Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*x - 191)
gp: K = bnfinit(y^18 - 7*y^17 - 23*y^16 + 238*y^15 + 67*y^14 - 3057*y^13 + 2147*y^12 + 18205*y^11 - 22172*y^10 - 48410*y^9 + 80696*y^8 + 38369*y^7 - 112670*y^6 + 27104*y^5 + 41513*y^4 - 25352*y^3 + 2633*y^2 + 1022*y - 191, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^17 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*x - 191);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 7*x^17 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*x - 191)
\( x^{18} - 7 x^{17} - 23 x^{16} + 238 x^{15} + 67 x^{14} - 3057 x^{13} + 2147 x^{12} + 18205 x^{11} + \cdots - 191 \)
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: |
\(3058776789325072365774692364013\)
\(\medspace = 13^{9}\cdot 19^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.39\) | 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: | $13^{1/2}19^{8/9}\approx 49.39028630333105$ | ||
| Ramified primes: |
\(13\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(247=13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{247}(64,·)$, $\chi_{247}(1,·)$, $\chi_{247}(66,·)$, $\chi_{247}(131,·)$, $\chi_{247}(196,·)$, $\chi_{247}(194,·)$, $\chi_{247}(142,·)$, $\chi_{247}(77,·)$, $\chi_{247}(144,·)$, $\chi_{247}(92,·)$, $\chi_{247}(25,·)$, $\chi_{247}(207,·)$, $\chi_{247}(220,·)$, $\chi_{247}(157,·)$, $\chi_{247}(168,·)$, $\chi_{247}(233,·)$, $\chi_{247}(235,·)$, $\chi_{247}(118,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{33\!\cdots\!31}a^{17}-\frac{18\!\cdots\!72}{33\!\cdots\!31}a^{16}+\frac{14\!\cdots\!74}{33\!\cdots\!31}a^{15}+\frac{11\!\cdots\!52}{33\!\cdots\!31}a^{14}-\frac{14\!\cdots\!11}{33\!\cdots\!31}a^{13}-\frac{15\!\cdots\!56}{33\!\cdots\!31}a^{12}-\frac{10\!\cdots\!16}{33\!\cdots\!31}a^{11}+\frac{52\!\cdots\!13}{33\!\cdots\!31}a^{10}-\frac{79\!\cdots\!60}{33\!\cdots\!31}a^{9}+\frac{31\!\cdots\!74}{33\!\cdots\!31}a^{8}-\frac{16\!\cdots\!76}{33\!\cdots\!31}a^{7}-\frac{10\!\cdots\!87}{33\!\cdots\!31}a^{6}-\frac{14\!\cdots\!27}{33\!\cdots\!31}a^{5}+\frac{65\!\cdots\!42}{33\!\cdots\!31}a^{4}+\frac{16\!\cdots\!71}{33\!\cdots\!31}a^{3}+\frac{10\!\cdots\!82}{33\!\cdots\!31}a^{2}+\frac{81\!\cdots\!57}{33\!\cdots\!31}a+\frac{97\!\cdots\!95}{33\!\cdots\!31}$
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: |
$\frac{19\!\cdots\!52}{33\!\cdots\!31}a^{17}-\frac{12\!\cdots\!07}{33\!\cdots\!31}a^{16}-\frac{52\!\cdots\!06}{33\!\cdots\!31}a^{15}+\frac{43\!\cdots\!43}{33\!\cdots\!31}a^{14}+\frac{38\!\cdots\!59}{33\!\cdots\!31}a^{13}-\frac{57\!\cdots\!41}{33\!\cdots\!31}a^{12}+\frac{83\!\cdots\!63}{33\!\cdots\!31}a^{11}+\frac{36\!\cdots\!62}{33\!\cdots\!31}a^{10}-\frac{22\!\cdots\!90}{33\!\cdots\!31}a^{9}-\frac{10\!\cdots\!78}{33\!\cdots\!31}a^{8}+\frac{95\!\cdots\!69}{33\!\cdots\!31}a^{7}+\frac{13\!\cdots\!72}{33\!\cdots\!31}a^{6}-\frac{14\!\cdots\!20}{33\!\cdots\!31}a^{5}-\frac{33\!\cdots\!58}{33\!\cdots\!31}a^{4}+\frac{63\!\cdots\!57}{33\!\cdots\!31}a^{3}-\frac{11\!\cdots\!82}{33\!\cdots\!31}a^{2}-\frac{26\!\cdots\!53}{33\!\cdots\!31}a+\frac{56\!\cdots\!71}{33\!\cdots\!31}$, $\frac{11\!\cdots\!86}{33\!\cdots\!31}a^{17}-\frac{82\!\cdots\!58}{33\!\cdots\!31}a^{16}-\frac{26\!\cdots\!76}{33\!\cdots\!31}a^{15}+\frac{28\!\cdots\!21}{33\!\cdots\!31}a^{14}+\frac{64\!\cdots\!93}{33\!\cdots\!31}a^{13}-\frac{36\!\cdots\!56}{33\!\cdots\!31}a^{12}+\frac{27\!\cdots\!49}{33\!\cdots\!31}a^{11}+\frac{21\!\cdots\!24}{33\!\cdots\!31}a^{10}-\frac{27\!\cdots\!07}{33\!\cdots\!31}a^{9}-\frac{59\!\cdots\!79}{33\!\cdots\!31}a^{8}+\frac{10\!\cdots\!91}{33\!\cdots\!31}a^{7}+\frac{51\!\cdots\!82}{33\!\cdots\!31}a^{6}-\frac{14\!\cdots\!97}{33\!\cdots\!31}a^{5}+\frac{24\!\cdots\!72}{33\!\cdots\!31}a^{4}+\frac{60\!\cdots\!69}{33\!\cdots\!31}a^{3}-\frac{26\!\cdots\!59}{33\!\cdots\!31}a^{2}-\frac{11\!\cdots\!61}{33\!\cdots\!31}a+\frac{13\!\cdots\!61}{33\!\cdots\!31}$, $\frac{63\!\cdots\!70}{33\!\cdots\!31}a^{17}-\frac{44\!\cdots\!58}{33\!\cdots\!31}a^{16}-\frac{14\!\cdots\!72}{33\!\cdots\!31}a^{15}+\frac{15\!\cdots\!34}{33\!\cdots\!31}a^{14}+\frac{44\!\cdots\!88}{33\!\cdots\!31}a^{13}-\frac{19\!\cdots\!62}{33\!\cdots\!31}a^{12}+\frac{13\!\cdots\!08}{33\!\cdots\!31}a^{11}+\frac{11\!\cdots\!35}{33\!\cdots\!31}a^{10}-\frac{14\!\cdots\!53}{33\!\cdots\!31}a^{9}-\frac{32\!\cdots\!76}{33\!\cdots\!31}a^{8}+\frac{51\!\cdots\!46}{33\!\cdots\!31}a^{7}+\frac{31\!\cdots\!62}{33\!\cdots\!31}a^{6}-\frac{74\!\cdots\!42}{33\!\cdots\!31}a^{5}+\frac{78\!\cdots\!44}{33\!\cdots\!31}a^{4}+\frac{31\!\cdots\!97}{33\!\cdots\!31}a^{3}-\frac{12\!\cdots\!14}{33\!\cdots\!31}a^{2}-\frac{86\!\cdots\!09}{33\!\cdots\!31}a+\frac{56\!\cdots\!84}{33\!\cdots\!31}$, $\frac{32\!\cdots\!94}{33\!\cdots\!31}a^{17}-\frac{21\!\cdots\!46}{33\!\cdots\!31}a^{16}-\frac{82\!\cdots\!10}{33\!\cdots\!31}a^{15}+\frac{73\!\cdots\!15}{33\!\cdots\!31}a^{14}+\frac{51\!\cdots\!89}{33\!\cdots\!31}a^{13}-\frac{96\!\cdots\!36}{33\!\cdots\!31}a^{12}+\frac{30\!\cdots\!35}{33\!\cdots\!31}a^{11}+\frac{59\!\cdots\!19}{33\!\cdots\!31}a^{10}-\frac{47\!\cdots\!42}{33\!\cdots\!31}a^{9}-\frac{17\!\cdots\!92}{33\!\cdots\!31}a^{8}+\frac{18\!\cdots\!03}{33\!\cdots\!31}a^{7}+\frac{19\!\cdots\!75}{33\!\cdots\!31}a^{6}-\frac{28\!\cdots\!56}{33\!\cdots\!31}a^{5}-\frac{23\!\cdots\!24}{33\!\cdots\!31}a^{4}+\frac{12\!\cdots\!95}{33\!\cdots\!31}a^{3}-\frac{33\!\cdots\!25}{33\!\cdots\!31}a^{2}-\frac{39\!\cdots\!60}{33\!\cdots\!31}a+\frac{15\!\cdots\!86}{33\!\cdots\!31}$, $\frac{34\!\cdots\!40}{33\!\cdots\!31}a^{17}-\frac{22\!\cdots\!51}{33\!\cdots\!31}a^{16}-\frac{90\!\cdots\!94}{33\!\cdots\!31}a^{15}+\frac{78\!\cdots\!71}{33\!\cdots\!31}a^{14}+\frac{60\!\cdots\!91}{33\!\cdots\!31}a^{13}-\frac{10\!\cdots\!91}{33\!\cdots\!31}a^{12}+\frac{26\!\cdots\!75}{33\!\cdots\!31}a^{11}+\frac{64\!\cdots\!15}{33\!\cdots\!31}a^{10}-\frac{47\!\cdots\!91}{33\!\cdots\!31}a^{9}-\frac{18\!\cdots\!10}{33\!\cdots\!31}a^{8}+\frac{19\!\cdots\!57}{33\!\cdots\!31}a^{7}+\frac{21\!\cdots\!16}{33\!\cdots\!31}a^{6}-\frac{28\!\cdots\!26}{33\!\cdots\!31}a^{5}-\frac{34\!\cdots\!09}{33\!\cdots\!31}a^{4}+\frac{12\!\cdots\!72}{33\!\cdots\!31}a^{3}-\frac{32\!\cdots\!92}{33\!\cdots\!31}a^{2}-\frac{40\!\cdots\!37}{33\!\cdots\!31}a+\frac{15\!\cdots\!06}{33\!\cdots\!31}$, $\frac{34\!\cdots\!00}{33\!\cdots\!31}a^{17}-\frac{22\!\cdots\!86}{33\!\cdots\!31}a^{16}-\frac{87\!\cdots\!64}{33\!\cdots\!31}a^{15}+\frac{78\!\cdots\!22}{33\!\cdots\!31}a^{14}+\frac{53\!\cdots\!90}{33\!\cdots\!31}a^{13}-\frac{10\!\cdots\!82}{33\!\cdots\!31}a^{12}+\frac{33\!\cdots\!74}{33\!\cdots\!31}a^{11}+\frac{63\!\cdots\!88}{33\!\cdots\!31}a^{10}-\frac{50\!\cdots\!76}{33\!\cdots\!31}a^{9}-\frac{18\!\cdots\!39}{33\!\cdots\!31}a^{8}+\frac{20\!\cdots\!54}{33\!\cdots\!31}a^{7}+\frac{21\!\cdots\!40}{33\!\cdots\!31}a^{6}-\frac{30\!\cdots\!50}{33\!\cdots\!31}a^{5}-\frac{24\!\cdots\!11}{33\!\cdots\!31}a^{4}+\frac{13\!\cdots\!84}{33\!\cdots\!31}a^{3}-\frac{35\!\cdots\!62}{33\!\cdots\!31}a^{2}-\frac{45\!\cdots\!38}{33\!\cdots\!31}a+\frac{17\!\cdots\!94}{33\!\cdots\!31}$, $\frac{96\!\cdots\!98}{33\!\cdots\!31}a^{17}-\frac{60\!\cdots\!98}{33\!\cdots\!31}a^{16}-\frac{27\!\cdots\!68}{33\!\cdots\!31}a^{15}+\frac{20\!\cdots\!37}{33\!\cdots\!31}a^{14}+\frac{23\!\cdots\!53}{33\!\cdots\!31}a^{13}-\frac{27\!\cdots\!73}{33\!\cdots\!31}a^{12}-\frac{13\!\cdots\!49}{33\!\cdots\!31}a^{11}+\frac{17\!\cdots\!31}{33\!\cdots\!31}a^{10}-\frac{74\!\cdots\!05}{33\!\cdots\!31}a^{9}-\frac{53\!\cdots\!65}{33\!\cdots\!31}a^{8}+\frac{35\!\cdots\!21}{33\!\cdots\!31}a^{7}+\frac{69\!\cdots\!19}{33\!\cdots\!31}a^{6}-\frac{56\!\cdots\!53}{33\!\cdots\!31}a^{5}-\frac{23\!\cdots\!72}{33\!\cdots\!31}a^{4}+\frac{24\!\cdots\!73}{33\!\cdots\!31}a^{3}-\frac{25\!\cdots\!54}{33\!\cdots\!31}a^{2}-\frac{88\!\cdots\!09}{33\!\cdots\!31}a+\frac{12\!\cdots\!17}{33\!\cdots\!31}$, $\frac{79\!\cdots\!48}{33\!\cdots\!31}a^{17}-\frac{28\!\cdots\!48}{33\!\cdots\!31}a^{16}-\frac{35\!\cdots\!90}{33\!\cdots\!31}a^{15}+\frac{11\!\cdots\!40}{33\!\cdots\!31}a^{14}+\frac{64\!\cdots\!34}{33\!\cdots\!31}a^{13}-\frac{17\!\cdots\!79}{33\!\cdots\!31}a^{12}-\frac{62\!\cdots\!46}{33\!\cdots\!31}a^{11}+\frac{13\!\cdots\!73}{33\!\cdots\!31}a^{10}+\frac{33\!\cdots\!10}{33\!\cdots\!31}a^{9}-\frac{56\!\cdots\!68}{33\!\cdots\!31}a^{8}-\frac{92\!\cdots\!40}{33\!\cdots\!31}a^{7}+\frac{12\!\cdots\!82}{33\!\cdots\!31}a^{6}+\frac{11\!\cdots\!51}{33\!\cdots\!31}a^{5}-\frac{12\!\cdots\!53}{33\!\cdots\!31}a^{4}-\frac{47\!\cdots\!75}{33\!\cdots\!31}a^{3}+\frac{44\!\cdots\!76}{33\!\cdots\!31}a^{2}-\frac{40\!\cdots\!29}{33\!\cdots\!31}a-\frac{18\!\cdots\!91}{33\!\cdots\!31}$, $\frac{11\!\cdots\!02}{33\!\cdots\!31}a^{17}-\frac{66\!\cdots\!27}{33\!\cdots\!31}a^{16}-\frac{33\!\cdots\!86}{33\!\cdots\!31}a^{15}+\frac{23\!\cdots\!28}{33\!\cdots\!31}a^{14}+\frac{35\!\cdots\!24}{33\!\cdots\!31}a^{13}-\frac{31\!\cdots\!91}{33\!\cdots\!31}a^{12}-\frac{13\!\cdots\!72}{33\!\cdots\!31}a^{11}+\frac{20\!\cdots\!37}{33\!\cdots\!31}a^{10}-\frac{13\!\cdots\!40}{33\!\cdots\!31}a^{9}-\frac{65\!\cdots\!68}{33\!\cdots\!31}a^{8}+\frac{19\!\cdots\!24}{33\!\cdots\!31}a^{7}+\frac{95\!\cdots\!22}{33\!\cdots\!31}a^{6}-\frac{35\!\cdots\!41}{33\!\cdots\!31}a^{5}-\frac{50\!\cdots\!63}{33\!\cdots\!31}a^{4}+\frac{16\!\cdots\!67}{33\!\cdots\!31}a^{3}+\frac{71\!\cdots\!38}{33\!\cdots\!31}a^{2}-\frac{10\!\cdots\!53}{33\!\cdots\!31}a-\frac{31\!\cdots\!27}{33\!\cdots\!31}$, $\frac{34\!\cdots\!94}{33\!\cdots\!31}a^{17}-\frac{23\!\cdots\!84}{33\!\cdots\!31}a^{16}-\frac{85\!\cdots\!21}{33\!\cdots\!31}a^{15}+\frac{79\!\cdots\!24}{33\!\cdots\!31}a^{14}+\frac{45\!\cdots\!52}{33\!\cdots\!31}a^{13}-\frac{10\!\cdots\!04}{33\!\cdots\!31}a^{12}+\frac{44\!\cdots\!08}{33\!\cdots\!31}a^{11}+\frac{64\!\cdots\!60}{33\!\cdots\!31}a^{10}-\frac{58\!\cdots\!75}{33\!\cdots\!31}a^{9}-\frac{18\!\cdots\!81}{33\!\cdots\!31}a^{8}+\frac{22\!\cdots\!68}{33\!\cdots\!31}a^{7}+\frac{19\!\cdots\!32}{33\!\cdots\!31}a^{6}-\frac{33\!\cdots\!64}{33\!\cdots\!31}a^{5}+\frac{19\!\cdots\!67}{33\!\cdots\!31}a^{4}+\frac{14\!\cdots\!87}{33\!\cdots\!31}a^{3}-\frac{49\!\cdots\!36}{33\!\cdots\!31}a^{2}-\frac{44\!\cdots\!76}{33\!\cdots\!31}a+\frac{27\!\cdots\!35}{33\!\cdots\!31}$, $\frac{18\!\cdots\!40}{33\!\cdots\!31}a^{17}-\frac{13\!\cdots\!35}{33\!\cdots\!31}a^{16}-\frac{40\!\cdots\!66}{33\!\cdots\!31}a^{15}+\frac{45\!\cdots\!17}{33\!\cdots\!31}a^{14}+\frac{47\!\cdots\!09}{33\!\cdots\!31}a^{13}-\frac{58\!\cdots\!12}{33\!\cdots\!31}a^{12}+\frac{50\!\cdots\!51}{33\!\cdots\!31}a^{11}+\frac{34\!\cdots\!54}{33\!\cdots\!31}a^{10}-\frac{48\!\cdots\!55}{33\!\cdots\!31}a^{9}-\frac{93\!\cdots\!16}{33\!\cdots\!31}a^{8}+\frac{17\!\cdots\!13}{33\!\cdots\!31}a^{7}+\frac{76\!\cdots\!27}{33\!\cdots\!31}a^{6}-\frac{24\!\cdots\!82}{33\!\cdots\!31}a^{5}+\frac{46\!\cdots\!50}{33\!\cdots\!31}a^{4}+\frac{10\!\cdots\!06}{33\!\cdots\!31}a^{3}-\frac{46\!\cdots\!18}{33\!\cdots\!31}a^{2}-\frac{29\!\cdots\!94}{33\!\cdots\!31}a+\frac{23\!\cdots\!80}{33\!\cdots\!31}$, $\frac{77\!\cdots\!33}{33\!\cdots\!31}a^{17}-\frac{51\!\cdots\!54}{33\!\cdots\!31}a^{16}-\frac{19\!\cdots\!74}{33\!\cdots\!31}a^{15}+\frac{17\!\cdots\!05}{33\!\cdots\!31}a^{14}+\frac{11\!\cdots\!84}{33\!\cdots\!31}a^{13}-\frac{23\!\cdots\!61}{33\!\cdots\!31}a^{12}+\frac{86\!\cdots\!20}{33\!\cdots\!31}a^{11}+\frac{14\!\cdots\!05}{33\!\cdots\!31}a^{10}-\frac{12\!\cdots\!11}{33\!\cdots\!31}a^{9}-\frac{42\!\cdots\!44}{33\!\cdots\!31}a^{8}+\frac{48\!\cdots\!65}{33\!\cdots\!31}a^{7}+\frac{46\!\cdots\!45}{33\!\cdots\!31}a^{6}-\frac{71\!\cdots\!03}{33\!\cdots\!31}a^{5}-\frac{38\!\cdots\!17}{33\!\cdots\!31}a^{4}+\frac{31\!\cdots\!38}{33\!\cdots\!31}a^{3}-\frac{89\!\cdots\!72}{33\!\cdots\!31}a^{2}-\frac{10\!\cdots\!22}{33\!\cdots\!31}a+\frac{42\!\cdots\!43}{33\!\cdots\!31}$, $\frac{44\!\cdots\!72}{33\!\cdots\!31}a^{17}-\frac{44\!\cdots\!46}{33\!\cdots\!31}a^{16}-\frac{21\!\cdots\!49}{33\!\cdots\!31}a^{15}+\frac{14\!\cdots\!51}{33\!\cdots\!31}a^{14}-\frac{25\!\cdots\!66}{33\!\cdots\!31}a^{13}-\frac{17\!\cdots\!56}{33\!\cdots\!31}a^{12}+\frac{47\!\cdots\!39}{33\!\cdots\!31}a^{11}+\frac{87\!\cdots\!15}{33\!\cdots\!31}a^{10}-\frac{34\!\cdots\!48}{33\!\cdots\!31}a^{9}-\frac{14\!\cdots\!59}{33\!\cdots\!31}a^{8}+\frac{11\!\cdots\!29}{33\!\cdots\!31}a^{7}-\frac{22\!\cdots\!70}{33\!\cdots\!31}a^{6}-\frac{15\!\cdots\!19}{33\!\cdots\!31}a^{5}+\frac{75\!\cdots\!98}{33\!\cdots\!31}a^{4}+\frac{61\!\cdots\!88}{33\!\cdots\!31}a^{3}-\frac{40\!\cdots\!60}{33\!\cdots\!31}a^{2}-\frac{34\!\cdots\!55}{33\!\cdots\!31}a+\frac{19\!\cdots\!14}{33\!\cdots\!31}$, $\frac{11\!\cdots\!65}{33\!\cdots\!31}a^{17}-\frac{78\!\cdots\!86}{33\!\cdots\!31}a^{16}-\frac{28\!\cdots\!73}{33\!\cdots\!31}a^{15}+\frac{26\!\cdots\!02}{33\!\cdots\!31}a^{14}+\frac{14\!\cdots\!18}{33\!\cdots\!31}a^{13}-\frac{35\!\cdots\!45}{33\!\cdots\!31}a^{12}+\frac{16\!\cdots\!84}{33\!\cdots\!31}a^{11}+\frac{21\!\cdots\!90}{33\!\cdots\!31}a^{10}-\frac{20\!\cdots\!26}{33\!\cdots\!31}a^{9}-\frac{62\!\cdots\!67}{33\!\cdots\!31}a^{8}+\frac{79\!\cdots\!45}{33\!\cdots\!31}a^{7}+\frac{68\!\cdots\!90}{33\!\cdots\!31}a^{6}-\frac{11\!\cdots\!96}{33\!\cdots\!31}a^{5}-\frac{23\!\cdots\!55}{33\!\cdots\!31}a^{4}+\frac{52\!\cdots\!22}{33\!\cdots\!31}a^{3}-\frac{15\!\cdots\!70}{33\!\cdots\!31}a^{2}-\frac{25\!\cdots\!63}{33\!\cdots\!31}a+\frac{92\!\cdots\!47}{33\!\cdots\!31}$, $\frac{88\!\cdots\!32}{33\!\cdots\!31}a^{17}-\frac{58\!\cdots\!79}{33\!\cdots\!31}a^{16}-\frac{22\!\cdots\!31}{33\!\cdots\!31}a^{15}+\frac{20\!\cdots\!28}{33\!\cdots\!31}a^{14}+\frac{14\!\cdots\!64}{33\!\cdots\!31}a^{13}-\frac{26\!\cdots\!85}{33\!\cdots\!31}a^{12}+\frac{82\!\cdots\!74}{33\!\cdots\!31}a^{11}+\frac{16\!\cdots\!56}{33\!\cdots\!31}a^{10}-\frac{12\!\cdots\!47}{33\!\cdots\!31}a^{9}-\frac{47\!\cdots\!13}{33\!\cdots\!31}a^{8}+\frac{51\!\cdots\!50}{33\!\cdots\!31}a^{7}+\frac{54\!\cdots\!26}{33\!\cdots\!31}a^{6}-\frac{77\!\cdots\!05}{33\!\cdots\!31}a^{5}-\frac{64\!\cdots\!72}{33\!\cdots\!31}a^{4}+\frac{33\!\cdots\!77}{33\!\cdots\!31}a^{3}-\frac{91\!\cdots\!63}{33\!\cdots\!31}a^{2}-\frac{11\!\cdots\!80}{33\!\cdots\!31}a+\frac{44\!\cdots\!43}{33\!\cdots\!31}$, $\frac{22\!\cdots\!40}{33\!\cdots\!31}a^{17}-\frac{14\!\cdots\!93}{33\!\cdots\!31}a^{16}-\frac{60\!\cdots\!90}{33\!\cdots\!31}a^{15}+\frac{49\!\cdots\!61}{33\!\cdots\!31}a^{14}+\frac{45\!\cdots\!16}{33\!\cdots\!31}a^{13}-\frac{65\!\cdots\!43}{33\!\cdots\!31}a^{12}+\frac{74\!\cdots\!96}{33\!\cdots\!31}a^{11}+\frac{41\!\cdots\!28}{33\!\cdots\!31}a^{10}-\frac{24\!\cdots\!78}{33\!\cdots\!31}a^{9}-\frac{12\!\cdots\!26}{33\!\cdots\!31}a^{8}+\frac{10\!\cdots\!39}{33\!\cdots\!31}a^{7}+\frac{15\!\cdots\!83}{33\!\cdots\!31}a^{6}-\frac{15\!\cdots\!65}{33\!\cdots\!31}a^{5}-\frac{41\!\cdots\!57}{33\!\cdots\!31}a^{4}+\frac{70\!\cdots\!29}{33\!\cdots\!31}a^{3}-\frac{11\!\cdots\!95}{33\!\cdots\!31}a^{2}-\frac{28\!\cdots\!38}{33\!\cdots\!31}a+\frac{57\!\cdots\!39}{33\!\cdots\!31}$, $\frac{81\!\cdots\!12}{33\!\cdots\!31}a^{17}-\frac{52\!\cdots\!42}{33\!\cdots\!31}a^{16}-\frac{21\!\cdots\!28}{33\!\cdots\!31}a^{15}+\frac{18\!\cdots\!18}{33\!\cdots\!31}a^{14}+\frac{15\!\cdots\!49}{33\!\cdots\!31}a^{13}-\frac{24\!\cdots\!69}{33\!\cdots\!31}a^{12}+\frac{48\!\cdots\!27}{33\!\cdots\!31}a^{11}+\frac{15\!\cdots\!94}{33\!\cdots\!31}a^{10}-\frac{10\!\cdots\!86}{33\!\cdots\!31}a^{9}-\frac{46\!\cdots\!61}{33\!\cdots\!31}a^{8}+\frac{42\!\cdots\!19}{33\!\cdots\!31}a^{7}+\frac{57\!\cdots\!74}{33\!\cdots\!31}a^{6}-\frac{63\!\cdots\!39}{33\!\cdots\!31}a^{5}-\frac{15\!\cdots\!14}{33\!\cdots\!31}a^{4}+\frac{27\!\cdots\!89}{33\!\cdots\!31}a^{3}-\frac{46\!\cdots\!39}{33\!\cdots\!31}a^{2}-\frac{10\!\cdots\!62}{33\!\cdots\!31}a+\frac{22\!\cdots\!34}{33\!\cdots\!31}$
| 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: | \( 1710187464.86 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 1710187464.86 \cdot 1}{2\cdot\sqrt{3058776789325072365774692364013}}\cr\approx \mathstrut & 0.128168042715 \end{aligned}\] (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 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*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 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*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 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*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 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*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
| \(\Q(\sqrt{13}) \), 3.3.361.1, 6.6.286315237.1, \(\Q(\zeta_{19})^+\) |
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 | $18$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.18.9.1 | $x^{18} + 1170 x^{17} + 608517 x^{16} + 184669680 x^{15} + 36042230484 x^{14} + 4692692080464 x^{13} + 407793261316444 x^{12} + 22833205275255672 x^{11} + 750031142087897694 x^{10} + 11196577827794288770 x^{9} + 9750461205950186580 x^{8} + 3863714899398059352 x^{7} + 1170776365765219708 x^{6} + 9183655224695901156 x^{5} + 136076384268316458696 x^{4} + 146209355090752705280 x^{3} + 170259556431855716025 x^{2} + 163704388102720431984 x + 129853841096201133292$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
|
\(19\)
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |