Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209)
gp: K = bnfinit(y^18 - 333*y^16 + 39960*y^14 - 2147961*y^12 + 54976302*y^10 - 14845436*y^9 - 713903382*y^8 + 439970589*y^7 + 4664989341*y^6 - 4344963687*y^5 - 13662633690*y^4 + 15837166980*y^3 + 12296370321*y^2 - 15028897059*y - 873916209, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209)
\( x^{18} - 333 x^{16} + 39960 x^{14} - 2147961 x^{12} + 54976302 x^{10} - 14845436 x^{9} + \cdots - 873916209 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(328368182120667332857143332667211818298338333\)
\(\medspace = 3^{44}\cdot 37^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(297.26\) | 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: | $3^{22/9}37^{5/6}\approx 297.25637031384787$ | ||
| Ramified primes: |
\(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(999=3^{3}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(322,·)$, $\chi_{999}(196,·)$, $\chi_{999}(454,·)$, $\chi_{999}(73,·)$, $\chi_{999}(334,·)$, $\chi_{999}(655,·)$, $\chi_{999}(529,·)$, $\chi_{999}(787,·)$, $\chi_{999}(406,·)$, $\chi_{999}(667,·)$, $\chi_{999}(988,·)$, $\chi_{999}(862,·)$, $\chi_{999}(739,·)$, $\chi_{999}(175,·)$, $\chi_{999}(841,·)$, $\chi_{999}(121,·)$, $\chi_{999}(508,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{37}a^{6}$, $\frac{1}{37}a^{7}$, $\frac{1}{37}a^{8}$, $\frac{1}{37}a^{9}$, $\frac{1}{37}a^{10}$, $\frac{1}{111}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{4107}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{874791}a^{13}-\frac{8}{97199}a^{12}+\frac{5}{2627}a^{11}+\frac{17}{2627}a^{10}-\frac{32}{2627}a^{9}-\frac{34}{2627}a^{8}+\frac{95}{7881}a^{7}+\frac{17}{2627}a^{6}-\frac{34}{71}a^{5}-\frac{161}{639}a^{4}+\frac{5}{71}a^{3}-\frac{11}{71}a^{2}+\frac{35}{71}a$, $\frac{1}{874791}a^{14}+\frac{34}{291597}a^{12}-\frac{5}{7881}a^{11}-\frac{15}{2627}a^{10}+\frac{5}{2627}a^{9}-\frac{7}{7881}a^{8}+\frac{25}{2627}a^{7}-\frac{34}{2627}a^{6}+\frac{172}{639}a^{5}-\frac{5}{71}a^{4}-\frac{89}{213}a^{3}+\frac{1}{213}a^{2}+\frac{35}{71}a$, $\frac{1}{2624373}a^{15}-\frac{1}{97199}a^{12}-\frac{28}{7881}a^{11}-\frac{32}{2627}a^{10}-\frac{226}{23643}a^{9}-\frac{19}{2627}a^{8}+\frac{73}{7881}a^{7}+\frac{343}{70929}a^{6}-\frac{29}{71}a^{5}+\frac{91}{213}a^{4}-\frac{20}{71}a^{3}+\frac{92}{213}a^{2}+\frac{17}{71}a$, $\frac{1}{2624373}a^{16}+\frac{26}{291597}a^{12}-\frac{32}{7881}a^{11}-\frac{127}{23643}a^{10}-\frac{23}{2627}a^{9}+\frac{7}{7881}a^{8}+\frac{10}{1917}a^{7}+\frac{3}{2627}a^{6}+\frac{25}{213}a^{5}+\frac{32}{71}a^{4}+\frac{14}{213}a^{3}-\frac{104}{213}a^{2}+\frac{31}{71}a$, $\frac{1}{11\!\cdots\!53}a^{17}+\frac{10\!\cdots\!22}{12\!\cdots\!17}a^{16}+\frac{14\!\cdots\!36}{41\!\cdots\!39}a^{15}+\frac{19\!\cdots\!35}{12\!\cdots\!17}a^{14}-\frac{27\!\cdots\!00}{13\!\cdots\!13}a^{13}-\frac{39\!\cdots\!51}{41\!\cdots\!39}a^{12}-\frac{13\!\cdots\!70}{27\!\cdots\!79}a^{11}+\frac{11\!\cdots\!34}{11\!\cdots\!47}a^{10}-\frac{40\!\cdots\!94}{33\!\cdots\!41}a^{9}+\frac{22\!\cdots\!72}{30\!\cdots\!69}a^{8}-\frac{21\!\cdots\!20}{33\!\cdots\!41}a^{7}-\frac{23\!\cdots\!01}{33\!\cdots\!41}a^{6}-\frac{33\!\cdots\!40}{90\!\cdots\!93}a^{5}-\frac{70\!\cdots\!39}{30\!\cdots\!31}a^{4}-\frac{11\!\cdots\!57}{30\!\cdots\!31}a^{3}+\frac{80\!\cdots\!92}{30\!\cdots\!31}a^{2}+\frac{41\!\cdots\!97}{10\!\cdots\!77}a-\frac{39\!\cdots\!53}{14\!\cdots\!87}$
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
$C_{3}$, 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: | $17$ | 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: |
$\frac{21\!\cdots\!75}{37\!\cdots\!23}a^{17}+\frac{13\!\cdots\!50}{11\!\cdots\!69}a^{16}-\frac{18\!\cdots\!39}{10\!\cdots\!21}a^{15}-\frac{40\!\cdots\!25}{10\!\cdots\!79}a^{14}+\frac{81\!\cdots\!80}{37\!\cdots\!23}a^{13}+\frac{17\!\cdots\!12}{37\!\cdots\!23}a^{12}-\frac{11\!\cdots\!40}{10\!\cdots\!79}a^{11}-\frac{24\!\cdots\!32}{10\!\cdots\!79}a^{10}+\frac{23\!\cdots\!00}{90\!\cdots\!11}a^{9}+\frac{47\!\cdots\!34}{10\!\cdots\!79}a^{8}-\frac{83\!\cdots\!50}{27\!\cdots\!67}a^{7}-\frac{10\!\cdots\!53}{27\!\cdots\!33}a^{6}+\frac{49\!\cdots\!09}{27\!\cdots\!67}a^{5}+\frac{10\!\cdots\!63}{81\!\cdots\!01}a^{4}-\frac{13\!\cdots\!73}{27\!\cdots\!67}a^{3}-\frac{29\!\cdots\!77}{27\!\cdots\!67}a^{2}+\frac{11\!\cdots\!54}{27\!\cdots\!67}a+\frac{16\!\cdots\!63}{38\!\cdots\!77}$, $\frac{61\!\cdots\!75}{37\!\cdots\!23}a^{17}+\frac{40\!\cdots\!50}{11\!\cdots\!69}a^{16}-\frac{54\!\cdots\!50}{10\!\cdots\!21}a^{15}-\frac{12\!\cdots\!75}{10\!\cdots\!79}a^{14}+\frac{23\!\cdots\!80}{37\!\cdots\!23}a^{13}+\frac{15\!\cdots\!39}{11\!\cdots\!69}a^{12}-\frac{32\!\cdots\!80}{10\!\cdots\!79}a^{11}-\frac{71\!\cdots\!58}{10\!\cdots\!79}a^{10}+\frac{68\!\cdots\!00}{90\!\cdots\!11}a^{9}+\frac{14\!\cdots\!26}{10\!\cdots\!79}a^{8}-\frac{23\!\cdots\!50}{27\!\cdots\!67}a^{7}-\frac{32\!\cdots\!16}{27\!\cdots\!33}a^{6}+\frac{13\!\cdots\!95}{27\!\cdots\!67}a^{5}+\frac{32\!\cdots\!85}{81\!\cdots\!01}a^{4}-\frac{11\!\cdots\!50}{81\!\cdots\!01}a^{3}-\frac{10\!\cdots\!57}{27\!\cdots\!67}a^{2}+\frac{31\!\cdots\!90}{27\!\cdots\!67}a+\frac{23\!\cdots\!24}{38\!\cdots\!77}$, $\frac{69\!\cdots\!75}{13\!\cdots\!13}a^{17}+\frac{37\!\cdots\!05}{41\!\cdots\!39}a^{16}-\frac{61\!\cdots\!97}{37\!\cdots\!51}a^{15}-\frac{11\!\cdots\!25}{37\!\cdots\!49}a^{14}+\frac{26\!\cdots\!85}{13\!\cdots\!13}a^{13}+\frac{49\!\cdots\!32}{13\!\cdots\!13}a^{12}-\frac{37\!\cdots\!65}{37\!\cdots\!49}a^{11}-\frac{68\!\cdots\!12}{37\!\cdots\!49}a^{10}+\frac{81\!\cdots\!86}{33\!\cdots\!41}a^{9}+\frac{13\!\cdots\!14}{37\!\cdots\!49}a^{8}-\frac{29\!\cdots\!38}{10\!\cdots\!77}a^{7}-\frac{31\!\cdots\!79}{10\!\cdots\!23}a^{6}+\frac{18\!\cdots\!06}{10\!\cdots\!77}a^{5}+\frac{31\!\cdots\!60}{30\!\cdots\!31}a^{4}-\frac{50\!\cdots\!51}{10\!\cdots\!77}a^{3}-\frac{92\!\cdots\!18}{10\!\cdots\!77}a^{2}+\frac{43\!\cdots\!66}{10\!\cdots\!77}a+\frac{16\!\cdots\!97}{14\!\cdots\!87}$, $\frac{13\!\cdots\!75}{13\!\cdots\!13}a^{17}+\frac{86\!\cdots\!45}{41\!\cdots\!39}a^{16}-\frac{12\!\cdots\!29}{37\!\cdots\!51}a^{15}-\frac{25\!\cdots\!25}{37\!\cdots\!49}a^{14}+\frac{53\!\cdots\!25}{13\!\cdots\!13}a^{13}+\frac{33\!\cdots\!64}{41\!\cdots\!39}a^{12}-\frac{73\!\cdots\!35}{37\!\cdots\!49}a^{11}-\frac{15\!\cdots\!08}{37\!\cdots\!49}a^{10}+\frac{15\!\cdots\!14}{33\!\cdots\!41}a^{9}+\frac{30\!\cdots\!26}{37\!\cdots\!49}a^{8}-\frac{54\!\cdots\!62}{10\!\cdots\!77}a^{7}-\frac{69\!\cdots\!03}{10\!\cdots\!23}a^{6}+\frac{32\!\cdots\!60}{10\!\cdots\!77}a^{5}+\frac{70\!\cdots\!64}{30\!\cdots\!31}a^{4}-\frac{26\!\cdots\!37}{30\!\cdots\!31}a^{3}-\frac{22\!\cdots\!64}{10\!\cdots\!77}a^{2}+\frac{75\!\cdots\!94}{10\!\cdots\!77}a+\frac{56\!\cdots\!69}{14\!\cdots\!87}$, $\frac{14\!\cdots\!50}{37\!\cdots\!49}a^{17}+\frac{12\!\cdots\!30}{13\!\cdots\!13}a^{16}-\frac{47\!\cdots\!50}{37\!\cdots\!49}a^{15}-\frac{30\!\cdots\!50}{10\!\cdots\!77}a^{14}+\frac{14\!\cdots\!50}{10\!\cdots\!77}a^{13}+\frac{13\!\cdots\!40}{37\!\cdots\!49}a^{12}-\frac{75\!\cdots\!50}{10\!\cdots\!77}a^{11}-\frac{17\!\cdots\!20}{10\!\cdots\!77}a^{10}+\frac{63\!\cdots\!92}{37\!\cdots\!49}a^{9}+\frac{35\!\cdots\!00}{10\!\cdots\!77}a^{8}-\frac{19\!\cdots\!24}{10\!\cdots\!77}a^{7}-\frac{29\!\cdots\!90}{10\!\cdots\!77}a^{6}+\frac{11\!\cdots\!12}{10\!\cdots\!77}a^{5}+\frac{10\!\cdots\!00}{10\!\cdots\!77}a^{4}-\frac{29\!\cdots\!30}{10\!\cdots\!77}a^{3}-\frac{99\!\cdots\!20}{10\!\cdots\!77}a^{2}+\frac{24\!\cdots\!64}{10\!\cdots\!77}a+\frac{19\!\cdots\!69}{14\!\cdots\!87}$, $\frac{17\!\cdots\!32}{11\!\cdots\!53}a^{17}+\frac{13\!\cdots\!84}{37\!\cdots\!51}a^{16}-\frac{19\!\cdots\!97}{37\!\cdots\!51}a^{15}-\frac{14\!\cdots\!50}{12\!\cdots\!17}a^{14}+\frac{74\!\cdots\!64}{12\!\cdots\!17}a^{13}+\frac{18\!\cdots\!67}{13\!\cdots\!13}a^{12}-\frac{30\!\cdots\!61}{10\!\cdots\!23}a^{11}-\frac{23\!\cdots\!56}{33\!\cdots\!41}a^{10}+\frac{78\!\cdots\!01}{11\!\cdots\!47}a^{9}+\frac{11\!\cdots\!70}{81\!\cdots\!37}a^{8}-\frac{80\!\cdots\!31}{10\!\cdots\!23}a^{7}-\frac{11\!\cdots\!61}{10\!\cdots\!23}a^{6}+\frac{42\!\cdots\!76}{90\!\cdots\!93}a^{5}+\frac{35\!\cdots\!16}{90\!\cdots\!93}a^{4}-\frac{12\!\cdots\!43}{10\!\cdots\!77}a^{3}-\frac{11\!\cdots\!21}{30\!\cdots\!31}a^{2}+\frac{10\!\cdots\!43}{10\!\cdots\!77}a+\frac{84\!\cdots\!67}{14\!\cdots\!87}$, $\frac{77\!\cdots\!86}{12\!\cdots\!17}a^{17}-\frac{49\!\cdots\!62}{37\!\cdots\!51}a^{16}-\frac{78\!\cdots\!65}{37\!\cdots\!51}a^{15}+\frac{66\!\cdots\!14}{12\!\cdots\!17}a^{14}+\frac{10\!\cdots\!50}{41\!\cdots\!39}a^{13}-\frac{11\!\cdots\!93}{13\!\cdots\!13}a^{12}-\frac{51\!\cdots\!81}{37\!\cdots\!49}a^{11}+\frac{20\!\cdots\!40}{33\!\cdots\!41}a^{10}+\frac{12\!\cdots\!57}{33\!\cdots\!41}a^{9}-\frac{10\!\cdots\!40}{33\!\cdots\!41}a^{8}-\frac{46\!\cdots\!77}{10\!\cdots\!23}a^{7}+\frac{66\!\cdots\!37}{10\!\cdots\!23}a^{6}+\frac{23\!\cdots\!14}{90\!\cdots\!93}a^{5}-\frac{16\!\cdots\!02}{30\!\cdots\!31}a^{4}-\frac{37\!\cdots\!27}{10\!\cdots\!77}a^{3}+\frac{44\!\cdots\!65}{30\!\cdots\!31}a^{2}-\frac{83\!\cdots\!59}{10\!\cdots\!77}a-\frac{71\!\cdots\!75}{14\!\cdots\!87}$, $\frac{36\!\cdots\!27}{12\!\cdots\!17}a^{17}+\frac{23\!\cdots\!90}{37\!\cdots\!51}a^{16}-\frac{11\!\cdots\!77}{12\!\cdots\!17}a^{15}-\frac{28\!\cdots\!13}{13\!\cdots\!13}a^{14}+\frac{46\!\cdots\!60}{41\!\cdots\!39}a^{13}+\frac{10\!\cdots\!34}{41\!\cdots\!39}a^{12}-\frac{21\!\cdots\!75}{37\!\cdots\!49}a^{11}-\frac{41\!\cdots\!13}{33\!\cdots\!41}a^{10}+\frac{14\!\cdots\!39}{11\!\cdots\!47}a^{9}+\frac{82\!\cdots\!84}{33\!\cdots\!41}a^{8}-\frac{15\!\cdots\!04}{10\!\cdots\!23}a^{7}-\frac{69\!\cdots\!30}{33\!\cdots\!41}a^{6}+\frac{27\!\cdots\!78}{30\!\cdots\!31}a^{5}+\frac{21\!\cdots\!73}{30\!\cdots\!31}a^{4}-\frac{73\!\cdots\!81}{30\!\cdots\!31}a^{3}-\frac{20\!\cdots\!00}{30\!\cdots\!31}a^{2}+\frac{20\!\cdots\!02}{10\!\cdots\!77}a+\frac{16\!\cdots\!94}{14\!\cdots\!87}$, $\frac{20\!\cdots\!76}{11\!\cdots\!53}a^{17}+\frac{12\!\cdots\!33}{37\!\cdots\!51}a^{16}-\frac{22\!\cdots\!95}{37\!\cdots\!51}a^{15}-\frac{15\!\cdots\!39}{13\!\cdots\!13}a^{14}+\frac{89\!\cdots\!23}{12\!\cdots\!17}a^{13}+\frac{53\!\cdots\!20}{41\!\cdots\!39}a^{12}-\frac{37\!\cdots\!89}{10\!\cdots\!23}a^{11}-\frac{22\!\cdots\!14}{33\!\cdots\!41}a^{10}+\frac{30\!\cdots\!49}{33\!\cdots\!41}a^{9}+\frac{41\!\cdots\!50}{30\!\cdots\!69}a^{8}-\frac{11\!\cdots\!80}{10\!\cdots\!23}a^{7}-\frac{12\!\cdots\!96}{10\!\cdots\!23}a^{6}+\frac{20\!\cdots\!06}{30\!\cdots\!31}a^{5}+\frac{40\!\cdots\!46}{90\!\cdots\!93}a^{4}-\frac{59\!\cdots\!27}{30\!\cdots\!31}a^{3}-\frac{19\!\cdots\!26}{30\!\cdots\!31}a^{2}+\frac{18\!\cdots\!30}{10\!\cdots\!77}a+\frac{68\!\cdots\!99}{14\!\cdots\!87}$, $\frac{77\!\cdots\!07}{12\!\cdots\!17}a^{17}+\frac{42\!\cdots\!80}{37\!\cdots\!51}a^{16}-\frac{28\!\cdots\!08}{13\!\cdots\!13}a^{15}-\frac{46\!\cdots\!21}{12\!\cdots\!17}a^{14}+\frac{30\!\cdots\!56}{12\!\cdots\!17}a^{13}+\frac{60\!\cdots\!97}{13\!\cdots\!13}a^{12}-\frac{46\!\cdots\!79}{37\!\cdots\!49}a^{11}-\frac{20\!\cdots\!52}{90\!\cdots\!93}a^{10}+\frac{11\!\cdots\!50}{37\!\cdots\!49}a^{9}+\frac{15\!\cdots\!94}{33\!\cdots\!41}a^{8}-\frac{36\!\cdots\!33}{10\!\cdots\!23}a^{7}-\frac{14\!\cdots\!56}{37\!\cdots\!49}a^{6}+\frac{20\!\cdots\!73}{90\!\cdots\!93}a^{5}+\frac{11\!\cdots\!31}{90\!\cdots\!93}a^{4}-\frac{61\!\cdots\!92}{10\!\cdots\!77}a^{3}-\frac{38\!\cdots\!46}{30\!\cdots\!31}a^{2}+\frac{53\!\cdots\!29}{10\!\cdots\!77}a+\frac{37\!\cdots\!74}{14\!\cdots\!87}$, $\frac{11\!\cdots\!65}{12\!\cdots\!17}a^{17}+\frac{47\!\cdots\!74}{37\!\cdots\!51}a^{16}-\frac{11\!\cdots\!25}{37\!\cdots\!51}a^{15}-\frac{17\!\cdots\!71}{41\!\cdots\!39}a^{14}+\frac{44\!\cdots\!00}{12\!\cdots\!17}a^{13}+\frac{67\!\cdots\!43}{13\!\cdots\!13}a^{12}-\frac{20\!\cdots\!67}{11\!\cdots\!47}a^{11}-\frac{84\!\cdots\!13}{33\!\cdots\!41}a^{10}+\frac{14\!\cdots\!08}{33\!\cdots\!41}a^{9}+\frac{15\!\cdots\!05}{33\!\cdots\!41}a^{8}-\frac{50\!\cdots\!44}{10\!\cdots\!23}a^{7}-\frac{30\!\cdots\!87}{10\!\cdots\!23}a^{6}+\frac{87\!\cdots\!75}{30\!\cdots\!31}a^{5}+\frac{45\!\cdots\!70}{90\!\cdots\!93}a^{4}-\frac{72\!\cdots\!38}{10\!\cdots\!77}a^{3}+\frac{30\!\cdots\!66}{30\!\cdots\!31}a^{2}+\frac{55\!\cdots\!93}{10\!\cdots\!77}a-\frac{27\!\cdots\!90}{14\!\cdots\!87}$, $\frac{25\!\cdots\!01}{30\!\cdots\!69}a^{17}+\frac{29\!\cdots\!12}{12\!\cdots\!17}a^{16}-\frac{10\!\cdots\!06}{37\!\cdots\!51}a^{15}-\frac{10\!\cdots\!51}{13\!\cdots\!13}a^{14}+\frac{13\!\cdots\!38}{41\!\cdots\!39}a^{13}+\frac{37\!\cdots\!49}{41\!\cdots\!39}a^{12}-\frac{16\!\cdots\!75}{10\!\cdots\!23}a^{11}-\frac{51\!\cdots\!65}{11\!\cdots\!47}a^{10}+\frac{11\!\cdots\!72}{30\!\cdots\!31}a^{9}+\frac{28\!\cdots\!89}{30\!\cdots\!69}a^{8}-\frac{15\!\cdots\!40}{33\!\cdots\!41}a^{7}-\frac{83\!\cdots\!18}{10\!\cdots\!23}a^{6}+\frac{29\!\cdots\!33}{10\!\cdots\!77}a^{5}+\frac{92\!\cdots\!07}{30\!\cdots\!31}a^{4}-\frac{26\!\cdots\!02}{30\!\cdots\!31}a^{3}-\frac{92\!\cdots\!24}{30\!\cdots\!31}a^{2}+\frac{78\!\cdots\!29}{10\!\cdots\!77}a+\frac{62\!\cdots\!07}{14\!\cdots\!87}$, $\frac{41\!\cdots\!07}{37\!\cdots\!51}a^{17}+\frac{91\!\cdots\!83}{37\!\cdots\!51}a^{16}-\frac{45\!\cdots\!25}{12\!\cdots\!17}a^{15}-\frac{99\!\cdots\!51}{12\!\cdots\!17}a^{14}+\frac{53\!\cdots\!96}{12\!\cdots\!17}a^{13}+\frac{38\!\cdots\!75}{41\!\cdots\!39}a^{12}-\frac{73\!\cdots\!07}{33\!\cdots\!41}a^{11}-\frac{16\!\cdots\!46}{33\!\cdots\!41}a^{10}+\frac{18\!\cdots\!30}{37\!\cdots\!49}a^{9}+\frac{95\!\cdots\!98}{10\!\cdots\!23}a^{8}-\frac{59\!\cdots\!54}{10\!\cdots\!23}a^{7}-\frac{26\!\cdots\!19}{33\!\cdots\!41}a^{6}+\frac{31\!\cdots\!88}{90\!\cdots\!93}a^{5}+\frac{24\!\cdots\!99}{90\!\cdots\!93}a^{4}-\frac{28\!\cdots\!71}{30\!\cdots\!31}a^{3}-\frac{82\!\cdots\!86}{30\!\cdots\!31}a^{2}+\frac{79\!\cdots\!66}{10\!\cdots\!77}a+\frac{63\!\cdots\!61}{14\!\cdots\!87}$, $\frac{11\!\cdots\!98}{30\!\cdots\!69}a^{17}+\frac{44\!\cdots\!77}{37\!\cdots\!51}a^{16}-\frac{46\!\cdots\!40}{37\!\cdots\!51}a^{15}-\frac{15\!\cdots\!01}{41\!\cdots\!39}a^{14}+\frac{17\!\cdots\!71}{12\!\cdots\!17}a^{13}+\frac{18\!\cdots\!31}{41\!\cdots\!39}a^{12}-\frac{71\!\cdots\!63}{10\!\cdots\!23}a^{11}-\frac{71\!\cdots\!04}{33\!\cdots\!41}a^{10}+\frac{51\!\cdots\!25}{33\!\cdots\!41}a^{9}+\frac{11\!\cdots\!63}{30\!\cdots\!69}a^{8}-\frac{16\!\cdots\!00}{10\!\cdots\!23}a^{7}-\frac{30\!\cdots\!82}{10\!\cdots\!23}a^{6}+\frac{28\!\cdots\!30}{30\!\cdots\!31}a^{5}+\frac{79\!\cdots\!90}{90\!\cdots\!93}a^{4}-\frac{76\!\cdots\!94}{30\!\cdots\!31}a^{3}-\frac{17\!\cdots\!86}{30\!\cdots\!31}a^{2}+\frac{18\!\cdots\!82}{10\!\cdots\!77}a+\frac{13\!\cdots\!50}{14\!\cdots\!87}$, $\frac{18\!\cdots\!54}{11\!\cdots\!53}a^{17}+\frac{13\!\cdots\!58}{37\!\cdots\!51}a^{16}-\frac{19\!\cdots\!46}{37\!\cdots\!51}a^{15}-\frac{48\!\cdots\!47}{41\!\cdots\!39}a^{14}+\frac{77\!\cdots\!98}{12\!\cdots\!17}a^{13}+\frac{56\!\cdots\!63}{41\!\cdots\!39}a^{12}-\frac{31\!\cdots\!19}{10\!\cdots\!23}a^{11}-\frac{23\!\cdots\!14}{33\!\cdots\!41}a^{10}+\frac{27\!\cdots\!22}{37\!\cdots\!49}a^{9}+\frac{41\!\cdots\!33}{30\!\cdots\!69}a^{8}-\frac{85\!\cdots\!92}{10\!\cdots\!23}a^{7}-\frac{11\!\cdots\!24}{10\!\cdots\!23}a^{6}+\frac{15\!\cdots\!00}{30\!\cdots\!31}a^{5}+\frac{36\!\cdots\!42}{90\!\cdots\!93}a^{4}-\frac{40\!\cdots\!00}{30\!\cdots\!31}a^{3}-\frac{11\!\cdots\!52}{30\!\cdots\!31}a^{2}+\frac{11\!\cdots\!51}{10\!\cdots\!77}a+\frac{91\!\cdots\!14}{14\!\cdots\!87}$, $\frac{44\!\cdots\!46}{37\!\cdots\!51}a^{17}+\frac{12\!\cdots\!69}{12\!\cdots\!17}a^{16}-\frac{16\!\cdots\!87}{41\!\cdots\!39}a^{15}-\frac{42\!\cdots\!09}{12\!\cdots\!17}a^{14}+\frac{19\!\cdots\!95}{41\!\cdots\!39}a^{13}+\frac{58\!\cdots\!28}{13\!\cdots\!13}a^{12}-\frac{86\!\cdots\!21}{33\!\cdots\!41}a^{11}-\frac{27\!\cdots\!84}{11\!\cdots\!47}a^{10}+\frac{74\!\cdots\!13}{11\!\cdots\!47}a^{9}+\frac{50\!\cdots\!09}{10\!\cdots\!23}a^{8}-\frac{28\!\cdots\!44}{33\!\cdots\!41}a^{7}-\frac{49\!\cdots\!33}{11\!\cdots\!47}a^{6}+\frac{50\!\cdots\!48}{90\!\cdots\!93}a^{5}+\frac{46\!\cdots\!10}{30\!\cdots\!31}a^{4}-\frac{16\!\cdots\!84}{10\!\cdots\!77}a^{3}-\frac{49\!\cdots\!90}{30\!\cdots\!31}a^{2}+\frac{14\!\cdots\!44}{10\!\cdots\!77}a+\frac{12\!\cdots\!06}{14\!\cdots\!87}$, $\frac{62\!\cdots\!42}{37\!\cdots\!51}a^{17}+\frac{83\!\cdots\!07}{37\!\cdots\!51}a^{16}-\frac{20\!\cdots\!33}{37\!\cdots\!51}a^{15}-\frac{10\!\cdots\!62}{13\!\cdots\!13}a^{14}+\frac{27\!\cdots\!48}{41\!\cdots\!39}a^{13}+\frac{36\!\cdots\!52}{41\!\cdots\!39}a^{12}-\frac{11\!\cdots\!47}{33\!\cdots\!41}a^{11}-\frac{15\!\cdots\!50}{33\!\cdots\!41}a^{10}+\frac{28\!\cdots\!29}{33\!\cdots\!41}a^{9}+\frac{90\!\cdots\!16}{10\!\cdots\!23}a^{8}-\frac{10\!\cdots\!92}{10\!\cdots\!23}a^{7}-\frac{70\!\cdots\!14}{10\!\cdots\!23}a^{6}+\frac{20\!\cdots\!17}{30\!\cdots\!31}a^{5}+\frac{56\!\cdots\!42}{30\!\cdots\!31}a^{4}-\frac{61\!\cdots\!75}{30\!\cdots\!31}a^{3}-\frac{14\!\cdots\!95}{30\!\cdots\!31}a^{2}+\frac{19\!\cdots\!44}{10\!\cdots\!77}a+\frac{15\!\cdots\!39}{14\!\cdots\!87}$
| 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: | \( 18783533201005800 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 18783533201005800 \cdot 3}{2\cdot\sqrt{328368182120667332857143332667211818298338333}}\cr\approx \mathstrut & 0.407594141874199 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209)
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 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209, 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 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209);
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 - 333*x^16 + 39960*x^14 - 2147961*x^12 + 54976302*x^10 - 14845436*x^9 - 713903382*x^8 + 439970589*x^7 + 4664989341*x^6 - 4344963687*x^5 - 13662633690*x^4 + 15837166980*x^3 + 12296370321*x^2 - 15028897059*x - 873916209);
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)
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), \(\Q(\zeta_{9})^+\), 6.6.332334333.1, 9.9.80515213381214514081.2 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | $18$ | $18$ | $18$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(37\)
| 37.6.5.3 | $x^{6} + 333$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 37.6.5.3 | $x^{6} + 333$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 37.6.5.3 | $x^{6} + 333$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |