61\!\cdots\!09}a^{19}-\frac{51\!\cdots\!40}{61\!\cdots\!09}a^{18}+\frac{84\!\cdots\!36}{61\!\cdots\!09}a^{17}+\frac{47\!\cdots\!09}{12\!\cdots\!18}a^{16}-\frac{98\!\cdots\!68}{61\!\cdots\!09}a^{15}-\frac{64\!\cdots\!21}{61\!\cdots\!09}a^{14}-\frac{17\!\cdots\!12}{61\!\cdots\!09}a^{13}+\frac{54\!\cdots\!60}{47\!\cdots\!93}a^{12}+\frac{55\!\cdots\!31}{61\!\cdots\!09}a^{11}+\frac{30\!\cdots\!03}{12\!\cdots\!18}a^{10}-\frac{65\!\cdots\!44}{61\!\cdots\!09}a^{9}-\frac{29\!\cdots\!17}{12\!\cdots\!18}a^{8}-\frac{79\!\cdots\!83}{12\!\cdots\!18}a^{7}+\frac{18\!\cdots\!64}{61\!\cdots\!09}a^{6}+\frac{35\!\cdots\!37}{61\!\cdots\!09}a^{5}+\frac{19\!\cdots\!45}{12\!\cdots\!18}a^{4}-\frac{10\!\cdots\!05}{12\!\cdots\!18}a^{3}+\frac{74\!\cdots\!98}{61\!\cdots\!09}a^{2}+\frac{20\!\cdots\!34}{61\!\cdots\!09}a+\frac{86\!\cdots\!28}{61\!\cdots\!09}$, $\frac{109847055908923}{283759093798606}a^{35}-\frac{22\!\cdots\!18}{47\!\cdots\!93}a^{34}-\frac{64\!\cdots\!51}{12\!\cdots\!18}a^{33}-\frac{17\!\cdots\!51}{283759093798606}a^{32}+\frac{92\!\cdots\!63}{12\!\cdots\!18}a^{31}+\frac{51\!\cdots\!47}{61\!\cdots\!09}a^{30}+\frac{25\!\cdots\!03}{141879546899303}a^{29}-\frac{27\!\cdots\!49}{12\!\cdots\!18}a^{28}-\frac{15\!\cdots\!52}{61\!\cdots\!09}a^{27}+\frac{19\!\cdots\!91}{141879546899303}a^{26}-\frac{20\!\cdots\!47}{12\!\cdots\!18}a^{25}-\frac{23\!\cdots\!63}{12\!\cdots\!18}a^{24}+\frac{48\!\cdots\!45}{283759093798606}a^{23}-\frac{12\!\cdots\!63}{61\!\cdots\!09}a^{22}-\frac{28\!\cdots\!91}{12\!\cdots\!18}a^{21}+\frac{31\!\cdots\!73}{283759093798606}a^{20}-\frac{11\!\cdots\!41}{94\!\cdots\!86}a^{19}-\frac{16\!\cdots\!93}{12\!\cdots\!18}a^{18}+\frac{28\!\cdots\!20}{141879546899303}a^{17}-\frac{29\!\cdots\!21}{12\!\cdots\!18}a^{16}-\frac{31\!\cdots\!31}{12\!\cdots\!18}a^{15}-\frac{40\!\cdots\!11}{283759093798606}a^{14}+\frac{11\!\cdots\!32}{61\!\cdots\!09}a^{13}+\frac{26\!\cdots\!93}{12\!\cdots\!18}a^{12}+\frac{35\!\cdots\!49}{283759093798606}a^{11}-\frac{10\!\cdots\!33}{61\!\cdots\!09}a^{10}-\frac{20\!\cdots\!01}{12\!\cdots\!18}a^{9}-\frac{45\!\cdots\!09}{141879546899303}a^{8}+\frac{65\!\cdots\!09}{12\!\cdots\!18}a^{7}+\frac{56\!\cdots\!93}{12\!\cdots\!18}a^{6}+\frac{23\!\cdots\!51}{283759093798606}a^{5}-\frac{19\!\cdots\!55}{12\!\cdots\!18}a^{4}-\frac{60\!\cdots\!67}{61\!\cdots\!09}a^{3}+\frac{23\!\cdots\!65}{141879546899303}a^{2}+\frac{23\!\cdots\!31}{47\!\cdots\!93}a-\frac{13\!\cdots\!63}{12\!\cdots\!18}$, $\frac{37\!\cdots\!21}{12\!\cdots\!18}a^{35}-\frac{90\!\cdots\!65}{36\!\cdots\!77}a^{34}+\frac{19\!\cdots\!97}{61\!\cdots\!09}a^{33}-\frac{30\!\cdots\!69}{61\!\cdots\!09}a^{32}+\frac{14\!\cdots\!88}{36\!\cdots\!77}a^{31}-\frac{30\!\cdots\!61}{61\!\cdots\!09}a^{30}+\frac{89\!\cdots\!58}{61\!\cdots\!09}a^{29}-\frac{42\!\cdots\!46}{36\!\cdots\!77}a^{28}+\frac{18\!\cdots\!57}{12\!\cdots\!18}a^{27}+\frac{13\!\cdots\!37}{12\!\cdots\!18}a^{26}-\frac{32\!\cdots\!12}{36\!\cdots\!77}a^{25}+\frac{53\!\cdots\!50}{47\!\cdots\!93}a^{24}+\frac{83\!\cdots\!84}{61\!\cdots\!09}a^{23}-\frac{39\!\cdots\!91}{36\!\cdots\!77}a^{22}+\frac{16\!\cdots\!07}{12\!\cdots\!18}a^{21}+\frac{54\!\cdots\!55}{61\!\cdots\!09}a^{20}-\frac{25\!\cdots\!72}{36\!\cdots\!77}a^{19}+\frac{53\!\cdots\!60}{61\!\cdots\!09}a^{18}+\frac{97\!\cdots\!93}{61\!\cdots\!09}a^{17}-\frac{46\!\cdots\!99}{36\!\cdots\!77}a^{16}+\frac{20\!\cdots\!69}{12\!\cdots\!18}a^{15}-\frac{70\!\cdots\!93}{61\!\cdots\!09}a^{14}+\frac{33\!\cdots\!48}{36\!\cdots\!77}a^{13}-\frac{10\!\cdots\!55}{94\!\cdots\!86}a^{12}+\frac{59\!\cdots\!76}{61\!\cdots\!09}a^{11}-\frac{28\!\cdots\!19}{36\!\cdots\!77}a^{10}+\frac{13\!\cdots\!27}{12\!\cdots\!18}a^{9}-\frac{30\!\cdots\!45}{12\!\cdots\!18}a^{8}+\frac{75\!\cdots\!32}{36\!\cdots\!77}a^{7}-\frac{19\!\cdots\!76}{61\!\cdots\!09}a^{6}+\frac{35\!\cdots\!91}{61\!\cdots\!09}a^{5}-\frac{19\!\cdots\!29}{36\!\cdots\!77}a^{4}+\frac{52\!\cdots\!71}{61\!\cdots\!09}a^{3}+\frac{11\!\cdots\!42}{47\!\cdots\!93}a^{2}-\frac{39\!\cdots\!88}{36\!\cdots\!77}a+\frac{52\!\cdots\!87}{61\!\cdots\!09}$
| 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: | \( 13624539961495.691 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 13624539961495.691 \cdot 148}{18\cdot\sqrt{42497246625894555234552515412349089862939543796539306640625}}\cr\approx \mathstrut & 0.126580932629111 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1)
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 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1, 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 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1);
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 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1);
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
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
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 | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{12}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{12}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $36$ | $6$ | $6$ | $54$ | |||
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |