Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508)
gp: K = bnfinit(y^18 - 6*y^17 - 87*y^16 + 826*y^15 - 3753*y^14 - 972*y^13 + 204393*y^12 - 1520964*y^11 + 11658492*y^10 - 51881516*y^9 - 352803192*y^8 + 4612889568*y^7 - 9608361812*y^6 - 44908575264*y^5 + 256235575932*y^4 - 772567331760*y^3 - 64695851892*y^2 + 4666625012448*y - 4082274578508, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508)
\( x^{18} - 6 x^{17} - 87 x^{16} + 826 x^{15} - 3753 x^{14} - 972 x^{13} + 204393 x^{12} + \cdots - 4082274578508 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(12785972316065954992388499659827239408417915273216\)
\(\medspace = 2^{24}\cdot 3^{33}\cdot 7^{12}\cdot 17^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(534.75\) | 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: | $2^{4/3}3^{11/6}7^{2/3}17^{5/6}\approx 732.6025438679068$ | ||
| 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: | \(\Q(\sqrt{51}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}-\frac{1}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}$, $\frac{1}{1428}a^{12}+\frac{15}{238}a^{11}-\frac{173}{476}a^{10}+\frac{116}{357}a^{9}+\frac{1}{28}a^{8}-\frac{81}{238}a^{7}-\frac{617}{1428}a^{6}+\frac{37}{238}a^{5}-\frac{1}{14}a^{4}+\frac{5}{34}a^{3}+\frac{5}{34}a^{2}+\frac{8}{17}a-\frac{5}{34}$, $\frac{1}{1428}a^{13}-\frac{1}{28}a^{11}+\frac{25}{714}a^{10}-\frac{99}{476}a^{9}+\frac{53}{119}a^{8}+\frac{283}{1428}a^{7}+\frac{5}{119}a^{6}-\frac{15}{238}a^{5}-\frac{101}{238}a^{4}-\frac{3}{34}a^{3}+\frac{4}{17}a^{2}-\frac{1}{2}a+\frac{4}{17}$, $\frac{1}{2856}a^{14}-\frac{1}{2856}a^{12}-\frac{71}{1428}a^{11}-\frac{45}{952}a^{10}+\frac{349}{714}a^{9}-\frac{839}{2856}a^{8}+\frac{27}{119}a^{7}+\frac{2}{21}a^{6}-\frac{87}{476}a^{5}+\frac{31}{68}a^{4}-\frac{16}{119}a^{3}+\frac{29}{68}a^{2}+\frac{13}{34}a+\frac{11}{34}$, $\frac{1}{59976}a^{15}-\frac{1}{9996}a^{14}-\frac{1}{19992}a^{13}-\frac{271}{19992}a^{11}+\frac{1847}{9996}a^{10}+\frac{269}{952}a^{9}-\frac{2257}{9996}a^{8}-\frac{205}{588}a^{7}-\frac{6577}{29988}a^{6}-\frac{67}{476}a^{5}+\frac{59}{119}a^{4}+\frac{23}{84}a^{3}-\frac{6}{17}a^{2}-\frac{2}{17}a+\frac{2}{51}$, $\frac{1}{59976}a^{16}+\frac{1}{19992}a^{14}-\frac{1}{3332}a^{13}-\frac{5}{19992}a^{12}-\frac{53}{2499}a^{11}+\frac{2089}{6664}a^{10}+\frac{1157}{4998}a^{9}-\frac{370}{2499}a^{8}-\frac{1009}{4284}a^{7}-\frac{505}{9996}a^{6}-\frac{38}{119}a^{5}+\frac{433}{1428}a^{4}+\frac{25}{238}a^{3}-\frac{15}{34}a^{2}-\frac{25}{51}a-\frac{1}{17}$, $\frac{1}{95\!\cdots\!08}a^{17}-\frac{19\!\cdots\!35}{47\!\cdots\!04}a^{16}-\frac{56\!\cdots\!17}{11\!\cdots\!51}a^{15}-\frac{16\!\cdots\!99}{45\!\cdots\!48}a^{14}-\frac{11\!\cdots\!01}{39\!\cdots\!17}a^{13}-\frac{36\!\cdots\!03}{31\!\cdots\!36}a^{12}-\frac{16\!\cdots\!37}{56\!\cdots\!31}a^{11}-\frac{73\!\cdots\!15}{18\!\cdots\!08}a^{10}-\frac{55\!\cdots\!93}{31\!\cdots\!36}a^{9}+\frac{98\!\cdots\!83}{95\!\cdots\!08}a^{8}-\frac{35\!\cdots\!21}{97\!\cdots\!96}a^{7}-\frac{11\!\cdots\!27}{40\!\cdots\!16}a^{6}+\frac{96\!\cdots\!43}{22\!\cdots\!24}a^{5}+\frac{16\!\cdots\!77}{32\!\cdots\!32}a^{4}-\frac{16\!\cdots\!79}{32\!\cdots\!32}a^{3}+\frac{58\!\cdots\!41}{32\!\cdots\!32}a^{2}+\frac{32\!\cdots\!45}{11\!\cdots\!19}a+\frac{53\!\cdots\!79}{14\!\cdots\!26}$
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
$C_{3}\times C_{3}\times C_{3}\times C_{18}$, which has order $486$ (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: | $11$ | 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: |
$\frac{27\!\cdots\!44}{43\!\cdots\!43}a^{17}-\frac{30\!\cdots\!36}{43\!\cdots\!43}a^{16}-\frac{20\!\cdots\!96}{43\!\cdots\!43}a^{15}+\frac{99\!\cdots\!36}{43\!\cdots\!43}a^{14}-\frac{82\!\cdots\!24}{43\!\cdots\!43}a^{13}-\frac{14\!\cdots\!06}{43\!\cdots\!43}a^{12}+\frac{26\!\cdots\!66}{43\!\cdots\!43}a^{11}-\frac{43\!\cdots\!64}{88\!\cdots\!07}a^{10}+\frac{22\!\cdots\!76}{43\!\cdots\!43}a^{9}-\frac{66\!\cdots\!63}{43\!\cdots\!43}a^{8}-\frac{11\!\cdots\!38}{61\!\cdots\!49}a^{7}+\frac{55\!\cdots\!92}{43\!\cdots\!43}a^{6}+\frac{47\!\cdots\!56}{61\!\cdots\!49}a^{5}-\frac{37\!\cdots\!65}{61\!\cdots\!49}a^{4}+\frac{48\!\cdots\!32}{61\!\cdots\!49}a^{3}-\frac{10\!\cdots\!48}{88\!\cdots\!07}a^{2}-\frac{37\!\cdots\!64}{88\!\cdots\!07}a+\frac{25\!\cdots\!15}{54\!\cdots\!89}$, $\frac{41\!\cdots\!29}{10\!\cdots\!42}a^{17}-\frac{20\!\cdots\!91}{10\!\cdots\!42}a^{16}-\frac{54\!\cdots\!73}{15\!\cdots\!63}a^{15}+\frac{14\!\cdots\!97}{51\!\cdots\!21}a^{14}-\frac{68\!\cdots\!88}{51\!\cdots\!21}a^{13}-\frac{55\!\cdots\!57}{10\!\cdots\!42}a^{12}+\frac{36\!\cdots\!89}{51\!\cdots\!21}a^{11}-\frac{51\!\cdots\!35}{10\!\cdots\!42}a^{10}+\frac{44\!\cdots\!21}{10\!\cdots\!42}a^{9}-\frac{94\!\cdots\!00}{51\!\cdots\!21}a^{8}-\frac{69\!\cdots\!78}{51\!\cdots\!21}a^{7}+\frac{48\!\cdots\!61}{30\!\cdots\!26}a^{6}-\frac{16\!\cdots\!76}{73\!\cdots\!03}a^{5}-\frac{11\!\cdots\!81}{73\!\cdots\!03}a^{4}+\frac{51\!\cdots\!83}{73\!\cdots\!03}a^{3}-\frac{86\!\cdots\!44}{43\!\cdots\!59}a^{2}-\frac{15\!\cdots\!78}{73\!\cdots\!03}a+\frac{61\!\cdots\!32}{44\!\cdots\!81}$, $\frac{12\!\cdots\!13}{15\!\cdots\!68}a^{17}-\frac{36\!\cdots\!13}{39\!\cdots\!17}a^{16}-\frac{26\!\cdots\!47}{28\!\cdots\!12}a^{15}+\frac{79\!\cdots\!83}{11\!\cdots\!62}a^{14}-\frac{37\!\cdots\!13}{53\!\cdots\!56}a^{13}+\frac{67\!\cdots\!49}{15\!\cdots\!68}a^{12}-\frac{33\!\cdots\!53}{32\!\cdots\!32}a^{11}-\frac{29\!\cdots\!57}{53\!\cdots\!56}a^{10}+\frac{28\!\cdots\!78}{23\!\cdots\!01}a^{9}-\frac{60\!\cdots\!13}{53\!\cdots\!56}a^{8}+\frac{24\!\cdots\!51}{56\!\cdots\!31}a^{7}+\frac{59\!\cdots\!15}{68\!\cdots\!72}a^{6}-\frac{24\!\cdots\!14}{18\!\cdots\!77}a^{5}+\frac{24\!\cdots\!75}{54\!\cdots\!22}a^{4}-\frac{12\!\cdots\!71}{16\!\cdots\!66}a^{3}-\frac{60\!\cdots\!67}{54\!\cdots\!22}a^{2}+\frac{26\!\cdots\!34}{38\!\cdots\!73}a-\frac{77\!\cdots\!99}{14\!\cdots\!26}$, $\frac{35\!\cdots\!73}{79\!\cdots\!34}a^{17}-\frac{66\!\cdots\!11}{13\!\cdots\!39}a^{16}+\frac{46\!\cdots\!58}{39\!\cdots\!17}a^{15}+\frac{12\!\cdots\!33}{22\!\cdots\!24}a^{14}-\frac{11\!\cdots\!39}{39\!\cdots\!17}a^{13}+\frac{17\!\cdots\!25}{26\!\cdots\!78}a^{12}+\frac{53\!\cdots\!33}{81\!\cdots\!33}a^{11}-\frac{39\!\cdots\!64}{39\!\cdots\!17}a^{10}+\frac{12\!\cdots\!53}{26\!\cdots\!78}a^{9}-\frac{10\!\cdots\!97}{26\!\cdots\!78}a^{8}+\frac{86\!\cdots\!09}{11\!\cdots\!62}a^{7}+\frac{56\!\cdots\!73}{22\!\cdots\!24}a^{6}-\frac{32\!\cdots\!82}{18\!\cdots\!77}a^{5}-\frac{44\!\cdots\!91}{27\!\cdots\!11}a^{4}+\frac{29\!\cdots\!65}{77\!\cdots\!46}a^{3}-\frac{92\!\cdots\!59}{27\!\cdots\!11}a^{2}-\frac{62\!\cdots\!29}{38\!\cdots\!73}a+\frac{76\!\cdots\!57}{47\!\cdots\!42}$, $\frac{19\!\cdots\!65}{23\!\cdots\!02}a^{17}-\frac{31\!\cdots\!43}{47\!\cdots\!04}a^{16}-\frac{14\!\cdots\!03}{26\!\cdots\!78}a^{15}+\frac{18\!\cdots\!21}{22\!\cdots\!24}a^{14}-\frac{19\!\cdots\!58}{39\!\cdots\!17}a^{13}+\frac{15\!\cdots\!11}{15\!\cdots\!68}a^{12}+\frac{16\!\cdots\!49}{11\!\cdots\!62}a^{11}-\frac{24\!\cdots\!89}{15\!\cdots\!68}a^{10}+\frac{51\!\cdots\!21}{39\!\cdots\!17}a^{9}-\frac{84\!\cdots\!38}{11\!\cdots\!51}a^{8}-\frac{43\!\cdots\!07}{34\!\cdots\!86}a^{7}+\frac{22\!\cdots\!56}{56\!\cdots\!31}a^{6}-\frac{95\!\cdots\!47}{56\!\cdots\!31}a^{5}+\frac{14\!\cdots\!63}{16\!\cdots\!66}a^{4}+\frac{55\!\cdots\!08}{27\!\cdots\!11}a^{3}-\frac{88\!\cdots\!11}{81\!\cdots\!33}a^{2}+\frac{27\!\cdots\!98}{11\!\cdots\!19}a-\frac{35\!\cdots\!68}{23\!\cdots\!71}$, $\frac{78\!\cdots\!13}{23\!\cdots\!02}a^{17}-\frac{38\!\cdots\!75}{47\!\cdots\!04}a^{16}-\frac{80\!\cdots\!97}{23\!\cdots\!02}a^{15}+\frac{39\!\cdots\!31}{22\!\cdots\!24}a^{14}-\frac{38\!\cdots\!29}{78\!\cdots\!67}a^{13}-\frac{21\!\cdots\!27}{53\!\cdots\!56}a^{12}+\frac{11\!\cdots\!89}{16\!\cdots\!66}a^{11}-\frac{17\!\cdots\!43}{53\!\cdots\!56}a^{10}+\frac{33\!\cdots\!60}{13\!\cdots\!39}a^{9}-\frac{33\!\cdots\!67}{70\!\cdots\!03}a^{8}-\frac{58\!\cdots\!97}{34\!\cdots\!86}a^{7}+\frac{39\!\cdots\!41}{34\!\cdots\!86}a^{6}+\frac{38\!\cdots\!13}{56\!\cdots\!31}a^{5}-\frac{31\!\cdots\!41}{16\!\cdots\!66}a^{4}+\frac{42\!\cdots\!79}{81\!\cdots\!33}a^{3}-\frac{11\!\cdots\!35}{81\!\cdots\!33}a^{2}-\frac{53\!\cdots\!36}{11\!\cdots\!19}a+\frac{41\!\cdots\!75}{71\!\cdots\!13}$, $\frac{10\!\cdots\!49}{46\!\cdots\!02}a^{17}-\frac{93\!\cdots\!98}{11\!\cdots\!51}a^{16}-\frac{46\!\cdots\!93}{23\!\cdots\!02}a^{15}+\frac{82\!\cdots\!47}{11\!\cdots\!62}a^{14}-\frac{33\!\cdots\!71}{79\!\cdots\!34}a^{13}-\frac{71\!\cdots\!85}{26\!\cdots\!78}a^{12}+\frac{48\!\cdots\!49}{16\!\cdots\!66}a^{11}-\frac{13\!\cdots\!17}{79\!\cdots\!34}a^{10}+\frac{21\!\cdots\!50}{13\!\cdots\!39}a^{9}-\frac{17\!\cdots\!99}{79\!\cdots\!34}a^{8}-\frac{15\!\cdots\!82}{17\!\cdots\!93}a^{7}+\frac{87\!\cdots\!43}{17\!\cdots\!93}a^{6}+\frac{14\!\cdots\!45}{18\!\cdots\!77}a^{5}-\frac{70\!\cdots\!88}{11\!\cdots\!19}a^{4}+\frac{20\!\cdots\!42}{81\!\cdots\!33}a^{3}-\frac{47\!\cdots\!77}{27\!\cdots\!11}a^{2}-\frac{16\!\cdots\!20}{11\!\cdots\!19}a+\frac{10\!\cdots\!63}{71\!\cdots\!13}$, $\frac{10\!\cdots\!75}{95\!\cdots\!08}a^{17}-\frac{39\!\cdots\!21}{47\!\cdots\!04}a^{16}-\frac{57\!\cdots\!01}{56\!\cdots\!24}a^{15}+\frac{12\!\cdots\!45}{10\!\cdots\!44}a^{14}-\frac{12\!\cdots\!69}{31\!\cdots\!36}a^{13}-\frac{41\!\cdots\!23}{53\!\cdots\!56}a^{12}+\frac{39\!\cdots\!67}{15\!\cdots\!16}a^{11}-\frac{29\!\cdots\!43}{15\!\cdots\!68}a^{10}+\frac{63\!\cdots\!79}{53\!\cdots\!56}a^{9}-\frac{26\!\cdots\!85}{47\!\cdots\!04}a^{8}-\frac{30\!\cdots\!71}{68\!\cdots\!72}a^{7}+\frac{24\!\cdots\!79}{40\!\cdots\!16}a^{6}-\frac{24\!\cdots\!71}{22\!\cdots\!24}a^{5}-\frac{50\!\cdots\!68}{47\!\cdots\!49}a^{4}+\frac{62\!\cdots\!69}{16\!\cdots\!66}a^{3}+\frac{10\!\cdots\!79}{81\!\cdots\!33}a^{2}-\frac{10\!\cdots\!09}{68\!\cdots\!07}a+\frac{17\!\cdots\!39}{14\!\cdots\!26}$, $\frac{66\!\cdots\!82}{17\!\cdots\!93}a^{17}-\frac{55\!\cdots\!79}{34\!\cdots\!86}a^{16}-\frac{12\!\cdots\!71}{34\!\cdots\!86}a^{15}+\frac{28\!\cdots\!99}{11\!\cdots\!62}a^{14}-\frac{12\!\cdots\!49}{11\!\cdots\!62}a^{13}-\frac{77\!\cdots\!50}{56\!\cdots\!31}a^{12}+\frac{81\!\cdots\!11}{11\!\cdots\!62}a^{11}-\frac{37\!\cdots\!41}{81\!\cdots\!33}a^{10}+\frac{44\!\cdots\!23}{11\!\cdots\!62}a^{9}-\frac{51\!\cdots\!61}{34\!\cdots\!86}a^{8}-\frac{35\!\cdots\!03}{24\!\cdots\!99}a^{7}+\frac{29\!\cdots\!35}{20\!\cdots\!58}a^{6}-\frac{12\!\cdots\!03}{81\!\cdots\!33}a^{5}-\frac{11\!\cdots\!87}{81\!\cdots\!33}a^{4}+\frac{49\!\cdots\!05}{81\!\cdots\!33}a^{3}-\frac{26\!\cdots\!41}{11\!\cdots\!19}a^{2}-\frac{19\!\cdots\!06}{11\!\cdots\!19}a+\frac{25\!\cdots\!62}{71\!\cdots\!13}$, $\frac{60\!\cdots\!03}{45\!\cdots\!48}a^{17}-\frac{30\!\cdots\!95}{68\!\cdots\!72}a^{16}-\frac{17\!\cdots\!77}{13\!\cdots\!44}a^{15}+\frac{17\!\cdots\!61}{22\!\cdots\!24}a^{14}-\frac{38\!\cdots\!91}{12\!\cdots\!64}a^{13}-\frac{10\!\cdots\!07}{11\!\cdots\!62}a^{12}+\frac{11\!\cdots\!85}{45\!\cdots\!48}a^{11}-\frac{25\!\cdots\!19}{18\!\cdots\!77}a^{10}+\frac{27\!\cdots\!75}{22\!\cdots\!24}a^{9}-\frac{21\!\cdots\!11}{56\!\cdots\!31}a^{8}-\frac{19\!\cdots\!25}{34\!\cdots\!86}a^{7}+\frac{79\!\cdots\!49}{17\!\cdots\!93}a^{6}-\frac{48\!\cdots\!21}{10\!\cdots\!44}a^{5}-\frac{29\!\cdots\!37}{47\!\cdots\!49}a^{4}+\frac{14\!\cdots\!47}{81\!\cdots\!33}a^{3}-\frac{42\!\cdots\!23}{77\!\cdots\!46}a^{2}-\frac{36\!\cdots\!71}{23\!\cdots\!38}a+\frac{14\!\cdots\!56}{71\!\cdots\!13}$, $\frac{91\!\cdots\!49}{13\!\cdots\!44}a^{17}+\frac{19\!\cdots\!99}{37\!\cdots\!54}a^{16}-\frac{29\!\cdots\!33}{65\!\cdots\!64}a^{15}-\frac{12\!\cdots\!89}{11\!\cdots\!62}a^{14}+\frac{54\!\cdots\!37}{45\!\cdots\!48}a^{13}-\frac{27\!\cdots\!43}{22\!\cdots\!24}a^{12}+\frac{26\!\cdots\!91}{65\!\cdots\!64}a^{11}-\frac{23\!\cdots\!25}{22\!\cdots\!24}a^{10}+\frac{30\!\cdots\!85}{32\!\cdots\!32}a^{9}+\frac{33\!\cdots\!50}{17\!\cdots\!93}a^{8}-\frac{17\!\cdots\!35}{56\!\cdots\!31}a^{7}-\frac{24\!\cdots\!95}{11\!\cdots\!62}a^{6}+\frac{36\!\cdots\!29}{32\!\cdots\!32}a^{5}-\frac{74\!\cdots\!35}{54\!\cdots\!22}a^{4}-\frac{43\!\cdots\!77}{27\!\cdots\!11}a^{3}+\frac{52\!\cdots\!81}{23\!\cdots\!38}a^{2}+\frac{46\!\cdots\!71}{77\!\cdots\!46}a-\frac{16\!\cdots\!88}{23\!\cdots\!71}$
| 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: | \( 37304242818863384 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 37304242818863384 \cdot 486}{2\cdot\sqrt{12785972316065954992388499659827239408417915273216}}\cr\approx \mathstrut & 9.98290198686953 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508)
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^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508, 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^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508);
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^18 - 6*x^17 - 87*x^16 + 826*x^15 - 3753*x^14 - 972*x^13 + 204393*x^12 - 1520964*x^11 + 11658492*x^10 - 51881516*x^9 - 352803192*x^8 + 4612889568*x^7 - 9608361812*x^6 - 44908575264*x^5 + 256235575932*x^4 - 772567331760*x^3 - 64695851892*x^2 + 4666625012448*x - 4082274578508);
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
$C_3^2:D_6$ (as 18T52):
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)
| 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
| \(\Q(\sqrt{51}) \), 3.1.972.2, 6.2.222802742016.6, 9.3.9023659939580352192.1 |
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]
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.0.5214801830320385037285808398801662163390824448.1 |
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 | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ |
| 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
| 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
|
\(3\)
| 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |
| 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
| 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
|
\(7\)
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(17\)
| 17.6.3.2 | $x^{6} + 289 x^{2} - 68782$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 17.12.10.2 | $x^{12} - 3060 x^{6} - 197676$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ |