Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269)
gp: K = bnfinit(y^18 - 81*y^16 + 2457*y^14 - 450*y^13 - 35799*y^12 + 18036*y^11 + 270387*y^10 - 227047*y^9 - 1034208*y^8 + 1198881*y^7 + 1683324*y^6 - 2571156*y^5 - 445626*y^4 + 1440903*y^3 - 215973*y^2 - 79857*y + 14269, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269)
\( x^{18} - 81 x^{16} + 2457 x^{14} - 450 x^{13} - 35799 x^{12} + 18036 x^{11} + 270387 x^{10} + \cdots + 14269 \)
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: |
\(32665856832378527208340225685783203125\)
\(\medspace = 3^{44}\cdot 5^{9}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(121.37\) | 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: | $3^{232/81}5^{1/2}19^{8/9}\approx 712.4007850468525$ | ||
| Ramified primes: |
\(3\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{5}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{6}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{7}$, $\frac{1}{37\!\cdots\!89}a^{17}-\frac{10\!\cdots\!32}{37\!\cdots\!89}a^{16}-\frac{23\!\cdots\!51}{37\!\cdots\!89}a^{15}-\frac{26\!\cdots\!70}{37\!\cdots\!89}a^{14}-\frac{47\!\cdots\!70}{37\!\cdots\!89}a^{13}+\frac{32\!\cdots\!71}{37\!\cdots\!89}a^{12}-\frac{11\!\cdots\!69}{12\!\cdots\!63}a^{11}+\frac{51\!\cdots\!36}{37\!\cdots\!89}a^{10}-\frac{16\!\cdots\!47}{12\!\cdots\!63}a^{9}-\frac{11\!\cdots\!04}{37\!\cdots\!89}a^{8}-\frac{13\!\cdots\!24}{37\!\cdots\!89}a^{7}-\frac{12\!\cdots\!53}{37\!\cdots\!89}a^{6}-\frac{15\!\cdots\!99}{37\!\cdots\!89}a^{5}-\frac{17\!\cdots\!24}{37\!\cdots\!89}a^{4}+\frac{62\!\cdots\!34}{52\!\cdots\!59}a^{3}+\frac{46\!\cdots\!34}{12\!\cdots\!63}a^{2}+\frac{12\!\cdots\!48}{37\!\cdots\!89}a-\frac{77\!\cdots\!68}{16\!\cdots\!13}$
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: |
$\frac{15\!\cdots\!62}{11\!\cdots\!17}a^{17}+\frac{17\!\cdots\!31}{11\!\cdots\!17}a^{16}-\frac{12\!\cdots\!66}{11\!\cdots\!17}a^{15}-\frac{13\!\cdots\!97}{11\!\cdots\!17}a^{14}+\frac{36\!\cdots\!01}{11\!\cdots\!17}a^{13}+\frac{32\!\cdots\!17}{11\!\cdots\!17}a^{12}-\frac{51\!\cdots\!56}{11\!\cdots\!17}a^{11}-\frac{23\!\cdots\!46}{11\!\cdots\!17}a^{10}+\frac{11\!\cdots\!92}{35\!\cdots\!51}a^{9}+\frac{65\!\cdots\!82}{11\!\cdots\!17}a^{8}-\frac{14\!\cdots\!78}{11\!\cdots\!17}a^{7}+\frac{54\!\cdots\!45}{11\!\cdots\!17}a^{6}+\frac{25\!\cdots\!13}{11\!\cdots\!17}a^{5}-\frac{17\!\cdots\!79}{11\!\cdots\!17}a^{4}-\frac{13\!\cdots\!07}{11\!\cdots\!17}a^{3}+\frac{10\!\cdots\!81}{11\!\cdots\!17}a^{2}+\frac{11\!\cdots\!27}{11\!\cdots\!17}a-\frac{17\!\cdots\!49}{35\!\cdots\!51}$, $\frac{55\!\cdots\!68}{58\!\cdots\!47}a^{17}+\frac{61\!\cdots\!37}{58\!\cdots\!47}a^{16}-\frac{12\!\cdots\!64}{17\!\cdots\!41}a^{15}-\frac{46\!\cdots\!53}{58\!\cdots\!47}a^{14}+\frac{12\!\cdots\!04}{58\!\cdots\!47}a^{13}+\frac{10\!\cdots\!30}{58\!\cdots\!47}a^{12}-\frac{16\!\cdots\!58}{58\!\cdots\!47}a^{11}-\frac{60\!\cdots\!79}{58\!\cdots\!47}a^{10}+\frac{35\!\cdots\!14}{17\!\cdots\!41}a^{9}-\frac{17\!\cdots\!29}{58\!\cdots\!47}a^{8}-\frac{42\!\cdots\!71}{58\!\cdots\!47}a^{7}+\frac{81\!\cdots\!03}{17\!\cdots\!41}a^{6}+\frac{62\!\cdots\!95}{58\!\cdots\!47}a^{5}-\frac{72\!\cdots\!43}{58\!\cdots\!47}a^{4}-\frac{29\!\cdots\!92}{82\!\cdots\!57}a^{3}+\frac{35\!\cdots\!27}{58\!\cdots\!47}a^{2}-\frac{18\!\cdots\!63}{58\!\cdots\!47}a+\frac{66\!\cdots\!47}{23\!\cdots\!91}$, $\frac{94\!\cdots\!83}{58\!\cdots\!47}a^{17}+\frac{20\!\cdots\!74}{58\!\cdots\!47}a^{16}-\frac{21\!\cdots\!29}{17\!\cdots\!41}a^{15}-\frac{15\!\cdots\!68}{58\!\cdots\!47}a^{14}+\frac{19\!\cdots\!80}{58\!\cdots\!47}a^{13}+\frac{12\!\cdots\!23}{17\!\cdots\!41}a^{12}-\frac{25\!\cdots\!73}{58\!\cdots\!47}a^{11}-\frac{42\!\cdots\!57}{58\!\cdots\!47}a^{10}+\frac{17\!\cdots\!91}{58\!\cdots\!47}a^{9}+\frac{19\!\cdots\!76}{58\!\cdots\!47}a^{8}-\frac{62\!\cdots\!25}{58\!\cdots\!47}a^{7}-\frac{11\!\cdots\!76}{17\!\cdots\!41}a^{6}+\frac{10\!\cdots\!70}{58\!\cdots\!47}a^{5}+\frac{19\!\cdots\!94}{58\!\cdots\!47}a^{4}-\frac{22\!\cdots\!99}{24\!\cdots\!71}a^{3}-\frac{59\!\cdots\!64}{58\!\cdots\!47}a^{2}+\frac{37\!\cdots\!69}{58\!\cdots\!47}a-\frac{56\!\cdots\!05}{78\!\cdots\!97}$, $\frac{10\!\cdots\!90}{12\!\cdots\!63}a^{17}+\frac{31\!\cdots\!75}{37\!\cdots\!89}a^{16}-\frac{25\!\cdots\!31}{37\!\cdots\!89}a^{15}-\frac{24\!\cdots\!22}{37\!\cdots\!89}a^{14}+\frac{25\!\cdots\!78}{12\!\cdots\!63}a^{13}+\frac{18\!\cdots\!74}{12\!\cdots\!63}a^{12}-\frac{35\!\cdots\!20}{12\!\cdots\!63}a^{11}-\frac{11\!\cdots\!06}{12\!\cdots\!63}a^{10}+\frac{79\!\cdots\!48}{37\!\cdots\!89}a^{9}-\frac{25\!\cdots\!59}{12\!\cdots\!63}a^{8}-\frac{31\!\cdots\!96}{37\!\cdots\!89}a^{7}+\frac{14\!\cdots\!19}{37\!\cdots\!89}a^{6}+\frac{55\!\cdots\!27}{37\!\cdots\!89}a^{5}-\frac{14\!\cdots\!25}{12\!\cdots\!63}a^{4}-\frac{13\!\cdots\!57}{17\!\cdots\!53}a^{3}+\frac{85\!\cdots\!30}{12\!\cdots\!63}a^{2}+\frac{85\!\cdots\!67}{12\!\cdots\!63}a-\frac{15\!\cdots\!25}{49\!\cdots\!39}$, $\frac{80\!\cdots\!69}{12\!\cdots\!63}a^{17}+\frac{40\!\cdots\!05}{37\!\cdots\!89}a^{16}-\frac{62\!\cdots\!04}{12\!\cdots\!63}a^{15}-\frac{30\!\cdots\!04}{37\!\cdots\!89}a^{14}+\frac{52\!\cdots\!44}{37\!\cdots\!89}a^{13}+\frac{75\!\cdots\!80}{37\!\cdots\!89}a^{12}-\frac{70\!\cdots\!95}{37\!\cdots\!89}a^{11}-\frac{68\!\cdots\!03}{37\!\cdots\!89}a^{10}+\frac{50\!\cdots\!08}{37\!\cdots\!89}a^{9}+\frac{77\!\cdots\!12}{12\!\cdots\!63}a^{8}-\frac{19\!\cdots\!63}{37\!\cdots\!89}a^{7}-\frac{42\!\cdots\!55}{12\!\cdots\!63}a^{6}+\frac{33\!\cdots\!57}{37\!\cdots\!89}a^{5}-\frac{11\!\cdots\!71}{37\!\cdots\!89}a^{4}-\frac{25\!\cdots\!91}{52\!\cdots\!59}a^{3}+\frac{67\!\cdots\!50}{37\!\cdots\!89}a^{2}+\frac{91\!\cdots\!26}{37\!\cdots\!89}a-\frac{48\!\cdots\!89}{49\!\cdots\!39}$, $\frac{27\!\cdots\!14}{37\!\cdots\!89}a^{17}+\frac{57\!\cdots\!82}{37\!\cdots\!89}a^{16}-\frac{21\!\cdots\!48}{37\!\cdots\!89}a^{15}-\frac{43\!\cdots\!19}{37\!\cdots\!89}a^{14}+\frac{58\!\cdots\!42}{37\!\cdots\!89}a^{13}+\frac{10\!\cdots\!30}{37\!\cdots\!89}a^{12}-\frac{24\!\cdots\!03}{12\!\cdots\!63}a^{11}-\frac{33\!\cdots\!15}{12\!\cdots\!63}a^{10}+\frac{50\!\cdots\!59}{37\!\cdots\!89}a^{9}+\frac{38\!\cdots\!12}{37\!\cdots\!89}a^{8}-\frac{18\!\cdots\!80}{37\!\cdots\!89}a^{7}-\frac{30\!\cdots\!39}{37\!\cdots\!89}a^{6}+\frac{29\!\cdots\!97}{37\!\cdots\!89}a^{5}-\frac{91\!\cdots\!69}{37\!\cdots\!89}a^{4}-\frac{16\!\cdots\!68}{52\!\cdots\!59}a^{3}+\frac{22\!\cdots\!80}{12\!\cdots\!63}a^{2}-\frac{34\!\cdots\!67}{12\!\cdots\!63}a+\frac{89\!\cdots\!34}{49\!\cdots\!39}$, $\frac{49\!\cdots\!69}{12\!\cdots\!63}a^{17}+\frac{32\!\cdots\!81}{37\!\cdots\!89}a^{16}-\frac{11\!\cdots\!93}{37\!\cdots\!89}a^{15}-\frac{83\!\cdots\!60}{12\!\cdots\!63}a^{14}+\frac{10\!\cdots\!80}{12\!\cdots\!63}a^{13}+\frac{62\!\cdots\!21}{37\!\cdots\!89}a^{12}-\frac{40\!\cdots\!09}{37\!\cdots\!89}a^{11}-\frac{62\!\cdots\!38}{37\!\cdots\!89}a^{10}+\frac{27\!\cdots\!54}{37\!\cdots\!89}a^{9}+\frac{90\!\cdots\!17}{12\!\cdots\!63}a^{8}-\frac{98\!\cdots\!21}{37\!\cdots\!89}a^{7}-\frac{43\!\cdots\!84}{37\!\cdots\!89}a^{6}+\frac{56\!\cdots\!16}{12\!\cdots\!63}a^{5}-\frac{66\!\cdots\!24}{12\!\cdots\!63}a^{4}-\frac{12\!\cdots\!78}{52\!\cdots\!59}a^{3}+\frac{13\!\cdots\!53}{37\!\cdots\!89}a^{2}+\frac{49\!\cdots\!12}{37\!\cdots\!89}a-\frac{10\!\cdots\!07}{49\!\cdots\!39}$, $\frac{22\!\cdots\!70}{37\!\cdots\!89}a^{17}+\frac{26\!\cdots\!55}{37\!\cdots\!89}a^{16}-\frac{13\!\cdots\!15}{37\!\cdots\!89}a^{15}-\frac{67\!\cdots\!35}{12\!\cdots\!63}a^{14}+\frac{60\!\cdots\!59}{12\!\cdots\!63}a^{13}+\frac{57\!\cdots\!10}{37\!\cdots\!89}a^{12}+\frac{97\!\cdots\!01}{37\!\cdots\!89}a^{11}-\frac{74\!\cdots\!66}{37\!\cdots\!89}a^{10}-\frac{23\!\cdots\!86}{37\!\cdots\!89}a^{9}+\frac{50\!\cdots\!05}{37\!\cdots\!89}a^{8}+\frac{53\!\cdots\!32}{37\!\cdots\!89}a^{7}-\frac{17\!\cdots\!77}{37\!\cdots\!89}a^{6}+\frac{14\!\cdots\!06}{12\!\cdots\!63}a^{5}+\frac{87\!\cdots\!86}{12\!\cdots\!63}a^{4}-\frac{23\!\cdots\!24}{52\!\cdots\!59}a^{3}-\frac{21\!\cdots\!18}{37\!\cdots\!89}a^{2}+\frac{91\!\cdots\!76}{37\!\cdots\!89}a+\frac{28\!\cdots\!66}{49\!\cdots\!39}$, $\frac{10\!\cdots\!49}{37\!\cdots\!89}a^{17}+\frac{99\!\cdots\!59}{37\!\cdots\!89}a^{16}-\frac{86\!\cdots\!31}{37\!\cdots\!89}a^{15}-\frac{79\!\cdots\!91}{37\!\cdots\!89}a^{14}+\frac{85\!\cdots\!82}{12\!\cdots\!63}a^{13}+\frac{18\!\cdots\!12}{37\!\cdots\!89}a^{12}-\frac{35\!\cdots\!35}{37\!\cdots\!89}a^{11}-\frac{44\!\cdots\!61}{12\!\cdots\!63}a^{10}+\frac{26\!\cdots\!27}{37\!\cdots\!89}a^{9}+\frac{26\!\cdots\!71}{37\!\cdots\!89}a^{8}-\frac{10\!\cdots\!03}{37\!\cdots\!89}a^{7}+\frac{32\!\cdots\!04}{37\!\cdots\!89}a^{6}+\frac{17\!\cdots\!97}{37\!\cdots\!89}a^{5}-\frac{32\!\cdots\!41}{12\!\cdots\!63}a^{4}-\frac{12\!\cdots\!48}{52\!\cdots\!59}a^{3}+\frac{57\!\cdots\!01}{37\!\cdots\!89}a^{2}+\frac{14\!\cdots\!10}{12\!\cdots\!63}a-\frac{44\!\cdots\!27}{49\!\cdots\!39}$, $\frac{10\!\cdots\!84}{12\!\cdots\!63}a^{17}+\frac{65\!\cdots\!48}{37\!\cdots\!89}a^{16}-\frac{24\!\cdots\!36}{37\!\cdots\!89}a^{15}-\frac{50\!\cdots\!86}{37\!\cdots\!89}a^{14}+\frac{22\!\cdots\!36}{12\!\cdots\!63}a^{13}+\frac{12\!\cdots\!68}{37\!\cdots\!89}a^{12}-\frac{89\!\cdots\!54}{37\!\cdots\!89}a^{11}-\frac{13\!\cdots\!81}{37\!\cdots\!89}a^{10}+\frac{62\!\cdots\!89}{37\!\cdots\!89}a^{9}+\frac{19\!\cdots\!35}{12\!\cdots\!63}a^{8}-\frac{22\!\cdots\!19}{37\!\cdots\!89}a^{7}-\frac{10\!\cdots\!25}{37\!\cdots\!89}a^{6}+\frac{39\!\cdots\!84}{37\!\cdots\!89}a^{5}+\frac{81\!\cdots\!79}{12\!\cdots\!63}a^{4}-\frac{30\!\cdots\!58}{52\!\cdots\!59}a^{3}+\frac{26\!\cdots\!69}{37\!\cdots\!89}a^{2}+\frac{13\!\cdots\!97}{37\!\cdots\!89}a-\frac{28\!\cdots\!06}{49\!\cdots\!39}$, $\frac{78\!\cdots\!72}{12\!\cdots\!63}a^{17}+\frac{89\!\cdots\!78}{37\!\cdots\!89}a^{16}-\frac{58\!\cdots\!15}{12\!\cdots\!63}a^{15}-\frac{71\!\cdots\!55}{37\!\cdots\!89}a^{14}+\frac{46\!\cdots\!89}{37\!\cdots\!89}a^{13}+\frac{19\!\cdots\!91}{37\!\cdots\!89}a^{12}-\frac{55\!\cdots\!72}{37\!\cdots\!89}a^{11}-\frac{80\!\cdots\!35}{12\!\cdots\!63}a^{10}+\frac{33\!\cdots\!72}{37\!\cdots\!89}a^{9}+\frac{50\!\cdots\!09}{12\!\cdots\!63}a^{8}-\frac{10\!\cdots\!13}{37\!\cdots\!89}a^{7}-\frac{16\!\cdots\!35}{12\!\cdots\!63}a^{6}+\frac{17\!\cdots\!53}{37\!\cdots\!89}a^{5}+\frac{76\!\cdots\!98}{37\!\cdots\!89}a^{4}-\frac{15\!\cdots\!57}{52\!\cdots\!59}a^{3}-\frac{42\!\cdots\!17}{37\!\cdots\!89}a^{2}-\frac{65\!\cdots\!79}{12\!\cdots\!63}a+\frac{21\!\cdots\!03}{49\!\cdots\!39}$, $\frac{13\!\cdots\!87}{12\!\cdots\!63}a^{17}+\frac{72\!\cdots\!91}{37\!\cdots\!89}a^{16}-\frac{32\!\cdots\!41}{37\!\cdots\!89}a^{15}-\frac{55\!\cdots\!26}{37\!\cdots\!89}a^{14}+\frac{30\!\cdots\!87}{12\!\cdots\!63}a^{13}+\frac{13\!\cdots\!97}{37\!\cdots\!89}a^{12}-\frac{12\!\cdots\!40}{37\!\cdots\!89}a^{11}-\frac{12\!\cdots\!70}{37\!\cdots\!89}a^{10}+\frac{85\!\cdots\!24}{37\!\cdots\!89}a^{9}+\frac{14\!\cdots\!97}{12\!\cdots\!63}a^{8}-\frac{32\!\cdots\!80}{37\!\cdots\!89}a^{7}-\frac{80\!\cdots\!55}{37\!\cdots\!89}a^{6}+\frac{56\!\cdots\!97}{37\!\cdots\!89}a^{5}-\frac{61\!\cdots\!77}{12\!\cdots\!63}a^{4}-\frac{41\!\cdots\!85}{52\!\cdots\!59}a^{3}+\frac{12\!\cdots\!51}{37\!\cdots\!89}a^{2}+\frac{15\!\cdots\!53}{37\!\cdots\!89}a-\frac{91\!\cdots\!21}{49\!\cdots\!39}$, $\frac{88\!\cdots\!33}{12\!\cdots\!63}a^{17}+\frac{51\!\cdots\!95}{37\!\cdots\!89}a^{16}-\frac{69\!\cdots\!30}{12\!\cdots\!63}a^{15}-\frac{40\!\cdots\!13}{37\!\cdots\!89}a^{14}+\frac{59\!\cdots\!65}{37\!\cdots\!89}a^{13}+\frac{10\!\cdots\!03}{37\!\cdots\!89}a^{12}-\frac{80\!\cdots\!76}{37\!\cdots\!89}a^{11}-\frac{11\!\cdots\!46}{37\!\cdots\!89}a^{10}+\frac{56\!\cdots\!83}{37\!\cdots\!89}a^{9}+\frac{17\!\cdots\!41}{12\!\cdots\!63}a^{8}-\frac{21\!\cdots\!28}{37\!\cdots\!89}a^{7}-\frac{27\!\cdots\!79}{12\!\cdots\!63}a^{6}+\frac{36\!\cdots\!72}{37\!\cdots\!89}a^{5}-\frac{20\!\cdots\!99}{37\!\cdots\!89}a^{4}-\frac{26\!\cdots\!60}{52\!\cdots\!59}a^{3}+\frac{41\!\cdots\!53}{37\!\cdots\!89}a^{2}+\frac{13\!\cdots\!33}{37\!\cdots\!89}a-\frac{21\!\cdots\!30}{49\!\cdots\!39}$, $\frac{81\!\cdots\!99}{12\!\cdots\!63}a^{17}+\frac{49\!\cdots\!87}{37\!\cdots\!89}a^{16}-\frac{62\!\cdots\!79}{12\!\cdots\!63}a^{15}-\frac{38\!\cdots\!24}{37\!\cdots\!89}a^{14}+\frac{52\!\cdots\!52}{37\!\cdots\!89}a^{13}+\frac{94\!\cdots\!18}{37\!\cdots\!89}a^{12}-\frac{67\!\cdots\!87}{37\!\cdots\!89}a^{11}-\frac{91\!\cdots\!26}{37\!\cdots\!89}a^{10}+\frac{46\!\cdots\!65}{37\!\cdots\!89}a^{9}+\frac{12\!\cdots\!39}{12\!\cdots\!63}a^{8}-\frac{17\!\cdots\!08}{37\!\cdots\!89}a^{7}-\frac{14\!\cdots\!39}{12\!\cdots\!63}a^{6}+\frac{29\!\cdots\!66}{37\!\cdots\!89}a^{5}-\frac{45\!\cdots\!38}{37\!\cdots\!89}a^{4}-\frac{21\!\cdots\!17}{52\!\cdots\!59}a^{3}+\frac{42\!\cdots\!41}{37\!\cdots\!89}a^{2}+\frac{89\!\cdots\!50}{37\!\cdots\!89}a-\frac{30\!\cdots\!25}{49\!\cdots\!39}$, $\frac{18\!\cdots\!30}{37\!\cdots\!89}a^{17}+\frac{44\!\cdots\!36}{37\!\cdots\!89}a^{16}-\frac{48\!\cdots\!41}{12\!\cdots\!63}a^{15}-\frac{34\!\cdots\!05}{37\!\cdots\!89}a^{14}+\frac{13\!\cdots\!01}{12\!\cdots\!63}a^{13}+\frac{29\!\cdots\!47}{12\!\cdots\!63}a^{12}-\frac{51\!\cdots\!96}{37\!\cdots\!89}a^{11}-\frac{93\!\cdots\!72}{37\!\cdots\!89}a^{10}+\frac{35\!\cdots\!22}{37\!\cdots\!89}a^{9}+\frac{45\!\cdots\!28}{37\!\cdots\!89}a^{8}-\frac{13\!\cdots\!75}{37\!\cdots\!89}a^{7}-\frac{33\!\cdots\!54}{12\!\cdots\!63}a^{6}+\frac{24\!\cdots\!35}{37\!\cdots\!89}a^{5}+\frac{26\!\cdots\!87}{12\!\cdots\!63}a^{4}-\frac{82\!\cdots\!52}{17\!\cdots\!53}a^{3}-\frac{18\!\cdots\!52}{37\!\cdots\!89}a^{2}+\frac{34\!\cdots\!16}{37\!\cdots\!89}a-\frac{66\!\cdots\!57}{49\!\cdots\!39}$, $\frac{97\!\cdots\!03}{37\!\cdots\!89}a^{17}+\frac{17\!\cdots\!75}{37\!\cdots\!89}a^{16}-\frac{75\!\cdots\!84}{37\!\cdots\!89}a^{15}-\frac{13\!\cdots\!72}{37\!\cdots\!89}a^{14}+\frac{21\!\cdots\!66}{37\!\cdots\!89}a^{13}+\frac{33\!\cdots\!45}{37\!\cdots\!89}a^{12}-\frac{92\!\cdots\!66}{12\!\cdots\!63}a^{11}-\frac{31\!\cdots\!86}{37\!\cdots\!89}a^{10}+\frac{64\!\cdots\!84}{12\!\cdots\!63}a^{9}+\frac{12\!\cdots\!24}{37\!\cdots\!89}a^{8}-\frac{71\!\cdots\!56}{37\!\cdots\!89}a^{7}-\frac{10\!\cdots\!88}{37\!\cdots\!89}a^{6}+\frac{12\!\cdots\!25}{37\!\cdots\!89}a^{5}-\frac{27\!\cdots\!94}{37\!\cdots\!89}a^{4}-\frac{92\!\cdots\!65}{52\!\cdots\!59}a^{3}+\frac{68\!\cdots\!00}{12\!\cdots\!63}a^{2}+\frac{31\!\cdots\!94}{37\!\cdots\!89}a-\frac{46\!\cdots\!45}{16\!\cdots\!13}$, $\frac{34\!\cdots\!90}{37\!\cdots\!89}a^{17}+\frac{23\!\cdots\!71}{12\!\cdots\!63}a^{16}-\frac{26\!\cdots\!38}{37\!\cdots\!89}a^{15}-\frac{54\!\cdots\!50}{37\!\cdots\!89}a^{14}+\frac{72\!\cdots\!89}{37\!\cdots\!89}a^{13}+\frac{45\!\cdots\!66}{12\!\cdots\!63}a^{12}-\frac{95\!\cdots\!58}{37\!\cdots\!89}a^{11}-\frac{45\!\cdots\!45}{12\!\cdots\!63}a^{10}+\frac{21\!\cdots\!95}{12\!\cdots\!63}a^{9}+\frac{59\!\cdots\!71}{37\!\cdots\!89}a^{8}-\frac{79\!\cdots\!90}{12\!\cdots\!63}a^{7}-\frac{93\!\cdots\!01}{37\!\cdots\!89}a^{6}+\frac{41\!\cdots\!00}{37\!\cdots\!89}a^{5}-\frac{70\!\cdots\!11}{37\!\cdots\!89}a^{4}-\frac{10\!\cdots\!62}{17\!\cdots\!53}a^{3}+\frac{26\!\cdots\!96}{37\!\cdots\!89}a^{2}+\frac{36\!\cdots\!55}{12\!\cdots\!63}a-\frac{75\!\cdots\!22}{16\!\cdots\!13}$
| 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: | \( 9817904742040 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 9817904742040 \cdot 1}{2\cdot\sqrt{32665856832378527208340225685783203125}}\cr\approx \mathstrut & 0.225155060882672 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269)
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 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269, 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 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269);
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 - 81*x^16 + 2457*x^14 - 450*x^13 - 35799*x^12 + 18036*x^11 + 270387*x^10 - 227047*x^9 - 1034208*x^8 + 1198881*x^7 + 1683324*x^6 - 2571156*x^5 - 445626*x^4 + 1440903*x^3 - 215973*x^2 - 79857*x + 14269);
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
$C_9\times D_9$ (as 18T74):
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 solvable group of order 162 |
| The 54 conjugacy class representatives for $C_9\times D_9$ |
| Character table for $C_9\times D_9$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.296065125.2 |
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]
Sibling fields
| Degree 18 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | R | $18$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | $18$ | $18$ | R | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 $18$ | $9$ | $2$ | $44$ | |||
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.9.8.2 | $x^{9} + 57$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.0.1 | $x^{9} + 11 x^{3} + 14 x^{2} + 16 x + 17$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |