Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191)
gp: K = bnfinit(y^18 - 7*y^17 - 32*y^16 + 294*y^15 + 275*y^14 - 4779*y^13 + 1076*y^12 + 37663*y^11 - 29140*y^10 - 146783*y^9 + 161417*y^8 + 251979*y^7 - 350891*y^6 - 118682*y^5 + 269024*y^4 - 41471*y^3 - 43388*y^2 + 12154*y - 191, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191)
\( x^{18} - 7 x^{17} - 32 x^{16} + 294 x^{15} + 275 x^{14} - 4779 x^{13} + 1076 x^{12} + 37663 x^{11} + \cdots - 191 \)
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: |
\(34205654728777159191037355893457\)
\(\medspace = 17^{9}\cdot 19^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(56.48\) | 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: | $17^{1/2}19^{8/9}\approx 56.47995320269716$ | ||
| Ramified primes: |
\(17\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(323=17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{323}(256,·)$, $\chi_{323}(1,·)$, $\chi_{323}(137,·)$, $\chi_{323}(271,·)$, $\chi_{323}(16,·)$, $\chi_{323}(273,·)$, $\chi_{323}(220,·)$, $\chi_{323}(290,·)$, $\chi_{323}(35,·)$, $\chi_{323}(101,·)$, $\chi_{323}(169,·)$, $\chi_{323}(237,·)$, $\chi_{323}(239,·)$, $\chi_{323}(305,·)$, $\chi_{323}(118,·)$, $\chi_{323}(120,·)$, $\chi_{323}(188,·)$, $\chi_{323}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{26\!\cdots\!07}a^{17}-\frac{28\!\cdots\!81}{26\!\cdots\!07}a^{16}+\frac{71\!\cdots\!08}{26\!\cdots\!07}a^{15}+\frac{81\!\cdots\!61}{26\!\cdots\!07}a^{14}+\frac{12\!\cdots\!23}{26\!\cdots\!07}a^{13}-\frac{10\!\cdots\!70}{26\!\cdots\!07}a^{12}+\frac{10\!\cdots\!51}{26\!\cdots\!07}a^{11}-\frac{33\!\cdots\!14}{26\!\cdots\!07}a^{10}+\frac{12\!\cdots\!72}{26\!\cdots\!07}a^{9}-\frac{54\!\cdots\!74}{26\!\cdots\!07}a^{8}+\frac{35\!\cdots\!51}{26\!\cdots\!07}a^{7}+\frac{77\!\cdots\!73}{26\!\cdots\!07}a^{6}-\frac{12\!\cdots\!59}{26\!\cdots\!07}a^{5}-\frac{57\!\cdots\!48}{26\!\cdots\!07}a^{4}+\frac{30\!\cdots\!22}{26\!\cdots\!07}a^{3}+\frac{84\!\cdots\!98}{17\!\cdots\!57}a^{2}-\frac{75\!\cdots\!94}{26\!\cdots\!07}a+\frac{66\!\cdots\!08}{26\!\cdots\!07}$
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
Trivial group, which has order $1$ (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: |
$\frac{12\!\cdots\!08}{26\!\cdots\!07}a^{17}-\frac{73\!\cdots\!06}{26\!\cdots\!07}a^{16}-\frac{45\!\cdots\!36}{26\!\cdots\!07}a^{15}+\frac{31\!\cdots\!57}{26\!\cdots\!07}a^{14}+\frac{62\!\cdots\!23}{26\!\cdots\!07}a^{13}-\frac{51\!\cdots\!94}{26\!\cdots\!07}a^{12}-\frac{36\!\cdots\!17}{26\!\cdots\!07}a^{11}+\frac{41\!\cdots\!85}{26\!\cdots\!07}a^{10}+\frac{53\!\cdots\!82}{26\!\cdots\!07}a^{9}-\frac{17\!\cdots\!19}{26\!\cdots\!07}a^{8}+\frac{25\!\cdots\!57}{26\!\cdots\!07}a^{7}+\frac{32\!\cdots\!64}{26\!\cdots\!07}a^{6}-\frac{87\!\cdots\!03}{26\!\cdots\!07}a^{5}-\frac{23\!\cdots\!35}{26\!\cdots\!07}a^{4}+\frac{58\!\cdots\!34}{26\!\cdots\!07}a^{3}+\frac{23\!\cdots\!60}{17\!\cdots\!57}a^{2}-\frac{20\!\cdots\!40}{26\!\cdots\!07}a+\frac{43\!\cdots\!05}{26\!\cdots\!07}$, $\frac{23\!\cdots\!80}{26\!\cdots\!07}a^{17}-\frac{15\!\cdots\!53}{26\!\cdots\!07}a^{16}-\frac{84\!\cdots\!06}{26\!\cdots\!07}a^{15}+\frac{65\!\cdots\!13}{26\!\cdots\!07}a^{14}+\frac{99\!\cdots\!03}{26\!\cdots\!07}a^{13}-\frac{10\!\cdots\!88}{26\!\cdots\!07}a^{12}-\frac{30\!\cdots\!41}{26\!\cdots\!07}a^{11}+\frac{88\!\cdots\!00}{26\!\cdots\!07}a^{10}-\frac{23\!\cdots\!73}{26\!\cdots\!07}a^{9}-\frac{35\!\cdots\!69}{26\!\cdots\!07}a^{8}+\frac{19\!\cdots\!29}{26\!\cdots\!07}a^{7}+\frac{68\!\cdots\!69}{26\!\cdots\!07}a^{6}-\frac{47\!\cdots\!42}{26\!\cdots\!07}a^{5}-\frac{48\!\cdots\!11}{26\!\cdots\!07}a^{4}+\frac{36\!\cdots\!40}{26\!\cdots\!07}a^{3}+\frac{35\!\cdots\!03}{17\!\cdots\!57}a^{2}-\frac{54\!\cdots\!06}{26\!\cdots\!07}a+\frac{62\!\cdots\!75}{26\!\cdots\!07}$, $\frac{13\!\cdots\!00}{26\!\cdots\!07}a^{17}-\frac{97\!\cdots\!91}{26\!\cdots\!07}a^{16}-\frac{43\!\cdots\!70}{26\!\cdots\!07}a^{15}+\frac{40\!\cdots\!23}{26\!\cdots\!07}a^{14}+\frac{37\!\cdots\!09}{26\!\cdots\!07}a^{13}-\frac{66\!\cdots\!26}{26\!\cdots\!07}a^{12}+\frac{16\!\cdots\!13}{26\!\cdots\!07}a^{11}+\frac{52\!\cdots\!77}{26\!\cdots\!07}a^{10}-\frac{40\!\cdots\!76}{26\!\cdots\!07}a^{9}-\frac{20\!\cdots\!16}{26\!\cdots\!07}a^{8}+\frac{22\!\cdots\!87}{26\!\cdots\!07}a^{7}+\frac{36\!\cdots\!55}{26\!\cdots\!07}a^{6}-\frac{48\!\cdots\!52}{26\!\cdots\!07}a^{5}-\frac{20\!\cdots\!18}{26\!\cdots\!07}a^{4}+\frac{36\!\cdots\!63}{26\!\cdots\!07}a^{3}-\frac{10\!\cdots\!86}{17\!\cdots\!57}a^{2}-\frac{57\!\cdots\!16}{26\!\cdots\!07}a+\frac{60\!\cdots\!60}{26\!\cdots\!07}$, $\frac{19\!\cdots\!62}{26\!\cdots\!07}a^{17}-\frac{12\!\cdots\!14}{26\!\cdots\!07}a^{16}-\frac{69\!\cdots\!56}{26\!\cdots\!07}a^{15}+\frac{51\!\cdots\!90}{26\!\cdots\!07}a^{14}+\frac{87\!\cdots\!44}{26\!\cdots\!07}a^{13}-\frac{85\!\cdots\!36}{26\!\cdots\!07}a^{12}-\frac{37\!\cdots\!60}{26\!\cdots\!07}a^{11}+\frac{69\!\cdots\!45}{26\!\cdots\!07}a^{10}-\frac{87\!\cdots\!67}{26\!\cdots\!07}a^{9}-\frac{29\!\cdots\!51}{26\!\cdots\!07}a^{8}+\frac{11\!\cdots\!46}{26\!\cdots\!07}a^{7}+\frac{57\!\cdots\!68}{26\!\cdots\!07}a^{6}-\frac{31\!\cdots\!08}{26\!\cdots\!07}a^{5}-\frac{44\!\cdots\!30}{26\!\cdots\!07}a^{4}+\frac{26\!\cdots\!79}{26\!\cdots\!07}a^{3}+\frac{55\!\cdots\!43}{17\!\cdots\!57}a^{2}-\frac{45\!\cdots\!13}{26\!\cdots\!07}a-\frac{12\!\cdots\!52}{26\!\cdots\!07}$, $\frac{88\!\cdots\!32}{26\!\cdots\!07}a^{17}-\frac{51\!\cdots\!57}{26\!\cdots\!07}a^{16}-\frac{35\!\cdots\!44}{26\!\cdots\!07}a^{15}+\frac{22\!\cdots\!69}{26\!\cdots\!07}a^{14}+\frac{53\!\cdots\!55}{26\!\cdots\!07}a^{13}-\frac{37\!\cdots\!95}{26\!\cdots\!07}a^{12}-\frac{38\!\cdots\!59}{26\!\cdots\!07}a^{11}+\frac{31\!\cdots\!76}{26\!\cdots\!07}a^{10}+\frac{12\!\cdots\!10}{26\!\cdots\!07}a^{9}-\frac{13\!\cdots\!29}{26\!\cdots\!07}a^{8}-\frac{14\!\cdots\!49}{26\!\cdots\!07}a^{7}+\frac{30\!\cdots\!69}{26\!\cdots\!07}a^{6}-\frac{12\!\cdots\!43}{26\!\cdots\!07}a^{5}-\frac{29\!\cdots\!10}{26\!\cdots\!07}a^{4}+\frac{25\!\cdots\!76}{26\!\cdots\!07}a^{3}+\frac{65\!\cdots\!25}{17\!\cdots\!57}a^{2}-\frac{10\!\cdots\!32}{26\!\cdots\!07}a-\frac{27\!\cdots\!69}{26\!\cdots\!07}$, $\frac{23\!\cdots\!80}{26\!\cdots\!07}a^{17}-\frac{15\!\cdots\!53}{26\!\cdots\!07}a^{16}-\frac{84\!\cdots\!06}{26\!\cdots\!07}a^{15}+\frac{65\!\cdots\!13}{26\!\cdots\!07}a^{14}+\frac{99\!\cdots\!03}{26\!\cdots\!07}a^{13}-\frac{10\!\cdots\!88}{26\!\cdots\!07}a^{12}-\frac{30\!\cdots\!41}{26\!\cdots\!07}a^{11}+\frac{88\!\cdots\!00}{26\!\cdots\!07}a^{10}-\frac{23\!\cdots\!73}{26\!\cdots\!07}a^{9}-\frac{35\!\cdots\!69}{26\!\cdots\!07}a^{8}+\frac{19\!\cdots\!29}{26\!\cdots\!07}a^{7}+\frac{68\!\cdots\!69}{26\!\cdots\!07}a^{6}-\frac{47\!\cdots\!42}{26\!\cdots\!07}a^{5}-\frac{48\!\cdots\!11}{26\!\cdots\!07}a^{4}+\frac{36\!\cdots\!40}{26\!\cdots\!07}a^{3}+\frac{35\!\cdots\!03}{17\!\cdots\!57}a^{2}-\frac{54\!\cdots\!06}{26\!\cdots\!07}a+\frac{35\!\cdots\!68}{26\!\cdots\!07}$, $\frac{13\!\cdots\!00}{26\!\cdots\!07}a^{17}-\frac{97\!\cdots\!91}{26\!\cdots\!07}a^{16}-\frac{43\!\cdots\!70}{26\!\cdots\!07}a^{15}+\frac{40\!\cdots\!23}{26\!\cdots\!07}a^{14}+\frac{37\!\cdots\!09}{26\!\cdots\!07}a^{13}-\frac{66\!\cdots\!26}{26\!\cdots\!07}a^{12}+\frac{16\!\cdots\!13}{26\!\cdots\!07}a^{11}+\frac{52\!\cdots\!77}{26\!\cdots\!07}a^{10}-\frac{40\!\cdots\!76}{26\!\cdots\!07}a^{9}-\frac{20\!\cdots\!16}{26\!\cdots\!07}a^{8}+\frac{22\!\cdots\!87}{26\!\cdots\!07}a^{7}+\frac{36\!\cdots\!55}{26\!\cdots\!07}a^{6}-\frac{48\!\cdots\!52}{26\!\cdots\!07}a^{5}-\frac{20\!\cdots\!18}{26\!\cdots\!07}a^{4}+\frac{36\!\cdots\!63}{26\!\cdots\!07}a^{3}-\frac{10\!\cdots\!86}{17\!\cdots\!57}a^{2}-\frac{57\!\cdots\!16}{26\!\cdots\!07}a+\frac{87\!\cdots\!67}{26\!\cdots\!07}$, $\frac{86\!\cdots\!38}{26\!\cdots\!07}a^{17}-\frac{47\!\cdots\!52}{26\!\cdots\!07}a^{16}-\frac{35\!\cdots\!30}{26\!\cdots\!07}a^{15}+\frac{20\!\cdots\!04}{26\!\cdots\!07}a^{14}+\frac{58\!\cdots\!42}{26\!\cdots\!07}a^{13}-\frac{35\!\cdots\!52}{26\!\cdots\!07}a^{12}-\frac{48\!\cdots\!66}{26\!\cdots\!07}a^{11}+\frac{30\!\cdots\!08}{26\!\cdots\!07}a^{10}+\frac{22\!\cdots\!76}{26\!\cdots\!07}a^{9}-\frac{13\!\cdots\!94}{26\!\cdots\!07}a^{8}-\frac{55\!\cdots\!72}{26\!\cdots\!07}a^{7}+\frac{30\!\cdots\!95}{26\!\cdots\!07}a^{6}+\frac{79\!\cdots\!37}{26\!\cdots\!07}a^{5}-\frac{31\!\cdots\!83}{26\!\cdots\!07}a^{4}-\frac{59\!\cdots\!57}{26\!\cdots\!07}a^{3}+\frac{88\!\cdots\!15}{17\!\cdots\!57}a^{2}+\frac{10\!\cdots\!91}{26\!\cdots\!07}a-\frac{13\!\cdots\!96}{26\!\cdots\!07}$, $\frac{25\!\cdots\!78}{26\!\cdots\!07}a^{17}-\frac{16\!\cdots\!35}{26\!\cdots\!07}a^{16}-\frac{88\!\cdots\!48}{26\!\cdots\!07}a^{15}+\frac{69\!\cdots\!25}{26\!\cdots\!07}a^{14}+\frac{10\!\cdots\!49}{26\!\cdots\!07}a^{13}-\frac{11\!\cdots\!23}{26\!\cdots\!07}a^{12}-\frac{28\!\cdots\!09}{26\!\cdots\!07}a^{11}+\frac{92\!\cdots\!39}{26\!\cdots\!07}a^{10}-\frac{28\!\cdots\!17}{26\!\cdots\!07}a^{9}-\frac{37\!\cdots\!00}{26\!\cdots\!07}a^{8}+\frac{22\!\cdots\!27}{26\!\cdots\!07}a^{7}+\frac{71\!\cdots\!71}{26\!\cdots\!07}a^{6}-\frac{52\!\cdots\!35}{26\!\cdots\!07}a^{5}-\frac{50\!\cdots\!81}{26\!\cdots\!07}a^{4}+\frac{41\!\cdots\!63}{26\!\cdots\!07}a^{3}+\frac{28\!\cdots\!03}{17\!\cdots\!57}a^{2}-\frac{81\!\cdots\!92}{26\!\cdots\!07}a+\frac{14\!\cdots\!45}{26\!\cdots\!07}$, $\frac{71\!\cdots\!78}{26\!\cdots\!07}a^{17}-\frac{46\!\cdots\!94}{26\!\cdots\!07}a^{16}-\frac{25\!\cdots\!81}{26\!\cdots\!07}a^{15}+\frac{19\!\cdots\!36}{26\!\cdots\!07}a^{14}+\frac{30\!\cdots\!21}{26\!\cdots\!07}a^{13}-\frac{32\!\cdots\!42}{26\!\cdots\!07}a^{12}-\frac{99\!\cdots\!98}{26\!\cdots\!07}a^{11}+\frac{26\!\cdots\!58}{26\!\cdots\!07}a^{10}-\frac{66\!\cdots\!91}{26\!\cdots\!07}a^{9}-\frac{10\!\cdots\!29}{26\!\cdots\!07}a^{8}+\frac{57\!\cdots\!83}{26\!\cdots\!07}a^{7}+\frac{20\!\cdots\!57}{26\!\cdots\!07}a^{6}-\frac{14\!\cdots\!43}{26\!\cdots\!07}a^{5}-\frac{15\!\cdots\!28}{26\!\cdots\!07}a^{4}+\frac{11\!\cdots\!77}{26\!\cdots\!07}a^{3}+\frac{16\!\cdots\!12}{17\!\cdots\!57}a^{2}-\frac{18\!\cdots\!51}{26\!\cdots\!07}a+\frac{55\!\cdots\!90}{26\!\cdots\!07}$, $\frac{21\!\cdots\!39}{26\!\cdots\!07}a^{17}-\frac{99\!\cdots\!97}{26\!\cdots\!07}a^{16}-\frac{10\!\cdots\!48}{26\!\cdots\!07}a^{15}+\frac{43\!\cdots\!06}{26\!\cdots\!07}a^{14}+\frac{18\!\cdots\!09}{26\!\cdots\!07}a^{13}-\frac{73\!\cdots\!66}{26\!\cdots\!07}a^{12}-\frac{18\!\cdots\!45}{26\!\cdots\!07}a^{11}+\frac{61\!\cdots\!05}{26\!\cdots\!07}a^{10}+\frac{10\!\cdots\!97}{26\!\cdots\!07}a^{9}-\frac{26\!\cdots\!23}{26\!\cdots\!07}a^{8}-\frac{32\!\cdots\!68}{26\!\cdots\!07}a^{7}+\frac{59\!\cdots\!04}{26\!\cdots\!07}a^{6}+\frac{48\!\cdots\!79}{26\!\cdots\!07}a^{5}-\frac{64\!\cdots\!94}{26\!\cdots\!07}a^{4}-\frac{28\!\cdots\!31}{26\!\cdots\!07}a^{3}+\frac{18\!\cdots\!48}{17\!\cdots\!57}a^{2}+\frac{22\!\cdots\!84}{26\!\cdots\!07}a-\frac{19\!\cdots\!40}{26\!\cdots\!07}$, $\frac{84\!\cdots\!93}{26\!\cdots\!07}a^{17}-\frac{55\!\cdots\!74}{26\!\cdots\!07}a^{16}-\frac{28\!\cdots\!19}{26\!\cdots\!07}a^{15}+\frac{23\!\cdots\!59}{26\!\cdots\!07}a^{14}+\frac{31\!\cdots\!04}{26\!\cdots\!07}a^{13}-\frac{38\!\cdots\!41}{26\!\cdots\!07}a^{12}-\frac{53\!\cdots\!19}{26\!\cdots\!07}a^{11}+\frac{31\!\cdots\!58}{26\!\cdots\!07}a^{10}-\frac{12\!\cdots\!16}{26\!\cdots\!07}a^{9}-\frac{12\!\cdots\!02}{26\!\cdots\!07}a^{8}+\frac{87\!\cdots\!45}{26\!\cdots\!07}a^{7}+\frac{23\!\cdots\!82}{26\!\cdots\!07}a^{6}-\frac{19\!\cdots\!72}{26\!\cdots\!07}a^{5}-\frac{16\!\cdots\!28}{26\!\cdots\!07}a^{4}+\frac{15\!\cdots\!20}{26\!\cdots\!07}a^{3}+\frac{11\!\cdots\!17}{17\!\cdots\!57}a^{2}-\frac{23\!\cdots\!99}{26\!\cdots\!07}a+\frac{39\!\cdots\!77}{26\!\cdots\!07}$, $\frac{20\!\cdots\!31}{26\!\cdots\!07}a^{17}-\frac{16\!\cdots\!90}{26\!\cdots\!07}a^{16}-\frac{45\!\cdots\!67}{26\!\cdots\!07}a^{15}+\frac{67\!\cdots\!85}{26\!\cdots\!07}a^{14}-\frac{23\!\cdots\!52}{26\!\cdots\!07}a^{13}-\frac{10\!\cdots\!43}{26\!\cdots\!07}a^{12}+\frac{14\!\cdots\!28}{26\!\cdots\!07}a^{11}+\frac{71\!\cdots\!88}{26\!\cdots\!07}a^{10}-\frac{15\!\cdots\!00}{26\!\cdots\!07}a^{9}-\frac{21\!\cdots\!44}{26\!\cdots\!07}a^{8}+\frac{68\!\cdots\!95}{26\!\cdots\!07}a^{7}+\frac{67\!\cdots\!78}{26\!\cdots\!07}a^{6}-\frac{12\!\cdots\!91}{26\!\cdots\!07}a^{5}+\frac{62\!\cdots\!21}{26\!\cdots\!07}a^{4}+\frac{61\!\cdots\!13}{26\!\cdots\!07}a^{3}-\frac{38\!\cdots\!86}{17\!\cdots\!57}a^{2}+\frac{12\!\cdots\!75}{26\!\cdots\!07}a-\frac{14\!\cdots\!26}{26\!\cdots\!07}$, $\frac{52\!\cdots\!65}{26\!\cdots\!07}a^{17}-\frac{23\!\cdots\!16}{26\!\cdots\!07}a^{16}-\frac{24\!\cdots\!16}{26\!\cdots\!07}a^{15}+\frac{10\!\cdots\!35}{26\!\cdots\!07}a^{14}+\frac{46\!\cdots\!16}{26\!\cdots\!07}a^{13}-\frac{17\!\cdots\!65}{26\!\cdots\!07}a^{12}-\frac{46\!\cdots\!32}{26\!\cdots\!07}a^{11}+\frac{14\!\cdots\!34}{26\!\cdots\!07}a^{10}+\frac{26\!\cdots\!32}{26\!\cdots\!07}a^{9}-\frac{64\!\cdots\!29}{26\!\cdots\!07}a^{8}-\frac{77\!\cdots\!34}{26\!\cdots\!07}a^{7}+\frac{14\!\cdots\!69}{26\!\cdots\!07}a^{6}+\frac{10\!\cdots\!61}{26\!\cdots\!07}a^{5}-\frac{16\!\cdots\!60}{26\!\cdots\!07}a^{4}-\frac{36\!\cdots\!58}{26\!\cdots\!07}a^{3}+\frac{45\!\cdots\!97}{17\!\cdots\!57}a^{2}-\frac{11\!\cdots\!97}{26\!\cdots\!07}a-\frac{10\!\cdots\!10}{26\!\cdots\!07}$, $\frac{19\!\cdots\!90}{26\!\cdots\!07}a^{17}-\frac{12\!\cdots\!59}{26\!\cdots\!07}a^{16}-\frac{65\!\cdots\!57}{26\!\cdots\!07}a^{15}+\frac{54\!\cdots\!05}{26\!\cdots\!07}a^{14}+\frac{68\!\cdots\!09}{26\!\cdots\!07}a^{13}-\frac{90\!\cdots\!34}{26\!\cdots\!07}a^{12}-\frac{55\!\cdots\!91}{26\!\cdots\!07}a^{11}+\frac{72\!\cdots\!67}{26\!\cdots\!07}a^{10}-\frac{34\!\cdots\!21}{26\!\cdots\!07}a^{9}-\frac{29\!\cdots\!31}{26\!\cdots\!07}a^{8}+\frac{22\!\cdots\!10}{26\!\cdots\!07}a^{7}+\frac{55\!\cdots\!85}{26\!\cdots\!07}a^{6}-\frac{50\!\cdots\!43}{26\!\cdots\!07}a^{5}-\frac{39\!\cdots\!55}{26\!\cdots\!07}a^{4}+\frac{39\!\cdots\!68}{26\!\cdots\!07}a^{3}+\frac{32\!\cdots\!26}{17\!\cdots\!57}a^{2}-\frac{70\!\cdots\!85}{26\!\cdots\!07}a-\frac{27\!\cdots\!85}{26\!\cdots\!07}$, $\frac{16\!\cdots\!00}{26\!\cdots\!07}a^{17}-\frac{11\!\cdots\!13}{26\!\cdots\!07}a^{16}-\frac{57\!\cdots\!30}{26\!\cdots\!07}a^{15}+\frac{48\!\cdots\!29}{26\!\cdots\!07}a^{14}+\frac{59\!\cdots\!59}{26\!\cdots\!07}a^{13}-\frac{79\!\cdots\!71}{26\!\cdots\!07}a^{12}-\frac{18\!\cdots\!03}{26\!\cdots\!07}a^{11}+\frac{64\!\cdots\!64}{26\!\cdots\!07}a^{10}-\frac{33\!\cdots\!48}{26\!\cdots\!07}a^{9}-\frac{26\!\cdots\!23}{26\!\cdots\!07}a^{8}+\frac{21\!\cdots\!75}{26\!\cdots\!07}a^{7}+\frac{49\!\cdots\!64}{26\!\cdots\!07}a^{6}-\frac{49\!\cdots\!28}{26\!\cdots\!07}a^{5}-\frac{34\!\cdots\!97}{26\!\cdots\!07}a^{4}+\frac{40\!\cdots\!07}{26\!\cdots\!07}a^{3}+\frac{22\!\cdots\!05}{17\!\cdots\!57}a^{2}-\frac{73\!\cdots\!29}{26\!\cdots\!07}a+\frac{10\!\cdots\!39}{26\!\cdots\!07}$, $\frac{38\!\cdots\!40}{26\!\cdots\!07}a^{17}-\frac{25\!\cdots\!74}{26\!\cdots\!07}a^{16}-\frac{13\!\cdots\!88}{26\!\cdots\!07}a^{15}+\frac{10\!\cdots\!44}{26\!\cdots\!07}a^{14}+\frac{15\!\cdots\!64}{26\!\cdots\!07}a^{13}-\frac{17\!\cdots\!14}{26\!\cdots\!07}a^{12}-\frac{36\!\cdots\!88}{26\!\cdots\!07}a^{11}+\frac{14\!\cdots\!10}{26\!\cdots\!07}a^{10}-\frac{48\!\cdots\!04}{26\!\cdots\!07}a^{9}-\frac{58\!\cdots\!02}{26\!\cdots\!07}a^{8}+\frac{35\!\cdots\!92}{26\!\cdots\!07}a^{7}+\frac{11\!\cdots\!12}{26\!\cdots\!07}a^{6}-\frac{84\!\cdots\!50}{26\!\cdots\!07}a^{5}-\frac{85\!\cdots\!28}{26\!\cdots\!07}a^{4}+\frac{65\!\cdots\!90}{26\!\cdots\!07}a^{3}+\frac{10\!\cdots\!34}{17\!\cdots\!57}a^{2}-\frac{93\!\cdots\!68}{26\!\cdots\!07}a+\frac{11\!\cdots\!75}{26\!\cdots\!07}$
| 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: | \( 6552525312.42 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 6552525312.42 \cdot 1}{2\cdot\sqrt{34205654728777159191037355893457}}\cr\approx \mathstrut & 0.146848555802 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191)
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 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191, 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 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191);
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 - 7*x^17 - 32*x^16 + 294*x^15 + 275*x^14 - 4779*x^13 + 1076*x^12 + 37663*x^11 - 29140*x^10 - 146783*x^9 + 161417*x^8 + 251979*x^7 - 350891*x^6 - 118682*x^5 + 269024*x^4 - 41471*x^3 - 43388*x^2 + 12154*x - 191);
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
| \(\Q(\sqrt{17}) \), 3.3.361.1, 6.6.640267073.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | R | R | $18$ | $18$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.18.9.1 | $x^{18} + 1377 x^{17} + 842877 x^{16} + 301039740 x^{15} + 69145935966 x^{14} + 10594681229538 x^{13} + 1083400274206194 x^{12} + 71374592916053006 x^{11} + 2757704886031296688 x^{10} + 48363988493127014880 x^{9} + 46880983064641450752 x^{8} + 20627257837536901014 x^{7} + 5322819889168442136 x^{6} + 892489846394695920 x^{5} + 600661719561810931 x^{4} + 19503284078495824915 x^{3} + 346236970687224298200 x^{2} + 476441885309573581545 x + 683564390300988440775$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
|
\(19\)
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |