Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121)
gp: K = bnfinit(y^18 - 9*y^17 - 117*y^16 + 1140*y^15 + 4986*y^14 - 55566*y^13 - 92820*y^12 + 1333800*y^11 + 657540*y^10 - 16970004*y^9 + 658701*y^8 + 116979840*y^7 - 31572429*y^6 - 423568593*y^5 + 157061538*y^4 + 730986108*y^3 - 317010105*y^2 - 451758969*y + 250762121, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121)
\( x^{18} - 9 x^{17} - 117 x^{16} + 1140 x^{15} + 4986 x^{14} - 55566 x^{13} - 92820 x^{12} + \cdots + 250762121 \)
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: |
\(144544595663357009883982710483379948137213\)
\(\medspace = 3^{44}\cdot 7^{12}\cdot 13^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(193.49\) | 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: | $3^{22/9}7^{2/3}13^{1/2}\approx 193.4936735430652$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2457=3^{3}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2457}(64,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(844,·)$, $\chi_{2457}(781,·)$, $\chi_{2457}(2326,·)$, $\chi_{2457}(2263,·)$, $\chi_{2457}(25,·)$, $\chi_{2457}(883,·)$, $\chi_{2457}(1507,·)$, $\chi_{2457}(1444,·)$, $\chi_{2457}(1702,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(688,·)$, $\chi_{2457}(625,·)$, $\chi_{2457}(2419,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1600,·)$, $\chi_{2457}(1663,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{7}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{8}+\frac{3}{7}a^{5}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{9}+\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{133}a^{10}-\frac{8}{133}a^{9}-\frac{1}{19}a^{8}+\frac{9}{133}a^{7}-\frac{27}{133}a^{5}+\frac{59}{133}a^{4}-\frac{1}{133}a^{3}+\frac{59}{133}a^{2}-\frac{44}{133}a-\frac{2}{19}$, $\frac{1}{133}a^{11}+\frac{5}{133}a^{9}-\frac{9}{133}a^{8}-\frac{4}{133}a^{7}-\frac{8}{133}a^{6}-\frac{5}{133}a^{5}-\frac{6}{19}a^{4}+\frac{51}{133}a^{3}+\frac{10}{133}a^{2}+\frac{2}{19}a+\frac{3}{19}$, $\frac{1}{931}a^{12}+\frac{1}{931}a^{11}-\frac{3}{931}a^{10}-\frac{5}{133}a^{9}+\frac{5}{931}a^{8}-\frac{65}{931}a^{7}+\frac{6}{931}a^{6}+\frac{321}{931}a^{5}-\frac{235}{931}a^{4}+\frac{18}{133}a^{3}-\frac{201}{931}a^{2}+\frac{387}{931}a+\frac{3}{49}$, $\frac{1}{931}a^{13}+\frac{3}{931}a^{11}+\frac{3}{931}a^{10}+\frac{61}{931}a^{9}+\frac{3}{133}a^{8}-\frac{41}{931}a^{7}-\frac{1}{133}a^{6}+\frac{459}{931}a^{5}+\frac{137}{931}a^{4}+\frac{128}{931}a^{3}+\frac{47}{133}a^{2}+\frac{223}{931}a-\frac{400}{931}$, $\frac{1}{931}a^{14}+\frac{3}{133}a^{9}+\frac{5}{133}a^{8}-\frac{43}{931}a^{7}+\frac{6}{133}a^{6}+\frac{2}{7}a^{5}+\frac{6}{19}a^{4}+\frac{22}{133}a^{3}-\frac{54}{133}a^{2}-\frac{7}{19}a+\frac{410}{931}$, $\frac{1}{15827}a^{15}+\frac{1}{15827}a^{14}+\frac{3}{15827}a^{13}+\frac{5}{15827}a^{12}+\frac{4}{2261}a^{11}-\frac{13}{15827}a^{10}+\frac{757}{15827}a^{9}+\frac{1}{931}a^{8}+\frac{307}{15827}a^{7}+\frac{205}{15827}a^{6}-\frac{954}{2261}a^{5}-\frac{6812}{15827}a^{4}-\frac{1128}{15827}a^{3}-\frac{1985}{15827}a^{2}-\frac{6009}{15827}a+\frac{394}{931}$, $\frac{1}{15827}a^{16}+\frac{2}{15827}a^{14}+\frac{2}{15827}a^{13}+\frac{6}{15827}a^{12}-\frac{58}{15827}a^{11}-\frac{12}{15827}a^{10}-\frac{264}{15827}a^{9}-\frac{747}{15827}a^{8}+\frac{17}{931}a^{7}-\frac{202}{15827}a^{6}+\frac{3334}{15827}a^{5}-\frac{1031}{15827}a^{4}+\frac{243}{833}a^{3}+\frac{2249}{15827}a^{2}-\frac{179}{15827}a-\frac{164}{931}$, $\frac{1}{12\!\cdots\!69}a^{17}+\frac{25\!\cdots\!21}{12\!\cdots\!69}a^{16}-\frac{39\!\cdots\!91}{12\!\cdots\!69}a^{15}+\frac{23\!\cdots\!22}{12\!\cdots\!69}a^{14}+\frac{29\!\cdots\!64}{12\!\cdots\!69}a^{13}-\frac{61\!\cdots\!34}{12\!\cdots\!69}a^{12}+\frac{94\!\cdots\!86}{75\!\cdots\!57}a^{11}+\frac{25\!\cdots\!85}{12\!\cdots\!69}a^{10}+\frac{28\!\cdots\!24}{12\!\cdots\!69}a^{9}+\frac{72\!\cdots\!49}{12\!\cdots\!69}a^{8}-\frac{44\!\cdots\!03}{18\!\cdots\!67}a^{7}-\frac{57\!\cdots\!71}{12\!\cdots\!69}a^{6}+\frac{81\!\cdots\!22}{12\!\cdots\!69}a^{5}+\frac{17\!\cdots\!49}{12\!\cdots\!69}a^{4}+\frac{28\!\cdots\!90}{75\!\cdots\!57}a^{3}+\frac{38\!\cdots\!25}{12\!\cdots\!69}a^{2}-\frac{37\!\cdots\!48}{12\!\cdots\!69}a-\frac{33\!\cdots\!46}{75\!\cdots\!57}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $17$ |
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: |
$\frac{16\!\cdots\!98}{29\!\cdots\!83}a^{17}-\frac{56\!\cdots\!84}{12\!\cdots\!93}a^{16}-\frac{14\!\cdots\!68}{20\!\cdots\!81}a^{15}+\frac{12\!\cdots\!67}{20\!\cdots\!81}a^{14}+\frac{48\!\cdots\!31}{15\!\cdots\!57}a^{13}-\frac{59\!\cdots\!09}{20\!\cdots\!81}a^{12}-\frac{14\!\cdots\!99}{20\!\cdots\!81}a^{11}+\frac{14\!\cdots\!17}{20\!\cdots\!81}a^{10}+\frac{16\!\cdots\!71}{20\!\cdots\!81}a^{9}-\frac{10\!\cdots\!14}{12\!\cdots\!93}a^{8}-\frac{11\!\cdots\!09}{20\!\cdots\!81}a^{7}+\frac{12\!\cdots\!83}{20\!\cdots\!81}a^{6}+\frac{47\!\cdots\!11}{20\!\cdots\!81}a^{5}-\frac{45\!\cdots\!62}{20\!\cdots\!81}a^{4}-\frac{27\!\cdots\!91}{42\!\cdots\!69}a^{3}+\frac{74\!\cdots\!65}{20\!\cdots\!81}a^{2}+\frac{10\!\cdots\!17}{12\!\cdots\!93}a-\frac{22\!\cdots\!15}{12\!\cdots\!93}$, $\frac{96\!\cdots\!04}{29\!\cdots\!83}a^{17}-\frac{32\!\cdots\!37}{12\!\cdots\!93}a^{16}-\frac{84\!\cdots\!74}{20\!\cdots\!81}a^{15}+\frac{69\!\cdots\!30}{20\!\cdots\!81}a^{14}+\frac{30\!\cdots\!84}{15\!\cdots\!57}a^{13}-\frac{33\!\cdots\!80}{20\!\cdots\!81}a^{12}-\frac{93\!\cdots\!22}{20\!\cdots\!81}a^{11}+\frac{81\!\cdots\!71}{20\!\cdots\!81}a^{10}+\frac{11\!\cdots\!61}{20\!\cdots\!81}a^{9}-\frac{61\!\cdots\!24}{12\!\cdots\!93}a^{8}-\frac{91\!\cdots\!46}{20\!\cdots\!81}a^{7}+\frac{71\!\cdots\!30}{20\!\cdots\!81}a^{6}+\frac{43\!\cdots\!30}{20\!\cdots\!81}a^{5}-\frac{25\!\cdots\!60}{20\!\cdots\!81}a^{4}-\frac{24\!\cdots\!19}{42\!\cdots\!69}a^{3}+\frac{41\!\cdots\!54}{20\!\cdots\!81}a^{2}+\frac{76\!\cdots\!46}{12\!\cdots\!93}a-\frac{13\!\cdots\!16}{12\!\cdots\!93}$, $\frac{11\!\cdots\!16}{13\!\cdots\!99}a^{17}-\frac{95\!\cdots\!35}{13\!\cdots\!99}a^{16}-\frac{14\!\cdots\!34}{13\!\cdots\!99}a^{15}+\frac{12\!\cdots\!02}{13\!\cdots\!99}a^{14}+\frac{69\!\cdots\!24}{13\!\cdots\!99}a^{13}-\frac{60\!\cdots\!67}{13\!\cdots\!99}a^{12}-\frac{95\!\cdots\!56}{80\!\cdots\!47}a^{11}+\frac{14\!\cdots\!12}{13\!\cdots\!99}a^{10}+\frac{20\!\cdots\!15}{13\!\cdots\!99}a^{9}-\frac{18\!\cdots\!04}{13\!\cdots\!99}a^{8}-\frac{14\!\cdots\!60}{13\!\cdots\!99}a^{7}+\frac{13\!\cdots\!47}{13\!\cdots\!99}a^{6}+\frac{68\!\cdots\!32}{13\!\cdots\!99}a^{5}-\frac{48\!\cdots\!79}{13\!\cdots\!99}a^{4}-\frac{10\!\cdots\!95}{80\!\cdots\!47}a^{3}+\frac{80\!\cdots\!72}{13\!\cdots\!99}a^{2}+\frac{22\!\cdots\!07}{13\!\cdots\!99}a-\frac{25\!\cdots\!18}{80\!\cdots\!47}$, $\frac{13\!\cdots\!98}{12\!\cdots\!69}a^{17}-\frac{10\!\cdots\!87}{12\!\cdots\!69}a^{16}-\frac{16\!\cdots\!67}{12\!\cdots\!69}a^{15}+\frac{13\!\cdots\!57}{12\!\cdots\!69}a^{14}+\frac{79\!\cdots\!99}{12\!\cdots\!69}a^{13}-\frac{66\!\cdots\!94}{12\!\cdots\!69}a^{12}-\frac{11\!\cdots\!03}{75\!\cdots\!57}a^{11}+\frac{16\!\cdots\!75}{12\!\cdots\!69}a^{10}+\frac{34\!\cdots\!47}{18\!\cdots\!67}a^{9}-\frac{20\!\cdots\!39}{12\!\cdots\!69}a^{8}-\frac{19\!\cdots\!57}{12\!\cdots\!69}a^{7}+\frac{14\!\cdots\!54}{12\!\cdots\!69}a^{6}+\frac{93\!\cdots\!46}{12\!\cdots\!69}a^{5}-\frac{53\!\cdots\!61}{12\!\cdots\!69}a^{4}-\frac{21\!\cdots\!10}{10\!\cdots\!51}a^{3}+\frac{90\!\cdots\!10}{12\!\cdots\!69}a^{2}+\frac{28\!\cdots\!74}{12\!\cdots\!69}a-\frac{42\!\cdots\!84}{10\!\cdots\!51}$, $\frac{16\!\cdots\!15}{12\!\cdots\!69}a^{17}-\frac{74\!\cdots\!80}{67\!\cdots\!51}a^{16}-\frac{20\!\cdots\!89}{12\!\cdots\!69}a^{15}+\frac{18\!\cdots\!52}{12\!\cdots\!69}a^{14}+\frac{96\!\cdots\!30}{12\!\cdots\!69}a^{13}-\frac{88\!\cdots\!26}{12\!\cdots\!69}a^{12}-\frac{12\!\cdots\!97}{75\!\cdots\!57}a^{11}+\frac{21\!\cdots\!26}{12\!\cdots\!69}a^{10}+\frac{35\!\cdots\!30}{18\!\cdots\!67}a^{9}-\frac{27\!\cdots\!05}{12\!\cdots\!69}a^{8}-\frac{16\!\cdots\!20}{12\!\cdots\!69}a^{7}+\frac{19\!\cdots\!61}{12\!\cdots\!69}a^{6}+\frac{73\!\cdots\!26}{12\!\cdots\!69}a^{5}-\frac{70\!\cdots\!93}{12\!\cdots\!69}a^{4}-\frac{24\!\cdots\!98}{15\!\cdots\!93}a^{3}+\frac{11\!\cdots\!40}{12\!\cdots\!69}a^{2}+\frac{27\!\cdots\!83}{12\!\cdots\!69}a-\frac{50\!\cdots\!67}{10\!\cdots\!51}$, $\frac{90\!\cdots\!12}{12\!\cdots\!69}a^{17}-\frac{15\!\cdots\!29}{26\!\cdots\!81}a^{16}-\frac{15\!\cdots\!46}{18\!\cdots\!67}a^{15}+\frac{95\!\cdots\!95}{12\!\cdots\!69}a^{14}+\frac{10\!\cdots\!91}{26\!\cdots\!81}a^{13}-\frac{46\!\cdots\!42}{12\!\cdots\!69}a^{12}-\frac{68\!\cdots\!17}{75\!\cdots\!57}a^{11}+\frac{11\!\cdots\!17}{12\!\cdots\!69}a^{10}+\frac{19\!\cdots\!56}{18\!\cdots\!67}a^{9}-\frac{14\!\cdots\!79}{12\!\cdots\!69}a^{8}-\frac{95\!\cdots\!85}{12\!\cdots\!69}a^{7}+\frac{97\!\cdots\!16}{12\!\cdots\!69}a^{6}+\frac{42\!\cdots\!53}{12\!\cdots\!69}a^{5}-\frac{34\!\cdots\!38}{12\!\cdots\!69}a^{4}-\frac{39\!\cdots\!38}{44\!\cdots\!21}a^{3}+\frac{56\!\cdots\!70}{12\!\cdots\!69}a^{2}+\frac{13\!\cdots\!82}{12\!\cdots\!69}a-\frac{17\!\cdots\!94}{75\!\cdots\!57}$, $\frac{15\!\cdots\!84}{12\!\cdots\!69}a^{17}-\frac{19\!\cdots\!71}{18\!\cdots\!67}a^{16}-\frac{37\!\cdots\!62}{26\!\cdots\!81}a^{15}+\frac{24\!\cdots\!24}{18\!\cdots\!67}a^{14}+\frac{11\!\cdots\!46}{18\!\cdots\!67}a^{13}-\frac{82\!\cdots\!51}{12\!\cdots\!69}a^{12}-\frac{82\!\cdots\!72}{75\!\cdots\!57}a^{11}+\frac{14\!\cdots\!17}{95\!\cdots\!93}a^{10}+\frac{11\!\cdots\!82}{18\!\cdots\!67}a^{9}-\frac{23\!\cdots\!92}{12\!\cdots\!69}a^{8}+\frac{45\!\cdots\!86}{12\!\cdots\!69}a^{7}+\frac{14\!\cdots\!58}{12\!\cdots\!69}a^{6}-\frac{66\!\cdots\!37}{12\!\cdots\!69}a^{5}-\frac{46\!\cdots\!33}{12\!\cdots\!69}a^{4}+\frac{12\!\cdots\!69}{75\!\cdots\!57}a^{3}+\frac{58\!\cdots\!28}{12\!\cdots\!69}a^{2}-\frac{18\!\cdots\!80}{12\!\cdots\!69}a-\frac{86\!\cdots\!87}{75\!\cdots\!57}$, $\frac{52\!\cdots\!58}{12\!\cdots\!69}a^{17}-\frac{46\!\cdots\!26}{12\!\cdots\!69}a^{16}-\frac{63\!\cdots\!30}{12\!\cdots\!69}a^{15}+\frac{59\!\cdots\!36}{12\!\cdots\!69}a^{14}+\frac{61\!\cdots\!16}{26\!\cdots\!81}a^{13}-\frac{29\!\cdots\!52}{12\!\cdots\!69}a^{12}-\frac{43\!\cdots\!94}{75\!\cdots\!57}a^{11}+\frac{72\!\cdots\!34}{12\!\cdots\!69}a^{10}+\frac{11\!\cdots\!14}{12\!\cdots\!69}a^{9}-\frac{13\!\cdots\!54}{18\!\cdots\!67}a^{8}-\frac{12\!\cdots\!26}{12\!\cdots\!69}a^{7}+\frac{72\!\cdots\!99}{12\!\cdots\!69}a^{6}+\frac{86\!\cdots\!25}{12\!\cdots\!69}a^{5}-\frac{29\!\cdots\!68}{12\!\cdots\!69}a^{4}-\frac{85\!\cdots\!47}{39\!\cdots\!03}a^{3}+\frac{61\!\cdots\!34}{12\!\cdots\!69}a^{2}+\frac{30\!\cdots\!51}{12\!\cdots\!69}a-\frac{26\!\cdots\!35}{75\!\cdots\!57}$, $\frac{46\!\cdots\!84}{12\!\cdots\!69}a^{17}-\frac{40\!\cdots\!65}{12\!\cdots\!69}a^{16}-\frac{28\!\cdots\!84}{67\!\cdots\!51}a^{15}+\frac{51\!\cdots\!29}{12\!\cdots\!69}a^{14}+\frac{34\!\cdots\!21}{18\!\cdots\!67}a^{13}-\frac{24\!\cdots\!60}{12\!\cdots\!69}a^{12}-\frac{27\!\cdots\!19}{75\!\cdots\!57}a^{11}+\frac{58\!\cdots\!58}{12\!\cdots\!69}a^{10}+\frac{43\!\cdots\!39}{12\!\cdots\!69}a^{9}-\frac{71\!\cdots\!35}{12\!\cdots\!69}a^{8}-\frac{20\!\cdots\!91}{12\!\cdots\!69}a^{7}+\frac{46\!\cdots\!32}{12\!\cdots\!69}a^{6}+\frac{75\!\cdots\!39}{12\!\cdots\!69}a^{5}-\frac{15\!\cdots\!08}{12\!\cdots\!69}a^{4}-\frac{92\!\cdots\!43}{39\!\cdots\!03}a^{3}+\frac{34\!\cdots\!19}{18\!\cdots\!67}a^{2}+\frac{52\!\cdots\!52}{12\!\cdots\!69}a-\frac{72\!\cdots\!54}{75\!\cdots\!57}$, $\frac{96\!\cdots\!72}{12\!\cdots\!69}a^{17}-\frac{88\!\cdots\!82}{12\!\cdots\!69}a^{16}-\frac{11\!\cdots\!16}{12\!\cdots\!69}a^{15}+\frac{11\!\cdots\!97}{12\!\cdots\!69}a^{14}+\frac{67\!\cdots\!29}{18\!\cdots\!67}a^{13}-\frac{54\!\cdots\!76}{12\!\cdots\!69}a^{12}-\frac{51\!\cdots\!25}{75\!\cdots\!57}a^{11}+\frac{13\!\cdots\!49}{12\!\cdots\!69}a^{10}+\frac{34\!\cdots\!76}{67\!\cdots\!51}a^{9}-\frac{16\!\cdots\!23}{12\!\cdots\!69}a^{8}-\frac{11\!\cdots\!67}{12\!\cdots\!69}a^{7}+\frac{11\!\cdots\!85}{12\!\cdots\!69}a^{6}-\frac{36\!\cdots\!02}{12\!\cdots\!69}a^{5}-\frac{44\!\cdots\!36}{12\!\cdots\!69}a^{4}-\frac{10\!\cdots\!51}{75\!\cdots\!57}a^{3}+\frac{40\!\cdots\!58}{67\!\cdots\!51}a^{2}+\frac{10\!\cdots\!22}{12\!\cdots\!69}a-\frac{23\!\cdots\!23}{75\!\cdots\!57}$, $\frac{31\!\cdots\!04}{18\!\cdots\!67}a^{17}-\frac{17\!\cdots\!37}{12\!\cdots\!69}a^{16}-\frac{27\!\cdots\!12}{12\!\cdots\!69}a^{15}+\frac{22\!\cdots\!02}{12\!\cdots\!69}a^{14}+\frac{18\!\cdots\!74}{18\!\cdots\!67}a^{13}-\frac{57\!\cdots\!19}{67\!\cdots\!51}a^{12}-\frac{18\!\cdots\!38}{75\!\cdots\!57}a^{11}+\frac{26\!\cdots\!16}{12\!\cdots\!69}a^{10}+\frac{20\!\cdots\!85}{67\!\cdots\!51}a^{9}-\frac{33\!\cdots\!85}{12\!\cdots\!69}a^{8}-\frac{43\!\cdots\!72}{18\!\cdots\!67}a^{7}+\frac{23\!\cdots\!94}{12\!\cdots\!69}a^{6}+\frac{14\!\cdots\!71}{12\!\cdots\!69}a^{5}-\frac{83\!\cdots\!52}{12\!\cdots\!69}a^{4}-\frac{46\!\cdots\!34}{15\!\cdots\!93}a^{3}+\frac{13\!\cdots\!57}{12\!\cdots\!69}a^{2}+\frac{43\!\cdots\!16}{12\!\cdots\!69}a-\frac{62\!\cdots\!56}{10\!\cdots\!51}$, $\frac{20\!\cdots\!28}{12\!\cdots\!69}a^{17}-\frac{99\!\cdots\!61}{67\!\cdots\!51}a^{16}-\frac{22\!\cdots\!69}{12\!\cdots\!69}a^{15}+\frac{23\!\cdots\!17}{12\!\cdots\!69}a^{14}+\frac{87\!\cdots\!26}{12\!\cdots\!69}a^{13}-\frac{10\!\cdots\!81}{12\!\cdots\!69}a^{12}-\frac{79\!\cdots\!93}{75\!\cdots\!57}a^{11}+\frac{24\!\cdots\!97}{12\!\cdots\!69}a^{10}+\frac{28\!\cdots\!43}{67\!\cdots\!51}a^{9}-\frac{40\!\cdots\!34}{18\!\cdots\!67}a^{8}+\frac{37\!\cdots\!75}{12\!\cdots\!69}a^{7}+\frac{17\!\cdots\!17}{12\!\cdots\!69}a^{6}-\frac{28\!\cdots\!70}{12\!\cdots\!69}a^{5}-\frac{55\!\cdots\!34}{12\!\cdots\!69}a^{4}+\frac{19\!\cdots\!62}{75\!\cdots\!57}a^{3}+\frac{81\!\cdots\!31}{12\!\cdots\!69}a^{2}+\frac{45\!\cdots\!18}{12\!\cdots\!69}a-\frac{20\!\cdots\!47}{75\!\cdots\!57}$, $\frac{25\!\cdots\!94}{18\!\cdots\!67}a^{17}-\frac{15\!\cdots\!33}{12\!\cdots\!69}a^{16}-\frac{11\!\cdots\!33}{67\!\cdots\!51}a^{15}+\frac{19\!\cdots\!42}{12\!\cdots\!69}a^{14}+\frac{99\!\cdots\!87}{12\!\cdots\!69}a^{13}-\frac{96\!\cdots\!47}{12\!\cdots\!69}a^{12}-\frac{12\!\cdots\!07}{75\!\cdots\!57}a^{11}+\frac{23\!\cdots\!76}{12\!\cdots\!69}a^{10}+\frac{22\!\cdots\!89}{12\!\cdots\!69}a^{9}-\frac{30\!\cdots\!22}{12\!\cdots\!69}a^{8}-\frac{13\!\cdots\!38}{12\!\cdots\!69}a^{7}+\frac{21\!\cdots\!27}{12\!\cdots\!69}a^{6}+\frac{50\!\cdots\!61}{12\!\cdots\!69}a^{5}-\frac{80\!\cdots\!13}{12\!\cdots\!69}a^{4}-\frac{49\!\cdots\!75}{39\!\cdots\!03}a^{3}+\frac{13\!\cdots\!40}{12\!\cdots\!69}a^{2}+\frac{39\!\cdots\!52}{18\!\cdots\!67}a-\frac{42\!\cdots\!66}{75\!\cdots\!57}$, $\frac{41\!\cdots\!29}{67\!\cdots\!51}a^{17}-\frac{56\!\cdots\!78}{12\!\cdots\!69}a^{16}-\frac{10\!\cdots\!92}{12\!\cdots\!69}a^{15}+\frac{74\!\cdots\!06}{12\!\cdots\!69}a^{14}+\frac{55\!\cdots\!11}{12\!\cdots\!69}a^{13}-\frac{37\!\cdots\!10}{12\!\cdots\!69}a^{12}-\frac{88\!\cdots\!36}{75\!\cdots\!57}a^{11}+\frac{13\!\cdots\!66}{18\!\cdots\!67}a^{10}+\frac{22\!\cdots\!28}{12\!\cdots\!69}a^{9}-\frac{13\!\cdots\!52}{12\!\cdots\!69}a^{8}-\frac{19\!\cdots\!91}{12\!\cdots\!69}a^{7}+\frac{10\!\cdots\!81}{12\!\cdots\!69}a^{6}+\frac{98\!\cdots\!23}{12\!\cdots\!69}a^{5}-\frac{38\!\cdots\!95}{12\!\cdots\!69}a^{4}-\frac{14\!\cdots\!10}{75\!\cdots\!57}a^{3}+\frac{68\!\cdots\!38}{12\!\cdots\!69}a^{2}+\frac{24\!\cdots\!82}{12\!\cdots\!69}a-\frac{23\!\cdots\!01}{75\!\cdots\!57}$, $\frac{15\!\cdots\!94}{12\!\cdots\!69}a^{17}-\frac{13\!\cdots\!63}{12\!\cdots\!69}a^{16}-\frac{18\!\cdots\!72}{12\!\cdots\!69}a^{15}+\frac{16\!\cdots\!14}{12\!\cdots\!69}a^{14}+\frac{88\!\cdots\!92}{12\!\cdots\!69}a^{13}-\frac{82\!\cdots\!06}{12\!\cdots\!69}a^{12}-\frac{11\!\cdots\!46}{75\!\cdots\!57}a^{11}+\frac{20\!\cdots\!37}{12\!\cdots\!69}a^{10}+\frac{17\!\cdots\!59}{95\!\cdots\!93}a^{9}-\frac{26\!\cdots\!50}{12\!\cdots\!69}a^{8}-\frac{16\!\cdots\!29}{12\!\cdots\!69}a^{7}+\frac{18\!\cdots\!35}{12\!\cdots\!69}a^{6}+\frac{77\!\cdots\!60}{12\!\cdots\!69}a^{5}-\frac{66\!\cdots\!99}{12\!\cdots\!69}a^{4}-\frac{12\!\cdots\!31}{75\!\cdots\!57}a^{3}+\frac{10\!\cdots\!37}{12\!\cdots\!69}a^{2}+\frac{38\!\cdots\!29}{18\!\cdots\!67}a-\frac{33\!\cdots\!97}{75\!\cdots\!57}$, $\frac{20\!\cdots\!22}{12\!\cdots\!69}a^{17}-\frac{17\!\cdots\!07}{12\!\cdots\!69}a^{16}-\frac{24\!\cdots\!98}{12\!\cdots\!69}a^{15}+\frac{22\!\cdots\!49}{12\!\cdots\!69}a^{14}+\frac{10\!\cdots\!57}{12\!\cdots\!69}a^{13}-\frac{11\!\cdots\!92}{12\!\cdots\!69}a^{12}-\frac{10\!\cdots\!73}{75\!\cdots\!57}a^{11}+\frac{37\!\cdots\!73}{18\!\cdots\!67}a^{10}+\frac{57\!\cdots\!17}{12\!\cdots\!69}a^{9}-\frac{31\!\cdots\!97}{12\!\cdots\!69}a^{8}+\frac{92\!\cdots\!39}{67\!\cdots\!51}a^{7}+\frac{17\!\cdots\!58}{12\!\cdots\!69}a^{6}-\frac{17\!\cdots\!66}{12\!\cdots\!69}a^{5}-\frac{40\!\cdots\!05}{12\!\cdots\!69}a^{4}+\frac{29\!\cdots\!61}{75\!\cdots\!57}a^{3}+\frac{55\!\cdots\!61}{26\!\cdots\!81}a^{2}-\frac{42\!\cdots\!98}{12\!\cdots\!69}a+\frac{54\!\cdots\!57}{75\!\cdots\!57}$, $\frac{10\!\cdots\!42}{12\!\cdots\!69}a^{17}-\frac{11\!\cdots\!76}{12\!\cdots\!69}a^{16}-\frac{10\!\cdots\!28}{12\!\cdots\!69}a^{15}+\frac{14\!\cdots\!87}{12\!\cdots\!69}a^{14}+\frac{31\!\cdots\!62}{12\!\cdots\!69}a^{13}-\frac{65\!\cdots\!25}{12\!\cdots\!69}a^{12}+\frac{17\!\cdots\!68}{75\!\cdots\!57}a^{11}+\frac{20\!\cdots\!75}{18\!\cdots\!67}a^{10}-\frac{15\!\cdots\!24}{12\!\cdots\!69}a^{9}-\frac{32\!\cdots\!60}{26\!\cdots\!81}a^{8}+\frac{26\!\cdots\!21}{12\!\cdots\!69}a^{7}+\frac{86\!\cdots\!46}{12\!\cdots\!69}a^{6}-\frac{17\!\cdots\!06}{12\!\cdots\!69}a^{5}-\frac{19\!\cdots\!72}{12\!\cdots\!69}a^{4}+\frac{16\!\cdots\!24}{44\!\cdots\!21}a^{3}+\frac{44\!\cdots\!48}{12\!\cdots\!69}a^{2}-\frac{42\!\cdots\!81}{12\!\cdots\!69}a+\frac{10\!\cdots\!87}{75\!\cdots\!57}$
| 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: | \( 753247107887002.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 753247107887002.5 \cdot 3}{2\cdot\sqrt{144544595663357009883982710483379948137213}}\cr\approx \mathstrut & 0.779054292045667 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121)
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 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121, 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 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121);
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 - 9*x^17 - 117*x^16 + 1140*x^15 + 4986*x^14 - 55566*x^13 - 92820*x^12 + 1333800*x^11 + 657540*x^10 - 16970004*x^9 + 658701*x^8 + 116979840*x^7 - 31572429*x^6 - 423568593*x^5 + 157061538*x^4 + 730986108*x^3 - 317010105*x^2 - 451758969*x + 250762121);
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
| \(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 6.6.14414517.1, 9.9.3691950281939241.2 |
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 | $18$ | R | $18$ | R | ${\href{/padicField/17.1.0.1}{1} }^{18}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | $18$ |
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 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(7\)
| 7.18.12.2 | $x^{18} - 14 x^{15} + 441 x^{12} + 3773 x^{9} - 91238 x^{6} + 201684 x^{3} + 1058841$ | $3$ | $6$ | $12$ | $C_{18}$ | $[\ ]_{3}^{6}$ |
|
\(13\)
| 13.18.9.1 | $x^{18} + 1170 x^{17} + 608517 x^{16} + 184669680 x^{15} + 36042230484 x^{14} + 4692692080464 x^{13} + 407793261316444 x^{12} + 22833205275255672 x^{11} + 750031142087897694 x^{10} + 11196577827794288770 x^{9} + 9750461205950186580 x^{8} + 3863714899398059352 x^{7} + 1170776365765219708 x^{6} + 9183655224695901156 x^{5} + 136076384268316458696 x^{4} + 146209355090752705280 x^{3} + 170259556431855716025 x^{2} + 163704388102720431984 x + 129853841096201133292$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |