Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963)
gp: K = bnfinit(y^18 - 2*y^17 + 16*y^16 - 114*y^15 + 486*y^14 - 1617*y^13 + 5983*y^12 - 21824*y^11 + 61120*y^10 - 124458*y^9 + 193737*y^8 - 252336*y^7 + 307981*y^6 - 344420*y^5 + 321958*y^4 - 236085*y^3 + 139299*y^2 - 130761*y + 126963, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963)
\( x^{18} - 2 x^{17} + 16 x^{16} - 114 x^{15} + 486 x^{14} - 1617 x^{13} + 5983 x^{12} - 21824 x^{11} + \cdots + 126963 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-18665884703684652969692527276032\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 7^{10}\cdot 31^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.61\) | 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: | $2^{2/3}3^{1/2}7^{5/6}31^{5/6}\approx 243.38682724932116$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{5}$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{10}+\frac{1}{9}a^{7}-\frac{4}{27}a^{6}+\frac{1}{9}a^{5}-\frac{2}{27}a^{4}-\frac{2}{9}a^{3}-\frac{4}{9}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{81}a^{13}-\frac{1}{81}a^{12}-\frac{4}{81}a^{11}+\frac{1}{81}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}+\frac{2}{81}a^{7}+\frac{7}{81}a^{6}+\frac{25}{81}a^{5}-\frac{4}{81}a^{4}+\frac{2}{9}a^{3}-\frac{2}{27}a^{2}-\frac{1}{9}a-\frac{4}{9}$, $\frac{1}{81}a^{14}+\frac{1}{81}a^{12}-\frac{1}{27}a^{11}-\frac{2}{81}a^{10}-\frac{1}{27}a^{9}+\frac{5}{81}a^{8}+\frac{8}{81}a^{6}+\frac{13}{27}a^{5}+\frac{2}{81}a^{4}-\frac{5}{27}a^{3}-\frac{2}{27}a^{2}+\frac{4}{9}a+\frac{2}{9}$, $\frac{1}{243}a^{15}-\frac{1}{243}a^{13}+\frac{2}{243}a^{12}-\frac{4}{81}a^{11}+\frac{1}{243}a^{10}-\frac{10}{243}a^{9}-\frac{11}{81}a^{8}+\frac{13}{243}a^{7}-\frac{14}{243}a^{6}+\frac{29}{81}a^{5}+\frac{113}{243}a^{4}-\frac{8}{81}a^{3}+\frac{31}{81}a^{2}+\frac{10}{27}a-\frac{7}{27}$, $\frac{1}{243}a^{16}-\frac{1}{243}a^{14}-\frac{1}{243}a^{13}+\frac{13}{243}a^{11}+\frac{5}{243}a^{10}+\frac{4}{81}a^{9}+\frac{4}{243}a^{8}+\frac{7}{243}a^{7}+\frac{10}{81}a^{6}-\frac{97}{243}a^{5}-\frac{19}{81}a^{4}+\frac{4}{81}a^{3}-\frac{13}{27}a-\frac{2}{9}$, $\frac{1}{25\!\cdots\!39}a^{17}+\frac{42\!\cdots\!03}{25\!\cdots\!39}a^{16}+\frac{63\!\cdots\!49}{28\!\cdots\!71}a^{15}+\frac{18\!\cdots\!32}{85\!\cdots\!13}a^{14}-\frac{58\!\cdots\!38}{28\!\cdots\!71}a^{13}-\frac{19\!\cdots\!14}{28\!\cdots\!71}a^{12}-\frac{93\!\cdots\!07}{25\!\cdots\!39}a^{11}+\frac{21\!\cdots\!63}{25\!\cdots\!39}a^{10}+\frac{45\!\cdots\!69}{95\!\cdots\!57}a^{9}+\frac{85\!\cdots\!10}{85\!\cdots\!13}a^{8}+\frac{56\!\cdots\!65}{31\!\cdots\!19}a^{7}+\frac{26\!\cdots\!68}{28\!\cdots\!71}a^{6}-\frac{54\!\cdots\!50}{25\!\cdots\!39}a^{5}+\frac{12\!\cdots\!84}{25\!\cdots\!39}a^{4}+\frac{89\!\cdots\!07}{28\!\cdots\!71}a^{3}+\frac{22\!\cdots\!41}{85\!\cdots\!13}a^{2}-\frac{12\!\cdots\!06}{28\!\cdots\!71}a+\frac{66\!\cdots\!62}{28\!\cdots\!71}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{3}$, which has order $3$ (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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{55919525118659528}{19669355733881560089393} a^{17} + \frac{188573743120357555}{19669355733881560089393} a^{16} + \frac{299007753330834347}{6556451911293853363131} a^{15} - \frac{695969295971892493}{6556451911293853363131} a^{14} + \frac{638181969623069746}{6556451911293853363131} a^{13} - \frac{1158051162136249313}{6556451911293853363131} a^{12} + \frac{73519359275405503898}{19669355733881560089393} a^{11} - \frac{142429975104130465031}{19669355733881560089393} a^{10} - \frac{108077241736940251718}{6556451911293853363131} a^{9} + \frac{463477561192634118941}{6556451911293853363131} a^{8} - \frac{328240354406823678412}{6556451911293853363131} a^{7} - \frac{863601145902108423946}{6556451911293853363131} a^{6} + \frac{8563004073621646538261}{19669355733881560089393} a^{5} - \frac{9845432429977538068544}{19669355733881560089393} a^{4} + \frac{1580971888917259944313}{2185483970431284454377} a^{3} - \frac{4942857528174734940073}{6556451911293853363131} a^{2} + \frac{1313153521012351323796}{2185483970431284454377} a - \frac{27904668054178861310}{2185483970431284454377} \)
(order $6$)
| 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: |
$\frac{73\!\cdots\!96}{41\!\cdots\!33}a^{17}-\frac{10\!\cdots\!96}{41\!\cdots\!33}a^{16}+\frac{10\!\cdots\!51}{41\!\cdots\!33}a^{15}-\frac{73\!\cdots\!95}{41\!\cdots\!33}a^{14}+\frac{93\!\cdots\!94}{12\!\cdots\!99}a^{13}-\frac{28\!\cdots\!17}{12\!\cdots\!99}a^{12}+\frac{10\!\cdots\!06}{12\!\cdots\!99}a^{11}-\frac{37\!\cdots\!19}{12\!\cdots\!99}a^{10}+\frac{32\!\cdots\!34}{41\!\cdots\!33}a^{9}-\frac{53\!\cdots\!27}{41\!\cdots\!33}a^{8}+\frac{14\!\cdots\!82}{12\!\cdots\!99}a^{7}-\frac{22\!\cdots\!34}{12\!\cdots\!99}a^{6}-\frac{10\!\cdots\!45}{12\!\cdots\!99}a^{5}+\frac{20\!\cdots\!01}{12\!\cdots\!99}a^{4}-\frac{40\!\cdots\!31}{13\!\cdots\!11}a^{3}+\frac{15\!\cdots\!41}{41\!\cdots\!33}a^{2}-\frac{36\!\cdots\!02}{13\!\cdots\!11}a+\frac{11\!\cdots\!08}{13\!\cdots\!11}$, $\frac{18\!\cdots\!70}{11\!\cdots\!91}a^{17}+\frac{25\!\cdots\!98}{11\!\cdots\!91}a^{16}+\frac{28\!\cdots\!32}{12\!\cdots\!99}a^{15}-\frac{38\!\cdots\!88}{37\!\cdots\!97}a^{14}+\frac{35\!\cdots\!54}{12\!\cdots\!99}a^{13}-\frac{85\!\cdots\!75}{13\!\cdots\!11}a^{12}+\frac{37\!\cdots\!32}{11\!\cdots\!91}a^{11}-\frac{11\!\cdots\!90}{11\!\cdots\!91}a^{10}+\frac{45\!\cdots\!58}{41\!\cdots\!33}a^{9}+\frac{98\!\cdots\!87}{37\!\cdots\!97}a^{8}-\frac{14\!\cdots\!44}{13\!\cdots\!11}a^{7}+\frac{21\!\cdots\!46}{12\!\cdots\!99}a^{6}-\frac{19\!\cdots\!06}{11\!\cdots\!91}a^{5}+\frac{22\!\cdots\!41}{11\!\cdots\!91}a^{4}-\frac{23\!\cdots\!54}{12\!\cdots\!99}a^{3}+\frac{50\!\cdots\!36}{37\!\cdots\!97}a^{2}-\frac{27\!\cdots\!78}{12\!\cdots\!99}a+\frac{20\!\cdots\!13}{12\!\cdots\!99}$, $\frac{44\!\cdots\!18}{85\!\cdots\!13}a^{17}-\frac{53\!\cdots\!26}{31\!\cdots\!19}a^{16}+\frac{10\!\cdots\!55}{95\!\cdots\!57}a^{15}-\frac{61\!\cdots\!36}{85\!\cdots\!13}a^{14}+\frac{29\!\cdots\!52}{85\!\cdots\!13}a^{13}-\frac{36\!\cdots\!52}{28\!\cdots\!71}a^{12}+\frac{39\!\cdots\!48}{85\!\cdots\!13}a^{11}-\frac{14\!\cdots\!32}{85\!\cdots\!13}a^{10}+\frac{14\!\cdots\!93}{28\!\cdots\!71}a^{9}-\frac{10\!\cdots\!81}{85\!\cdots\!13}a^{8}+\frac{19\!\cdots\!19}{85\!\cdots\!13}a^{7}-\frac{33\!\cdots\!52}{95\!\cdots\!57}a^{6}+\frac{12\!\cdots\!22}{28\!\cdots\!71}a^{5}-\frac{42\!\cdots\!79}{85\!\cdots\!13}a^{4}+\frac{13\!\cdots\!85}{28\!\cdots\!71}a^{3}-\frac{10\!\cdots\!31}{28\!\cdots\!71}a^{2}+\frac{18\!\cdots\!46}{95\!\cdots\!57}a-\frac{38\!\cdots\!26}{95\!\cdots\!57}$, $\frac{32\!\cdots\!93}{85\!\cdots\!13}a^{17}+\frac{38\!\cdots\!63}{85\!\cdots\!13}a^{16}-\frac{41\!\cdots\!82}{85\!\cdots\!13}a^{15}+\frac{41\!\cdots\!57}{85\!\cdots\!13}a^{14}-\frac{32\!\cdots\!28}{85\!\cdots\!13}a^{13}+\frac{18\!\cdots\!31}{85\!\cdots\!13}a^{12}-\frac{63\!\cdots\!21}{95\!\cdots\!57}a^{11}+\frac{19\!\cdots\!38}{85\!\cdots\!13}a^{10}-\frac{82\!\cdots\!58}{85\!\cdots\!13}a^{9}+\frac{24\!\cdots\!31}{85\!\cdots\!13}a^{8}-\frac{50\!\cdots\!27}{85\!\cdots\!13}a^{7}+\frac{75\!\cdots\!75}{85\!\cdots\!13}a^{6}-\frac{88\!\cdots\!09}{85\!\cdots\!13}a^{5}+\frac{31\!\cdots\!18}{28\!\cdots\!71}a^{4}-\frac{20\!\cdots\!77}{28\!\cdots\!71}a^{3}+\frac{83\!\cdots\!44}{95\!\cdots\!57}a^{2}-\frac{26\!\cdots\!84}{95\!\cdots\!57}a+\frac{16\!\cdots\!66}{31\!\cdots\!19}$, $\frac{18\!\cdots\!24}{25\!\cdots\!39}a^{17}-\frac{31\!\cdots\!20}{25\!\cdots\!39}a^{16}+\frac{78\!\cdots\!09}{85\!\cdots\!13}a^{15}-\frac{66\!\cdots\!65}{95\!\cdots\!57}a^{14}+\frac{82\!\cdots\!30}{28\!\cdots\!71}a^{13}-\frac{65\!\cdots\!31}{85\!\cdots\!13}a^{12}+\frac{63\!\cdots\!77}{25\!\cdots\!39}a^{11}-\frac{23\!\cdots\!31}{25\!\cdots\!39}a^{10}+\frac{18\!\cdots\!12}{85\!\cdots\!13}a^{9}-\frac{80\!\cdots\!03}{85\!\cdots\!13}a^{8}-\frac{27\!\cdots\!93}{28\!\cdots\!71}a^{7}+\frac{27\!\cdots\!64}{85\!\cdots\!13}a^{6}-\frac{14\!\cdots\!70}{25\!\cdots\!39}a^{5}+\frac{17\!\cdots\!26}{25\!\cdots\!39}a^{4}-\frac{19\!\cdots\!04}{28\!\cdots\!71}a^{3}+\frac{55\!\cdots\!99}{85\!\cdots\!13}a^{2}-\frac{17\!\cdots\!33}{28\!\cdots\!71}a+\frac{93\!\cdots\!04}{28\!\cdots\!71}$, $\frac{26\!\cdots\!51}{25\!\cdots\!39}a^{17}+\frac{72\!\cdots\!62}{25\!\cdots\!39}a^{16}+\frac{13\!\cdots\!16}{85\!\cdots\!13}a^{15}-\frac{46\!\cdots\!17}{85\!\cdots\!13}a^{14}+\frac{53\!\cdots\!28}{85\!\cdots\!13}a^{13}-\frac{24\!\cdots\!15}{85\!\cdots\!13}a^{12}+\frac{49\!\cdots\!06}{25\!\cdots\!39}a^{11}-\frac{10\!\cdots\!66}{25\!\cdots\!39}a^{10}-\frac{42\!\cdots\!26}{85\!\cdots\!13}a^{9}+\frac{10\!\cdots\!05}{85\!\cdots\!13}a^{8}-\frac{14\!\cdots\!01}{85\!\cdots\!13}a^{7}+\frac{33\!\cdots\!86}{85\!\cdots\!13}a^{6}-\frac{11\!\cdots\!43}{25\!\cdots\!39}a^{5}+\frac{95\!\cdots\!10}{25\!\cdots\!39}a^{4}-\frac{27\!\cdots\!47}{28\!\cdots\!71}a^{3}+\frac{33\!\cdots\!21}{85\!\cdots\!13}a^{2}+\frac{92\!\cdots\!78}{28\!\cdots\!71}a+\frac{24\!\cdots\!79}{28\!\cdots\!71}$, $\frac{19\!\cdots\!04}{85\!\cdots\!13}a^{17}-\frac{15\!\cdots\!37}{28\!\cdots\!71}a^{16}+\frac{28\!\cdots\!73}{85\!\cdots\!13}a^{15}-\frac{22\!\cdots\!99}{85\!\cdots\!13}a^{14}+\frac{97\!\cdots\!99}{85\!\cdots\!13}a^{13}-\frac{30\!\cdots\!21}{85\!\cdots\!13}a^{12}+\frac{11\!\cdots\!52}{85\!\cdots\!13}a^{11}-\frac{42\!\cdots\!03}{85\!\cdots\!13}a^{10}+\frac{11\!\cdots\!11}{85\!\cdots\!13}a^{9}-\frac{22\!\cdots\!06}{85\!\cdots\!13}a^{8}+\frac{33\!\cdots\!33}{85\!\cdots\!13}a^{7}-\frac{43\!\cdots\!14}{85\!\cdots\!13}a^{6}+\frac{17\!\cdots\!35}{28\!\cdots\!71}a^{5}-\frac{53\!\cdots\!37}{85\!\cdots\!13}a^{4}+\frac{15\!\cdots\!69}{28\!\cdots\!71}a^{3}-\frac{79\!\cdots\!40}{28\!\cdots\!71}a^{2}+\frac{13\!\cdots\!98}{95\!\cdots\!57}a-\frac{17\!\cdots\!98}{95\!\cdots\!57}$, $\frac{24\!\cdots\!42}{25\!\cdots\!39}a^{17}+\frac{60\!\cdots\!19}{25\!\cdots\!39}a^{16}+\frac{12\!\cdots\!21}{85\!\cdots\!13}a^{15}-\frac{21\!\cdots\!23}{28\!\cdots\!71}a^{14}+\frac{24\!\cdots\!37}{85\!\cdots\!13}a^{13}-\frac{70\!\cdots\!44}{85\!\cdots\!13}a^{12}+\frac{90\!\cdots\!77}{25\!\cdots\!39}a^{11}-\frac{30\!\cdots\!84}{25\!\cdots\!39}a^{10}+\frac{23\!\cdots\!31}{85\!\cdots\!13}a^{9}-\frac{37\!\cdots\!15}{85\!\cdots\!13}a^{8}+\frac{49\!\cdots\!46}{85\!\cdots\!13}a^{7}-\frac{59\!\cdots\!61}{85\!\cdots\!13}a^{6}+\frac{22\!\cdots\!70}{25\!\cdots\!39}a^{5}-\frac{18\!\cdots\!02}{25\!\cdots\!39}a^{4}+\frac{67\!\cdots\!08}{95\!\cdots\!57}a^{3}-\frac{77\!\cdots\!21}{85\!\cdots\!13}a^{2}+\frac{14\!\cdots\!07}{28\!\cdots\!71}a-\frac{66\!\cdots\!09}{28\!\cdots\!71}$
| 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: | \( 4121877351.3792276 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 4121877351.3792276 \cdot 3}{6\cdot\sqrt{18665884703684652969692527276032}}\cr\approx \mathstrut & 7.28047299099802 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963)
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 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963, 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 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963);
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 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963);
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
$C_3\times S_3^2$ (as 18T46):
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 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.2604.1, 6.0.1271403.2, 6.0.20342448.2 |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.917583011822601339312384.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(31\)
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.6.3.1 | $x^{6} + 961 x^{2} - 834148$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 31.6.5.1 | $x^{6} + 465$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |