Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161)
gp: K = bnfinit(y^18 - 9*y^17 - 18*y^16 + 348*y^15 - 252*y^14 - 5040*y^13 + 8472*y^12 + 33174*y^11 - 76374*y^10 - 91106*y^9 + 292437*y^8 + 35910*y^7 - 455655*y^6 + 186741*y^5 + 192555*y^4 - 131820*y^3 - 2259*y^2 + 14355*y - 2161, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161)
\( x^{18} - 9 x^{17} - 18 x^{16} + 348 x^{15} - 252 x^{14} - 5040 x^{13} + 8472 x^{12} + 33174 x^{11} + \cdots - 2161 \)
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: |
\(116781890125989356502353933497857\)
\(\medspace = 3^{44}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.47\) | 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: | $3^{22/9}17^{1/2}\approx 60.467339582680516$ | ||
| Ramified primes: |
\(3\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(459=3^{3}\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{459}(256,·)$, $\chi_{459}(1,·)$, $\chi_{459}(322,·)$, $\chi_{459}(67,·)$, $\chi_{459}(205,·)$, $\chi_{459}(271,·)$, $\chi_{459}(16,·)$, $\chi_{459}(409,·)$, $\chi_{459}(154,·)$, $\chi_{459}(220,·)$, $\chi_{459}(358,·)$, $\chi_{459}(103,·)$, $\chi_{459}(424,·)$, $\chi_{459}(169,·)$, $\chi_{459}(307,·)$, $\chi_{459}(52,·)$, $\chi_{459}(373,·)$, $\chi_{459}(118,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{54\!\cdots\!53}a^{17}-\frac{19\!\cdots\!46}{54\!\cdots\!53}a^{16}+\frac{41\!\cdots\!30}{54\!\cdots\!53}a^{15}+\frac{49\!\cdots\!65}{54\!\cdots\!53}a^{14}+\frac{13\!\cdots\!38}{54\!\cdots\!53}a^{13}-\frac{14\!\cdots\!45}{54\!\cdots\!53}a^{12}-\frac{19\!\cdots\!20}{54\!\cdots\!53}a^{11}-\frac{24\!\cdots\!91}{54\!\cdots\!53}a^{10}-\frac{10\!\cdots\!71}{54\!\cdots\!53}a^{9}+\frac{63\!\cdots\!81}{54\!\cdots\!53}a^{8}+\frac{25\!\cdots\!57}{54\!\cdots\!53}a^{7}+\frac{23\!\cdots\!79}{54\!\cdots\!53}a^{6}+\frac{17\!\cdots\!06}{54\!\cdots\!53}a^{5}-\frac{21\!\cdots\!37}{54\!\cdots\!53}a^{4}+\frac{24\!\cdots\!97}{54\!\cdots\!53}a^{3}-\frac{74\!\cdots\!57}{54\!\cdots\!53}a^{2}-\frac{20\!\cdots\!12}{54\!\cdots\!53}a+\frac{21\!\cdots\!53}{54\!\cdots\!53}$
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: |
$\frac{82\!\cdots\!32}{49\!\cdots\!31}a^{17}-\frac{71\!\cdots\!83}{49\!\cdots\!31}a^{16}-\frac{16\!\cdots\!50}{49\!\cdots\!31}a^{15}+\frac{28\!\cdots\!38}{49\!\cdots\!31}a^{14}-\frac{13\!\cdots\!54}{49\!\cdots\!31}a^{13}-\frac{41\!\cdots\!80}{49\!\cdots\!31}a^{12}+\frac{59\!\cdots\!90}{49\!\cdots\!31}a^{11}+\frac{28\!\cdots\!43}{49\!\cdots\!31}a^{10}-\frac{56\!\cdots\!85}{49\!\cdots\!31}a^{9}-\frac{90\!\cdots\!37}{49\!\cdots\!31}a^{8}+\frac{22\!\cdots\!56}{49\!\cdots\!31}a^{7}+\frac{90\!\cdots\!46}{49\!\cdots\!31}a^{6}-\frac{36\!\cdots\!04}{49\!\cdots\!31}a^{5}+\frac{52\!\cdots\!14}{49\!\cdots\!31}a^{4}+\frac{18\!\cdots\!27}{49\!\cdots\!31}a^{3}-\frac{56\!\cdots\!27}{49\!\cdots\!31}a^{2}-\frac{19\!\cdots\!32}{49\!\cdots\!31}a+\frac{61\!\cdots\!25}{49\!\cdots\!31}$, $\frac{12\!\cdots\!54}{49\!\cdots\!31}a^{17}-\frac{88\!\cdots\!09}{49\!\cdots\!31}a^{16}-\frac{45\!\cdots\!34}{49\!\cdots\!31}a^{15}+\frac{37\!\cdots\!41}{49\!\cdots\!31}a^{14}+\frac{57\!\cdots\!45}{49\!\cdots\!31}a^{13}-\frac{63\!\cdots\!76}{49\!\cdots\!31}a^{12}-\frac{29\!\cdots\!47}{49\!\cdots\!31}a^{11}+\frac{53\!\cdots\!42}{49\!\cdots\!31}a^{10}+\frac{37\!\cdots\!81}{49\!\cdots\!31}a^{9}-\frac{23\!\cdots\!93}{49\!\cdots\!31}a^{8}+\frac{11\!\cdots\!91}{49\!\cdots\!31}a^{7}+\frac{54\!\cdots\!58}{49\!\cdots\!31}a^{6}-\frac{26\!\cdots\!03}{49\!\cdots\!31}a^{5}-\frac{56\!\cdots\!46}{49\!\cdots\!31}a^{4}+\frac{22\!\cdots\!67}{49\!\cdots\!31}a^{3}+\frac{22\!\cdots\!74}{49\!\cdots\!31}a^{2}-\frac{12\!\cdots\!63}{49\!\cdots\!31}a-\frac{21\!\cdots\!35}{49\!\cdots\!31}$, $\frac{20\!\cdots\!84}{54\!\cdots\!53}a^{17}-\frac{17\!\cdots\!28}{54\!\cdots\!53}a^{16}-\frac{46\!\cdots\!12}{54\!\cdots\!53}a^{15}+\frac{68\!\cdots\!37}{54\!\cdots\!53}a^{14}-\frac{14\!\cdots\!53}{54\!\cdots\!53}a^{13}-\frac{10\!\cdots\!47}{54\!\cdots\!53}a^{12}+\frac{11\!\cdots\!47}{54\!\cdots\!53}a^{11}+\frac{74\!\cdots\!78}{54\!\cdots\!53}a^{10}-\frac{11\!\cdots\!46}{54\!\cdots\!53}a^{9}-\frac{25\!\cdots\!48}{54\!\cdots\!53}a^{8}+\frac{47\!\cdots\!61}{54\!\cdots\!53}a^{7}+\frac{32\!\cdots\!59}{54\!\cdots\!53}a^{6}-\frac{78\!\cdots\!31}{54\!\cdots\!53}a^{5}-\frac{19\!\cdots\!38}{54\!\cdots\!53}a^{4}+\frac{40\!\cdots\!54}{54\!\cdots\!53}a^{3}-\frac{69\!\cdots\!39}{54\!\cdots\!53}a^{2}-\frac{38\!\cdots\!56}{54\!\cdots\!53}a+\frac{50\!\cdots\!56}{54\!\cdots\!53}$, $\frac{30\!\cdots\!94}{54\!\cdots\!53}a^{17}-\frac{24\!\cdots\!55}{54\!\cdots\!53}a^{16}-\frac{79\!\cdots\!42}{54\!\cdots\!53}a^{15}+\frac{99\!\cdots\!42}{54\!\cdots\!53}a^{14}+\frac{22\!\cdots\!04}{54\!\cdots\!53}a^{13}-\frac{15\!\cdots\!03}{54\!\cdots\!53}a^{12}+\frac{10\!\cdots\!24}{54\!\cdots\!53}a^{11}+\frac{11\!\cdots\!58}{54\!\cdots\!53}a^{10}-\frac{12\!\cdots\!87}{54\!\cdots\!53}a^{9}-\frac{42\!\cdots\!75}{54\!\cdots\!53}a^{8}+\frac{52\!\cdots\!56}{54\!\cdots\!53}a^{7}+\frac{70\!\cdots\!61}{54\!\cdots\!53}a^{6}-\frac{88\!\cdots\!78}{54\!\cdots\!53}a^{5}-\frac{39\!\cdots\!17}{54\!\cdots\!53}a^{4}+\frac{49\!\cdots\!73}{54\!\cdots\!53}a^{3}+\frac{57\!\cdots\!74}{54\!\cdots\!53}a^{2}-\frac{62\!\cdots\!01}{54\!\cdots\!53}a-\frac{21\!\cdots\!15}{54\!\cdots\!53}$, $\frac{11\!\cdots\!58}{54\!\cdots\!53}a^{17}-\frac{98\!\cdots\!89}{54\!\cdots\!53}a^{16}-\frac{24\!\cdots\!64}{54\!\cdots\!53}a^{15}+\frac{38\!\cdots\!91}{54\!\cdots\!53}a^{14}-\frac{13\!\cdots\!17}{54\!\cdots\!53}a^{13}-\frac{58\!\cdots\!02}{54\!\cdots\!53}a^{12}+\frac{74\!\cdots\!41}{54\!\cdots\!53}a^{11}+\frac{40\!\cdots\!54}{54\!\cdots\!53}a^{10}-\frac{71\!\cdots\!61}{54\!\cdots\!53}a^{9}-\frac{13\!\cdots\!19}{54\!\cdots\!53}a^{8}+\frac{28\!\cdots\!43}{54\!\cdots\!53}a^{7}+\frac{14\!\cdots\!25}{54\!\cdots\!53}a^{6}-\frac{46\!\cdots\!54}{54\!\cdots\!53}a^{5}+\frac{41\!\cdots\!91}{54\!\cdots\!53}a^{4}+\frac{23\!\cdots\!86}{54\!\cdots\!53}a^{3}-\frac{67\!\cdots\!53}{54\!\cdots\!53}a^{2}-\frac{26\!\cdots\!10}{54\!\cdots\!53}a+\frac{79\!\cdots\!64}{54\!\cdots\!53}$, $\frac{54\!\cdots\!26}{54\!\cdots\!53}a^{17}-\frac{54\!\cdots\!50}{54\!\cdots\!53}a^{16}-\frac{46\!\cdots\!08}{54\!\cdots\!53}a^{15}+\frac{19\!\cdots\!99}{54\!\cdots\!53}a^{14}-\frac{34\!\cdots\!85}{54\!\cdots\!53}a^{13}-\frac{26\!\cdots\!51}{54\!\cdots\!53}a^{12}+\frac{77\!\cdots\!87}{54\!\cdots\!53}a^{11}+\frac{13\!\cdots\!09}{54\!\cdots\!53}a^{10}-\frac{64\!\cdots\!05}{54\!\cdots\!53}a^{9}-\frac{76\!\cdots\!51}{54\!\cdots\!53}a^{8}+\frac{24\!\cdots\!41}{54\!\cdots\!53}a^{7}-\frac{14\!\cdots\!20}{54\!\cdots\!53}a^{6}-\frac{37\!\cdots\!44}{54\!\cdots\!53}a^{5}+\frac{37\!\cdots\!77}{54\!\cdots\!53}a^{4}+\frac{16\!\cdots\!60}{54\!\cdots\!53}a^{3}-\frac{20\!\cdots\!59}{54\!\cdots\!53}a^{2}-\frac{13\!\cdots\!94}{54\!\cdots\!53}a+\frac{17\!\cdots\!51}{54\!\cdots\!53}$, $\frac{91\!\cdots\!16}{54\!\cdots\!53}a^{17}-\frac{77\!\cdots\!41}{54\!\cdots\!53}a^{16}-\frac{20\!\cdots\!56}{54\!\cdots\!53}a^{15}+\frac{30\!\cdots\!08}{54\!\cdots\!53}a^{14}-\frac{82\!\cdots\!52}{54\!\cdots\!53}a^{13}-\frac{46\!\cdots\!88}{54\!\cdots\!53}a^{12}+\frac{54\!\cdots\!68}{54\!\cdots\!53}a^{11}+\frac{32\!\cdots\!92}{54\!\cdots\!53}a^{10}-\frac{53\!\cdots\!78}{54\!\cdots\!53}a^{9}-\frac{10\!\cdots\!60}{54\!\cdots\!53}a^{8}+\frac{21\!\cdots\!52}{54\!\cdots\!53}a^{7}+\frac{12\!\cdots\!96}{54\!\cdots\!53}a^{6}-\frac{33\!\cdots\!72}{54\!\cdots\!53}a^{5}+\frac{21\!\cdots\!72}{54\!\cdots\!53}a^{4}+\frac{16\!\cdots\!76}{54\!\cdots\!53}a^{3}-\frac{45\!\cdots\!37}{54\!\cdots\!53}a^{2}-\frac{14\!\cdots\!80}{54\!\cdots\!53}a+\frac{49\!\cdots\!02}{54\!\cdots\!53}$, $\frac{53\!\cdots\!30}{54\!\cdots\!53}a^{17}-\frac{33\!\cdots\!65}{54\!\cdots\!53}a^{16}-\frac{21\!\cdots\!62}{54\!\cdots\!53}a^{15}+\frac{14\!\cdots\!60}{54\!\cdots\!53}a^{14}+\frac{34\!\cdots\!70}{54\!\cdots\!53}a^{13}-\frac{26\!\cdots\!43}{54\!\cdots\!53}a^{12}-\frac{29\!\cdots\!18}{54\!\cdots\!53}a^{11}+\frac{23\!\cdots\!42}{54\!\cdots\!53}a^{10}+\frac{15\!\cdots\!95}{54\!\cdots\!53}a^{9}-\frac{11\!\cdots\!74}{54\!\cdots\!53}a^{8}-\frac{50\!\cdots\!02}{54\!\cdots\!53}a^{7}+\frac{27\!\cdots\!83}{54\!\cdots\!53}a^{6}+\frac{10\!\cdots\!88}{54\!\cdots\!53}a^{5}-\frac{32\!\cdots\!12}{54\!\cdots\!53}a^{4}-\frac{96\!\cdots\!93}{54\!\cdots\!53}a^{3}+\frac{14\!\cdots\!92}{54\!\cdots\!53}a^{2}+\frac{31\!\cdots\!85}{54\!\cdots\!53}a-\frac{15\!\cdots\!99}{54\!\cdots\!53}$, $\frac{77\!\cdots\!29}{54\!\cdots\!53}a^{17}-\frac{60\!\cdots\!18}{54\!\cdots\!53}a^{16}-\frac{21\!\cdots\!73}{54\!\cdots\!53}a^{15}+\frac{24\!\cdots\!93}{54\!\cdots\!53}a^{14}+\frac{11\!\cdots\!48}{54\!\cdots\!53}a^{13}-\frac{38\!\cdots\!51}{54\!\cdots\!53}a^{12}+\frac{17\!\cdots\!06}{54\!\cdots\!53}a^{11}+\frac{29\!\cdots\!22}{54\!\cdots\!53}a^{10}-\frac{23\!\cdots\!84}{54\!\cdots\!53}a^{9}-\frac{10\!\cdots\!00}{54\!\cdots\!53}a^{8}+\frac{98\!\cdots\!78}{54\!\cdots\!53}a^{7}+\frac{18\!\cdots\!14}{54\!\cdots\!53}a^{6}-\frac{15\!\cdots\!63}{54\!\cdots\!53}a^{5}-\frac{11\!\cdots\!99}{54\!\cdots\!53}a^{4}+\frac{65\!\cdots\!94}{54\!\cdots\!53}a^{3}+\frac{16\!\cdots\!58}{54\!\cdots\!53}a^{2}-\frac{42\!\cdots\!95}{54\!\cdots\!53}a-\frac{23\!\cdots\!84}{54\!\cdots\!53}$, $\frac{74\!\cdots\!99}{54\!\cdots\!53}a^{17}-\frac{61\!\cdots\!39}{54\!\cdots\!53}a^{16}-\frac{18\!\cdots\!68}{54\!\cdots\!53}a^{15}+\frac{24\!\cdots\!10}{54\!\cdots\!53}a^{14}-\frac{20\!\cdots\!10}{54\!\cdots\!53}a^{13}-\frac{37\!\cdots\!64}{54\!\cdots\!53}a^{12}+\frac{34\!\cdots\!37}{54\!\cdots\!53}a^{11}+\frac{27\!\cdots\!31}{54\!\cdots\!53}a^{10}-\frac{36\!\cdots\!97}{54\!\cdots\!53}a^{9}-\frac{92\!\cdots\!74}{54\!\cdots\!53}a^{8}+\frac{14\!\cdots\!62}{54\!\cdots\!53}a^{7}+\frac{12\!\cdots\!79}{54\!\cdots\!53}a^{6}-\frac{23\!\cdots\!81}{54\!\cdots\!53}a^{5}-\frac{15\!\cdots\!50}{54\!\cdots\!53}a^{4}+\frac{11\!\cdots\!12}{54\!\cdots\!53}a^{3}-\frac{23\!\cdots\!65}{54\!\cdots\!53}a^{2}-\frac{94\!\cdots\!18}{54\!\cdots\!53}a+\frac{25\!\cdots\!03}{54\!\cdots\!53}$, $\frac{78\!\cdots\!92}{54\!\cdots\!53}a^{17}-\frac{65\!\cdots\!80}{54\!\cdots\!53}a^{16}-\frac{18\!\cdots\!13}{54\!\cdots\!53}a^{15}+\frac{26\!\cdots\!32}{54\!\cdots\!53}a^{14}-\frac{37\!\cdots\!35}{54\!\cdots\!53}a^{13}-\frac{39\!\cdots\!68}{54\!\cdots\!53}a^{12}+\frac{42\!\cdots\!80}{54\!\cdots\!53}a^{11}+\frac{28\!\cdots\!04}{54\!\cdots\!53}a^{10}-\frac{42\!\cdots\!16}{54\!\cdots\!53}a^{9}-\frac{97\!\cdots\!96}{54\!\cdots\!53}a^{8}+\frac{17\!\cdots\!71}{54\!\cdots\!53}a^{7}+\frac{13\!\cdots\!28}{54\!\cdots\!53}a^{6}-\frac{28\!\cdots\!43}{54\!\cdots\!53}a^{5}-\frac{26\!\cdots\!38}{54\!\cdots\!53}a^{4}+\frac{15\!\cdots\!35}{54\!\cdots\!53}a^{3}-\frac{18\!\cdots\!84}{54\!\cdots\!53}a^{2}-\frac{25\!\cdots\!41}{54\!\cdots\!53}a+\frac{54\!\cdots\!73}{54\!\cdots\!53}$, $\frac{94\!\cdots\!38}{54\!\cdots\!53}a^{17}-\frac{76\!\cdots\!00}{54\!\cdots\!53}a^{16}-\frac{24\!\cdots\!22}{54\!\cdots\!53}a^{15}+\frac{31\!\cdots\!71}{54\!\cdots\!53}a^{14}+\frac{52\!\cdots\!39}{54\!\cdots\!53}a^{13}-\frac{48\!\cdots\!45}{54\!\cdots\!53}a^{12}+\frac{37\!\cdots\!19}{54\!\cdots\!53}a^{11}+\frac{37\!\cdots\!35}{54\!\cdots\!53}a^{10}-\frac{43\!\cdots\!89}{54\!\cdots\!53}a^{9}-\frac{13\!\cdots\!87}{54\!\cdots\!53}a^{8}+\frac{19\!\cdots\!72}{54\!\cdots\!53}a^{7}+\frac{23\!\cdots\!16}{54\!\cdots\!53}a^{6}-\frac{36\!\cdots\!08}{54\!\cdots\!53}a^{5}-\frac{10\!\cdots\!69}{54\!\cdots\!53}a^{4}+\frac{25\!\cdots\!40}{54\!\cdots\!53}a^{3}-\frac{41\!\cdots\!85}{54\!\cdots\!53}a^{2}-\frac{25\!\cdots\!03}{54\!\cdots\!53}a+\frac{55\!\cdots\!20}{54\!\cdots\!53}$, $\frac{18\!\cdots\!17}{54\!\cdots\!53}a^{17}-\frac{15\!\cdots\!30}{54\!\cdots\!53}a^{16}-\frac{38\!\cdots\!44}{54\!\cdots\!53}a^{15}+\frac{62\!\cdots\!51}{54\!\cdots\!53}a^{14}-\frac{24\!\cdots\!23}{54\!\cdots\!53}a^{13}-\frac{93\!\cdots\!81}{54\!\cdots\!53}a^{12}+\frac{12\!\cdots\!24}{54\!\cdots\!53}a^{11}+\frac{65\!\cdots\!58}{54\!\cdots\!53}a^{10}-\frac{11\!\cdots\!11}{54\!\cdots\!53}a^{9}-\frac{20\!\cdots\!16}{54\!\cdots\!53}a^{8}+\frac{47\!\cdots\!11}{54\!\cdots\!53}a^{7}+\frac{23\!\cdots\!66}{54\!\cdots\!53}a^{6}-\frac{78\!\cdots\!62}{54\!\cdots\!53}a^{5}+\frac{84\!\cdots\!03}{54\!\cdots\!53}a^{4}+\frac{40\!\cdots\!87}{54\!\cdots\!53}a^{3}-\frac{11\!\cdots\!79}{54\!\cdots\!53}a^{2}-\frac{46\!\cdots\!64}{54\!\cdots\!53}a+\frac{11\!\cdots\!93}{54\!\cdots\!53}$, $\frac{89\!\cdots\!83}{54\!\cdots\!53}a^{17}-\frac{73\!\cdots\!16}{54\!\cdots\!53}a^{16}-\frac{22\!\cdots\!59}{54\!\cdots\!53}a^{15}+\frac{29\!\cdots\!60}{54\!\cdots\!53}a^{14}+\frac{23\!\cdots\!88}{54\!\cdots\!53}a^{13}-\frac{44\!\cdots\!38}{54\!\cdots\!53}a^{12}+\frac{36\!\cdots\!85}{54\!\cdots\!53}a^{11}+\frac{32\!\cdots\!59}{54\!\cdots\!53}a^{10}-\frac{38\!\cdots\!00}{54\!\cdots\!53}a^{9}-\frac{11\!\cdots\!33}{54\!\cdots\!53}a^{8}+\frac{15\!\cdots\!68}{54\!\cdots\!53}a^{7}+\frac{17\!\cdots\!34}{54\!\cdots\!53}a^{6}-\frac{25\!\cdots\!78}{54\!\cdots\!53}a^{5}-\frac{79\!\cdots\!10}{54\!\cdots\!53}a^{4}+\frac{12\!\cdots\!02}{54\!\cdots\!53}a^{3}+\frac{13\!\cdots\!80}{54\!\cdots\!53}a^{2}-\frac{22\!\cdots\!18}{54\!\cdots\!53}a+\frac{29\!\cdots\!94}{54\!\cdots\!53}$, $\frac{58\!\cdots\!41}{54\!\cdots\!53}a^{17}-\frac{50\!\cdots\!44}{54\!\cdots\!53}a^{16}-\frac{12\!\cdots\!25}{54\!\cdots\!53}a^{15}+\frac{20\!\cdots\!41}{54\!\cdots\!53}a^{14}-\frac{76\!\cdots\!74}{54\!\cdots\!53}a^{13}-\frac{29\!\cdots\!10}{54\!\cdots\!53}a^{12}+\frac{39\!\cdots\!50}{54\!\cdots\!53}a^{11}+\frac{20\!\cdots\!55}{54\!\cdots\!53}a^{10}-\frac{37\!\cdots\!97}{54\!\cdots\!53}a^{9}-\frac{67\!\cdots\!83}{54\!\cdots\!53}a^{8}+\frac{14\!\cdots\!73}{54\!\cdots\!53}a^{7}+\frac{74\!\cdots\!05}{54\!\cdots\!53}a^{6}-\frac{24\!\cdots\!55}{54\!\cdots\!53}a^{5}+\frac{22\!\cdots\!03}{54\!\cdots\!53}a^{4}+\frac{12\!\cdots\!06}{54\!\cdots\!53}a^{3}-\frac{34\!\cdots\!07}{54\!\cdots\!53}a^{2}-\frac{13\!\cdots\!45}{54\!\cdots\!53}a+\frac{36\!\cdots\!27}{54\!\cdots\!53}$, $\frac{17\!\cdots\!04}{54\!\cdots\!53}a^{17}-\frac{16\!\cdots\!78}{54\!\cdots\!53}a^{16}-\frac{27\!\cdots\!69}{54\!\cdots\!53}a^{15}+\frac{61\!\cdots\!17}{54\!\cdots\!53}a^{14}-\frac{59\!\cdots\!82}{54\!\cdots\!53}a^{13}-\frac{86\!\cdots\!67}{54\!\cdots\!53}a^{12}+\frac{16\!\cdots\!12}{54\!\cdots\!53}a^{11}+\frac{53\!\cdots\!42}{54\!\cdots\!53}a^{10}-\frac{14\!\cdots\!41}{54\!\cdots\!53}a^{9}-\frac{12\!\cdots\!19}{54\!\cdots\!53}a^{8}+\frac{53\!\cdots\!78}{54\!\cdots\!53}a^{7}-\frac{74\!\cdots\!33}{54\!\cdots\!53}a^{6}-\frac{76\!\cdots\!73}{54\!\cdots\!53}a^{5}+\frac{50\!\cdots\!72}{54\!\cdots\!53}a^{4}+\frac{19\!\cdots\!93}{54\!\cdots\!53}a^{3}-\frac{25\!\cdots\!82}{54\!\cdots\!53}a^{2}+\frac{60\!\cdots\!88}{54\!\cdots\!53}a-\frac{17\!\cdots\!33}{54\!\cdots\!53}$, $\frac{92\!\cdots\!20}{54\!\cdots\!53}a^{17}-\frac{73\!\cdots\!58}{54\!\cdots\!53}a^{16}-\frac{24\!\cdots\!58}{54\!\cdots\!53}a^{15}+\frac{29\!\cdots\!48}{54\!\cdots\!53}a^{14}+\frac{65\!\cdots\!45}{54\!\cdots\!53}a^{13}-\frac{46\!\cdots\!79}{54\!\cdots\!53}a^{12}+\frac{32\!\cdots\!70}{54\!\cdots\!53}a^{11}+\frac{34\!\cdots\!78}{54\!\cdots\!53}a^{10}-\frac{37\!\cdots\!80}{54\!\cdots\!53}a^{9}-\frac{12\!\cdots\!16}{54\!\cdots\!53}a^{8}+\frac{15\!\cdots\!85}{54\!\cdots\!53}a^{7}+\frac{19\!\cdots\!21}{54\!\cdots\!53}a^{6}-\frac{24\!\cdots\!69}{54\!\cdots\!53}a^{5}-\frac{81\!\cdots\!70}{54\!\cdots\!53}a^{4}+\frac{12\!\cdots\!43}{54\!\cdots\!53}a^{3}-\frac{80\!\cdots\!12}{54\!\cdots\!53}a^{2}-\frac{14\!\cdots\!77}{54\!\cdots\!53}a+\frac{25\!\cdots\!03}{54\!\cdots\!53}$
| 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: | \( 13605878233.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 13605878233.0 \cdot 1}{2\cdot\sqrt{116781890125989356502353933497857}}\cr\approx \mathstrut & 0.165024625608 \end{aligned}\] (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 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161)
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 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161, 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 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161);
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 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161);
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
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 6.6.32234193.1, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | $18$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $9$ | $2$ | $44$ | |||
|
\(17\)
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |