Normalized defining polynomial
\( x^{18} - 6 x^{17} - 375 x^{16} + 2368 x^{15} + 55431 x^{14} - 341010 x^{13} - 4247625 x^{12} + \cdots + 6790409053632 \)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(242459919475028479855663400956723947300369356292096\)
\(\medspace = 2^{33}\cdot 3^{30}\cdot 7^{12}\cdot 17^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(629.72\) | 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: | $2^{11/6}3^{11/6}7^{2/3}17^{5/6}\approx 1036.056453367024$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{34}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{24}a^{9}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{5}{24}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{96}a^{10}-\frac{1}{48}a^{9}+\frac{3}{32}a^{8}+\frac{1}{8}a^{7}-\frac{3}{32}a^{6}+\frac{3}{16}a^{5}+\frac{47}{96}a^{4}-\frac{7}{24}a^{3}-\frac{1}{4}a^{2}-\frac{3}{8}$, $\frac{1}{96}a^{11}+\frac{1}{96}a^{9}+\frac{1}{16}a^{8}+\frac{1}{32}a^{7}-\frac{1}{96}a^{5}+\frac{7}{16}a^{4}-\frac{1}{24}a^{3}+\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{91392}a^{12}+\frac{13}{15232}a^{11}-\frac{19}{13056}a^{10}+\frac{65}{3808}a^{9}+\frac{2803}{30464}a^{8}-\frac{2809}{15232}a^{7}+\frac{10685}{91392}a^{6}-\frac{17}{224}a^{5}+\frac{6179}{45696}a^{4}+\frac{1671}{3808}a^{3}-\frac{1759}{7616}a^{2}-\frac{183}{476}a+\frac{677}{3808}$, $\frac{1}{274176}a^{13}+\frac{1}{274176}a^{12}-\frac{197}{39168}a^{11}+\frac{443}{91392}a^{10}+\frac{1795}{91392}a^{9}+\frac{367}{91392}a^{8}+\frac{54659}{274176}a^{7}+\frac{41399}{274176}a^{6}-\frac{6197}{137088}a^{5}-\frac{17837}{45696}a^{4}+\frac{11275}{22848}a^{3}-\frac{605}{1344}a^{2}-\frac{437}{3808}a-\frac{57}{544}$, $\frac{1}{469389312}a^{14}+\frac{409}{234694656}a^{13}-\frac{331}{469389312}a^{12}-\frac{37115}{7334208}a^{11}-\frac{238537}{52154368}a^{10}-\frac{386985}{26077184}a^{9}-\frac{34125673}{469389312}a^{8}+\frac{110545}{29336832}a^{7}+\frac{649889}{14668416}a^{6}-\frac{3495379}{29336832}a^{5}+\frac{508159}{6519296}a^{4}+\frac{1019369}{2444736}a^{3}-\frac{252283}{1396992}a^{2}-\frac{429557}{1629824}a+\frac{164753}{465664}$, $\frac{1}{1408167936}a^{15}-\frac{1}{22351872}a^{13}+\frac{205}{100583424}a^{12}+\frac{199917}{52154368}a^{11}+\frac{14401}{7334208}a^{10}+\frac{2105099}{1408167936}a^{9}+\frac{639565}{33527808}a^{8}+\frac{4519517}{29336832}a^{7}+\frac{3951331}{22002624}a^{6}-\frac{12016555}{58673664}a^{5}+\frac{4818095}{9778944}a^{4}+\frac{12402487}{29336832}a^{3}+\frac{216719}{1222368}a^{2}-\frac{77599}{3259648}a-\frac{85343}{232832}$, $\frac{1}{16875484545024}a^{16}-\frac{1081}{8437742272512}a^{15}-\frac{1385}{1875053838336}a^{14}+\frac{999863}{1054717784064}a^{13}+\frac{65556473}{16875484545024}a^{12}-\frac{6695885141}{2812580757504}a^{11}-\frac{7200092485}{2410783506432}a^{10}-\frac{8774392387}{1054717784064}a^{9}+\frac{205155725309}{2812580757504}a^{8}-\frac{9388334705}{1054717784064}a^{7}-\frac{524874552889}{2109435568128}a^{6}-\frac{304473887}{2092694016}a^{5}+\frac{3136147057}{21973287168}a^{4}+\frac{17454583775}{175786297344}a^{3}-\frac{5795913525}{39063621632}a^{2}-\frac{606647203}{2441476352}a-\frac{9529092883}{19531810816}$, $\frac{1}{76\!\cdots\!12}a^{17}-\frac{58\!\cdots\!65}{38\!\cdots\!56}a^{16}-\frac{11\!\cdots\!97}{76\!\cdots\!12}a^{15}-\frac{11\!\cdots\!71}{11\!\cdots\!08}a^{14}+\frac{18\!\cdots\!15}{10\!\cdots\!16}a^{13}+\frac{20\!\cdots\!29}{38\!\cdots\!56}a^{12}-\frac{23\!\cdots\!51}{76\!\cdots\!12}a^{11}-\frac{99\!\cdots\!41}{11\!\cdots\!08}a^{10}-\frac{50\!\cdots\!61}{38\!\cdots\!56}a^{9}-\frac{40\!\cdots\!07}{68\!\cdots\!76}a^{8}+\frac{22\!\cdots\!43}{95\!\cdots\!64}a^{7}-\frac{53\!\cdots\!55}{29\!\cdots\!52}a^{6}-\frac{14\!\cdots\!45}{99\!\cdots\!84}a^{5}+\frac{73\!\cdots\!83}{79\!\cdots\!72}a^{4}-\frac{49\!\cdots\!95}{22\!\cdots\!92}a^{3}-\frac{74\!\cdots\!19}{23\!\cdots\!52}a^{2}+\frac{42\!\cdots\!29}{88\!\cdots\!08}a-\frac{90\!\cdots\!69}{27\!\cdots\!44}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}\times C_{6}$, which has order $18$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{46\!\cdots\!45}{28\!\cdots\!44}a^{17}-\frac{11\!\cdots\!33}{14\!\cdots\!72}a^{16}-\frac{10\!\cdots\!33}{16\!\cdots\!32}a^{15}+\frac{11\!\cdots\!49}{35\!\cdots\!68}a^{14}+\frac{15\!\cdots\!69}{16\!\cdots\!32}a^{13}-\frac{67\!\cdots\!23}{14\!\cdots\!72}a^{12}-\frac{21\!\cdots\!87}{28\!\cdots\!44}a^{11}+\frac{82\!\cdots\!11}{35\!\cdots\!68}a^{10}+\frac{49\!\cdots\!07}{14\!\cdots\!72}a^{9}+\frac{45\!\cdots\!77}{17\!\cdots\!84}a^{8}-\frac{37\!\cdots\!25}{35\!\cdots\!68}a^{7}-\frac{20\!\cdots\!65}{44\!\cdots\!96}a^{6}+\frac{22\!\cdots\!73}{14\!\cdots\!32}a^{5}+\frac{26\!\cdots\!43}{29\!\cdots\!64}a^{4}-\frac{19\!\cdots\!13}{59\!\cdots\!28}a^{3}-\frac{34\!\cdots\!81}{24\!\cdots\!72}a^{2}+\frac{11\!\cdots\!83}{98\!\cdots\!88}a-\frac{51\!\cdots\!51}{61\!\cdots\!68}$, $\frac{59\!\cdots\!61}{59\!\cdots\!96}a^{17}-\frac{64\!\cdots\!51}{11\!\cdots\!92}a^{16}-\frac{11\!\cdots\!57}{29\!\cdots\!48}a^{15}+\frac{24\!\cdots\!31}{11\!\cdots\!92}a^{14}+\frac{69\!\cdots\!33}{12\!\cdots\!16}a^{13}-\frac{72\!\cdots\!55}{25\!\cdots\!32}a^{12}-\frac{27\!\cdots\!09}{62\!\cdots\!08}a^{11}+\frac{10\!\cdots\!27}{83\!\cdots\!44}a^{10}+\frac{41\!\cdots\!51}{20\!\cdots\!36}a^{9}+\frac{10\!\cdots\!01}{41\!\cdots\!72}a^{8}-\frac{48\!\cdots\!23}{78\!\cdots\!76}a^{7}-\frac{91\!\cdots\!17}{31\!\cdots\!04}a^{6}+\frac{30\!\cdots\!45}{39\!\cdots\!88}a^{5}+\frac{75\!\cdots\!33}{13\!\cdots\!96}a^{4}+\frac{81\!\cdots\!87}{18\!\cdots\!28}a^{3}-\frac{89\!\cdots\!97}{52\!\cdots\!84}a^{2}-\frac{11\!\cdots\!97}{13\!\cdots\!96}a+\frac{33\!\cdots\!05}{26\!\cdots\!92}$, $\frac{51\!\cdots\!43}{38\!\cdots\!56}a^{17}-\frac{11\!\cdots\!85}{11\!\cdots\!08}a^{16}-\frac{19\!\cdots\!91}{38\!\cdots\!56}a^{15}+\frac{11\!\cdots\!15}{27\!\cdots\!04}a^{14}+\frac{28\!\cdots\!95}{38\!\cdots\!56}a^{13}-\frac{60\!\cdots\!61}{95\!\cdots\!64}a^{12}-\frac{20\!\cdots\!61}{38\!\cdots\!56}a^{11}+\frac{81\!\cdots\!03}{19\!\cdots\!28}a^{10}+\frac{38\!\cdots\!53}{19\!\cdots\!28}a^{9}-\frac{10\!\cdots\!47}{95\!\cdots\!64}a^{8}-\frac{34\!\cdots\!33}{68\!\cdots\!76}a^{7}-\frac{34\!\cdots\!83}{23\!\cdots\!16}a^{6}+\frac{11\!\cdots\!89}{19\!\cdots\!68}a^{5}+\frac{10\!\cdots\!37}{39\!\cdots\!36}a^{4}+\frac{15\!\cdots\!41}{79\!\cdots\!72}a^{3}-\frac{15\!\cdots\!23}{44\!\cdots\!04}a^{2}+\frac{14\!\cdots\!75}{44\!\cdots\!04}a-\frac{54\!\cdots\!17}{22\!\cdots\!52}$, $\frac{40\!\cdots\!73}{35\!\cdots\!32}a^{17}-\frac{58\!\cdots\!17}{95\!\cdots\!64}a^{16}-\frac{15\!\cdots\!99}{35\!\cdots\!32}a^{15}+\frac{52\!\cdots\!47}{21\!\cdots\!04}a^{14}+\frac{64\!\cdots\!39}{95\!\cdots\!64}a^{13}-\frac{37\!\cdots\!41}{10\!\cdots\!96}a^{12}-\frac{17\!\cdots\!33}{31\!\cdots\!88}a^{11}+\frac{17\!\cdots\!05}{95\!\cdots\!64}a^{10}+\frac{41\!\cdots\!51}{15\!\cdots\!44}a^{9}+\frac{18\!\cdots\!01}{15\!\cdots\!44}a^{8}-\frac{14\!\cdots\!63}{17\!\cdots\!44}a^{7}-\frac{12\!\cdots\!07}{39\!\cdots\!36}a^{6}+\frac{74\!\cdots\!99}{55\!\cdots\!88}a^{5}+\frac{75\!\cdots\!55}{99\!\cdots\!84}a^{4}-\frac{33\!\cdots\!47}{66\!\cdots\!56}a^{3}-\frac{79\!\cdots\!17}{22\!\cdots\!52}a^{2}+\frac{54\!\cdots\!39}{11\!\cdots\!76}a-\frac{19\!\cdots\!35}{11\!\cdots\!76}$, $\frac{12\!\cdots\!35}{38\!\cdots\!56}a^{17}-\frac{63\!\cdots\!73}{38\!\cdots\!56}a^{16}-\frac{15\!\cdots\!67}{12\!\cdots\!52}a^{15}+\frac{36\!\cdots\!33}{54\!\cdots\!08}a^{14}+\frac{69\!\cdots\!23}{38\!\cdots\!56}a^{13}-\frac{70\!\cdots\!63}{75\!\cdots\!56}a^{12}-\frac{54\!\cdots\!07}{38\!\cdots\!56}a^{11}+\frac{17\!\cdots\!33}{38\!\cdots\!56}a^{10}+\frac{14\!\cdots\!05}{21\!\cdots\!92}a^{9}+\frac{79\!\cdots\!71}{19\!\cdots\!28}a^{8}-\frac{13\!\cdots\!15}{68\!\cdots\!76}a^{7}-\frac{44\!\cdots\!21}{53\!\cdots\!48}a^{6}+\frac{75\!\cdots\!79}{24\!\cdots\!96}a^{5}+\frac{67\!\cdots\!17}{39\!\cdots\!36}a^{4}-\frac{20\!\cdots\!19}{26\!\cdots\!24}a^{3}-\frac{56\!\cdots\!99}{26\!\cdots\!24}a^{2}+\frac{93\!\cdots\!63}{44\!\cdots\!04}a-\frac{12\!\cdots\!71}{44\!\cdots\!04}$, $\frac{92\!\cdots\!19}{95\!\cdots\!64}a^{17}-\frac{28\!\cdots\!49}{63\!\cdots\!76}a^{16}-\frac{34\!\cdots\!49}{26\!\cdots\!24}a^{15}+\frac{36\!\cdots\!03}{23\!\cdots\!16}a^{14}-\frac{34\!\cdots\!97}{10\!\cdots\!96}a^{13}-\frac{13\!\cdots\!53}{63\!\cdots\!76}a^{12}+\frac{12\!\cdots\!57}{18\!\cdots\!72}a^{11}+\frac{79\!\cdots\!35}{63\!\cdots\!76}a^{10}-\frac{32\!\cdots\!03}{15\!\cdots\!44}a^{9}-\frac{37\!\cdots\!37}{95\!\cdots\!64}a^{8}-\frac{33\!\cdots\!91}{20\!\cdots\!84}a^{7}+\frac{77\!\cdots\!41}{79\!\cdots\!72}a^{6}+\frac{18\!\cdots\!37}{33\!\cdots\!28}a^{5}-\frac{78\!\cdots\!35}{19\!\cdots\!84}a^{4}-\frac{12\!\cdots\!63}{83\!\cdots\!32}a^{3}+\frac{12\!\cdots\!77}{13\!\cdots\!12}a^{2}-\frac{18\!\cdots\!27}{11\!\cdots\!76}a+\frac{18\!\cdots\!49}{22\!\cdots\!52}$, $\frac{22\!\cdots\!25}{76\!\cdots\!12}a^{17}+\frac{14\!\cdots\!49}{76\!\cdots\!12}a^{16}-\frac{95\!\cdots\!61}{10\!\cdots\!16}a^{15}-\frac{29\!\cdots\!87}{76\!\cdots\!12}a^{14}+\frac{89\!\cdots\!29}{76\!\cdots\!12}a^{13}+\frac{33\!\cdots\!93}{76\!\cdots\!12}a^{12}-\frac{54\!\cdots\!53}{76\!\cdots\!12}a^{11}-\frac{26\!\cdots\!69}{76\!\cdots\!12}a^{10}+\frac{56\!\cdots\!79}{38\!\cdots\!56}a^{9}+\frac{66\!\cdots\!73}{38\!\cdots\!56}a^{8}+\frac{12\!\cdots\!77}{56\!\cdots\!92}a^{7}-\frac{33\!\cdots\!63}{95\!\cdots\!64}a^{6}-\frac{48\!\cdots\!01}{49\!\cdots\!92}a^{5}+\frac{12\!\cdots\!67}{79\!\cdots\!72}a^{4}+\frac{91\!\cdots\!09}{15\!\cdots\!44}a^{3}-\frac{76\!\cdots\!13}{53\!\cdots\!48}a^{2}+\frac{40\!\cdots\!11}{12\!\cdots\!44}a-\frac{14\!\cdots\!61}{88\!\cdots\!08}$, $\frac{44\!\cdots\!37}{54\!\cdots\!08}a^{17}-\frac{11\!\cdots\!55}{54\!\cdots\!08}a^{16}-\frac{55\!\cdots\!73}{18\!\cdots\!36}a^{15}+\frac{49\!\cdots\!61}{54\!\cdots\!08}a^{14}+\frac{24\!\cdots\!57}{54\!\cdots\!08}a^{13}-\frac{24\!\cdots\!37}{18\!\cdots\!36}a^{12}-\frac{18\!\cdots\!21}{54\!\cdots\!08}a^{11}+\frac{32\!\cdots\!99}{54\!\cdots\!08}a^{10}+\frac{14\!\cdots\!69}{10\!\cdots\!52}a^{9}+\frac{37\!\cdots\!05}{16\!\cdots\!12}a^{8}-\frac{26\!\cdots\!07}{68\!\cdots\!76}a^{7}-\frac{52\!\cdots\!21}{25\!\cdots\!88}a^{6}+\frac{36\!\cdots\!25}{79\!\cdots\!12}a^{5}+\frac{21\!\cdots\!87}{57\!\cdots\!48}a^{4}+\frac{69\!\cdots\!67}{10\!\cdots\!76}a^{3}-\frac{11\!\cdots\!57}{38\!\cdots\!32}a^{2}+\frac{28\!\cdots\!61}{63\!\cdots\!72}a-\frac{21\!\cdots\!77}{63\!\cdots\!72}$, $\frac{58\!\cdots\!51}{54\!\cdots\!08}a^{17}-\frac{15\!\cdots\!41}{18\!\cdots\!36}a^{16}-\frac{19\!\cdots\!25}{54\!\cdots\!08}a^{15}+\frac{21\!\cdots\!15}{78\!\cdots\!44}a^{14}+\frac{97\!\cdots\!85}{20\!\cdots\!04}a^{13}-\frac{18\!\cdots\!55}{54\!\cdots\!08}a^{12}-\frac{19\!\cdots\!07}{54\!\cdots\!08}a^{11}+\frac{95\!\cdots\!99}{60\!\cdots\!12}a^{10}+\frac{48\!\cdots\!09}{27\!\cdots\!04}a^{9}+\frac{53\!\cdots\!29}{27\!\cdots\!04}a^{8}-\frac{19\!\cdots\!65}{32\!\cdots\!56}a^{7}-\frac{13\!\cdots\!51}{68\!\cdots\!76}a^{6}+\frac{63\!\cdots\!91}{71\!\cdots\!56}a^{5}+\frac{81\!\cdots\!87}{19\!\cdots\!16}a^{4}-\frac{25\!\cdots\!09}{11\!\cdots\!96}a^{3}-\frac{14\!\cdots\!65}{38\!\cdots\!32}a^{2}+\frac{32\!\cdots\!67}{63\!\cdots\!72}a-\frac{45\!\cdots\!25}{63\!\cdots\!72}$, $\frac{54\!\cdots\!07}{25\!\cdots\!04}a^{17}-\frac{12\!\cdots\!25}{12\!\cdots\!52}a^{16}-\frac{14\!\cdots\!49}{17\!\cdots\!32}a^{15}+\frac{84\!\cdots\!11}{21\!\cdots\!92}a^{14}+\frac{31\!\cdots\!75}{25\!\cdots\!04}a^{13}-\frac{71\!\cdots\!79}{12\!\cdots\!52}a^{12}-\frac{14\!\cdots\!57}{15\!\cdots\!12}a^{11}+\frac{16\!\cdots\!31}{63\!\cdots\!76}a^{10}+\frac{19\!\cdots\!47}{42\!\cdots\!84}a^{9}+\frac{19\!\cdots\!23}{35\!\cdots\!32}a^{8}-\frac{42\!\cdots\!79}{31\!\cdots\!88}a^{7}-\frac{50\!\cdots\!19}{79\!\cdots\!72}a^{6}+\frac{26\!\cdots\!23}{16\!\cdots\!64}a^{5}+\frac{10\!\cdots\!95}{88\!\cdots\!08}a^{4}+\frac{10\!\cdots\!65}{53\!\cdots\!48}a^{3}-\frac{34\!\cdots\!97}{44\!\cdots\!04}a^{2}+\frac{25\!\cdots\!61}{18\!\cdots\!92}a-\frac{24\!\cdots\!83}{22\!\cdots\!52}$, $\frac{18\!\cdots\!41}{76\!\cdots\!12}a^{17}+\frac{41\!\cdots\!45}{76\!\cdots\!12}a^{16}-\frac{10\!\cdots\!45}{10\!\cdots\!16}a^{15}+\frac{84\!\cdots\!13}{76\!\cdots\!12}a^{14}+\frac{10\!\cdots\!37}{76\!\cdots\!12}a^{13}+\frac{10\!\cdots\!49}{76\!\cdots\!12}a^{12}-\frac{79\!\cdots\!57}{76\!\cdots\!12}a^{11}-\frac{15\!\cdots\!85}{76\!\cdots\!12}a^{10}+\frac{13\!\cdots\!19}{38\!\cdots\!56}a^{9}+\frac{83\!\cdots\!73}{38\!\cdots\!56}a^{8}-\frac{24\!\cdots\!83}{95\!\cdots\!64}a^{7}-\frac{66\!\cdots\!19}{95\!\cdots\!64}a^{6}-\frac{96\!\cdots\!97}{66\!\cdots\!56}a^{5}+\frac{19\!\cdots\!87}{79\!\cdots\!72}a^{4}-\frac{20\!\cdots\!75}{15\!\cdots\!44}a^{3}-\frac{10\!\cdots\!37}{53\!\cdots\!48}a^{2}+\frac{64\!\cdots\!75}{12\!\cdots\!44}a-\frac{24\!\cdots\!01}{88\!\cdots\!08}$
| 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: | \( 7037583519854197000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 7037583519854197000 \cdot 18}{2\cdot\sqrt{242459919475028479855663400956723947300369356292096}}\cr\approx \mathstrut & 16.0178889061164 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:D_6$ (as 18T52):
| A solvable group of order 108 |
| The 20 conjugacy class representatives for $C_3^2:D_6$ |
| Character table for $C_3^2:D_6$ |
Intermediate fields
| \(\Q(\sqrt{34}) \), 3.1.972.2, 6.2.594140645376.1, 9.3.9023659939580352192.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.6.2669978537124037139090333900186451027656102117376.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ |
| 2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
| 2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
|
\(3\)
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
|
\(7\)
| 7.6.4.2 | $x^{6} - 42 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 17.12.10.2 | $x^{12} - 3060 x^{6} - 197676$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ |