Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512)
gp: K = bnfinit(y^18 + 27*y^16 + 351*y^14 + 2646*y^12 + 12465*y^10 - 5*y^9 + 35964*y^8 + 468*y^7 + 61488*y^6 + 4752*y^5 + 51840*y^4 + 9600*y^3 + 20736*y^2 + 2304*y + 512, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512)
\( x^{18} + 27 x^{16} + 351 x^{14} + 2646 x^{12} + 12465 x^{10} - 5 x^{9} + 35964 x^{8} + 468 x^{7} + \cdots + 512 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-39739057971752889532465351767\)
\(\medspace = -\,3^{44}\cdot 7^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.80\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{22/9}7^{1/2}\approx 38.80122352031639$ | ||
| Ramified primes: |
\(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(189=3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(139,·)$, $\chi_{189}(76,·)$, $\chi_{189}(13,·)$, $\chi_{189}(148,·)$, $\chi_{189}(85,·)$, $\chi_{189}(22,·)$, $\chi_{189}(160,·)$, $\chi_{189}(97,·)$, $\chi_{189}(34,·)$, $\chi_{189}(169,·)$, $\chi_{189}(106,·)$, $\chi_{189}(43,·)$, $\chi_{189}(181,·)$, $\chi_{189}(118,·)$, $\chi_{189}(55,·)$, $\chi_{189}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}+\frac{1}{4}a^{6}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{5}{16}a^{9}+\frac{1}{8}a^{7}-\frac{7}{16}a^{5}-\frac{5}{16}a^{4}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}-\frac{5}{32}a^{10}+\frac{1}{16}a^{8}-\frac{7}{32}a^{6}-\frac{5}{32}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{13}-\frac{5}{64}a^{11}-\frac{15}{32}a^{9}-\frac{7}{64}a^{7}-\frac{5}{64}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{2571136}a^{16}-\frac{1177}{321392}a^{15}+\frac{7691}{2571136}a^{14}-\frac{2879}{160696}a^{13}+\frac{71295}{2571136}a^{12}+\frac{7463}{80348}a^{11}+\frac{215979}{1285568}a^{10}+\frac{41113}{321392}a^{9}+\frac{413313}{2571136}a^{8}-\frac{866365}{2571136}a^{7}-\frac{181171}{642784}a^{6}-\frac{223733}{642784}a^{5}+\frac{68235}{321392}a^{4}+\frac{2360}{20087}a^{3}+\frac{2697}{20087}a^{2}-\frac{12337}{40174}a+\frac{5414}{20087}$, $\frac{1}{74\!\cdots\!44}a^{17}-\frac{831127728464451}{37\!\cdots\!72}a^{16}+\frac{37\!\cdots\!19}{74\!\cdots\!44}a^{15}-\frac{27\!\cdots\!49}{37\!\cdots\!72}a^{14}-\frac{29\!\cdots\!01}{14\!\cdots\!48}a^{13}+\frac{19\!\cdots\!71}{37\!\cdots\!72}a^{12}+\frac{10\!\cdots\!09}{37\!\cdots\!72}a^{11}-\frac{96\!\cdots\!23}{18\!\cdots\!36}a^{10}-\frac{96\!\cdots\!87}{74\!\cdots\!44}a^{9}+\frac{11\!\cdots\!05}{74\!\cdots\!44}a^{8}-\frac{81\!\cdots\!09}{37\!\cdots\!72}a^{7}-\frac{40\!\cdots\!67}{92\!\cdots\!68}a^{6}+\frac{71\!\cdots\!73}{46\!\cdots\!84}a^{5}+\frac{23\!\cdots\!61}{57\!\cdots\!98}a^{4}+\frac{10\!\cdots\!35}{23\!\cdots\!92}a^{3}+\frac{32\!\cdots\!81}{11\!\cdots\!96}a^{2}-\frac{13\!\cdots\!35}{57\!\cdots\!98}a+\frac{14\!\cdots\!77}{28\!\cdots\!49}$
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
$C_{163}$, which has order $163$ (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: | $8$ | 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: |
$\frac{62335972121157}{33\!\cdots\!36}a^{17}+\frac{38712211866657}{66\!\cdots\!72}a^{16}+\frac{32\!\cdots\!33}{66\!\cdots\!72}a^{15}+\frac{125635739055498}{82\!\cdots\!59}a^{14}+\frac{49\!\cdots\!82}{82\!\cdots\!59}a^{13}+\frac{12\!\cdots\!47}{66\!\cdots\!72}a^{12}+\frac{69\!\cdots\!51}{16\!\cdots\!18}a^{11}+\frac{11\!\cdots\!58}{82\!\cdots\!59}a^{10}+\frac{12\!\cdots\!79}{66\!\cdots\!72}a^{9}+\frac{48\!\cdots\!72}{82\!\cdots\!59}a^{8}+\frac{73\!\cdots\!17}{16\!\cdots\!18}a^{7}+\frac{13\!\cdots\!24}{82\!\cdots\!59}a^{6}+\frac{47\!\cdots\!14}{82\!\cdots\!59}a^{5}+\frac{23\!\cdots\!04}{82\!\cdots\!59}a^{4}+\frac{10\!\cdots\!59}{66\!\cdots\!72}a^{3}+\frac{14\!\cdots\!88}{82\!\cdots\!59}a^{2}+\frac{35\!\cdots\!07}{16\!\cdots\!18}a-\frac{25\!\cdots\!12}{82\!\cdots\!59}$, $\frac{263827502621109}{52\!\cdots\!76}a^{17}-\frac{715392581865255}{52\!\cdots\!76}a^{16}+\frac{17\!\cdots\!77}{13\!\cdots\!44}a^{15}-\frac{19\!\cdots\!21}{52\!\cdots\!76}a^{14}+\frac{42\!\cdots\!87}{26\!\cdots\!88}a^{13}-\frac{25\!\cdots\!85}{52\!\cdots\!76}a^{12}+\frac{60\!\cdots\!73}{52\!\cdots\!76}a^{11}-\frac{95\!\cdots\!31}{26\!\cdots\!88}a^{10}+\frac{26\!\cdots\!87}{52\!\cdots\!76}a^{9}-\frac{11\!\cdots\!77}{66\!\cdots\!72}a^{8}+\frac{16\!\cdots\!05}{13\!\cdots\!44}a^{7}-\frac{25\!\cdots\!37}{52\!\cdots\!76}a^{6}+\frac{54\!\cdots\!73}{33\!\cdots\!36}a^{5}-\frac{54\!\cdots\!05}{66\!\cdots\!72}a^{4}+\frac{10\!\cdots\!02}{82\!\cdots\!59}a^{3}-\frac{23\!\cdots\!49}{33\!\cdots\!36}a^{2}-\frac{64\!\cdots\!96}{82\!\cdots\!59}a-\frac{17\!\cdots\!62}{82\!\cdots\!59}$, $\frac{40\!\cdots\!53}{57\!\cdots\!98}a^{17}+\frac{12\!\cdots\!69}{57\!\cdots\!98}a^{16}+\frac{10\!\cdots\!49}{57\!\cdots\!98}a^{15}+\frac{16\!\cdots\!81}{28\!\cdots\!49}a^{14}+\frac{65\!\cdots\!89}{28\!\cdots\!49}a^{13}+\frac{40\!\cdots\!99}{57\!\cdots\!98}a^{12}+\frac{45\!\cdots\!59}{28\!\cdots\!49}a^{11}+\frac{28\!\cdots\!03}{57\!\cdots\!98}a^{10}+\frac{39\!\cdots\!73}{57\!\cdots\!98}a^{9}+\frac{11\!\cdots\!98}{54\!\cdots\!33}a^{8}+\frac{48\!\cdots\!70}{28\!\cdots\!49}a^{7}+\frac{16\!\cdots\!00}{28\!\cdots\!49}a^{6}+\frac{63\!\cdots\!12}{28\!\cdots\!49}a^{5}+\frac{29\!\cdots\!20}{28\!\cdots\!49}a^{4}+\frac{16\!\cdots\!76}{28\!\cdots\!49}a^{3}+\frac{18\!\cdots\!16}{28\!\cdots\!49}a^{2}+\frac{45\!\cdots\!35}{57\!\cdots\!98}a-\frac{29\!\cdots\!72}{28\!\cdots\!49}$, $\frac{54\!\cdots\!29}{74\!\cdots\!44}a^{17}+\frac{13\!\cdots\!55}{37\!\cdots\!72}a^{16}+\frac{54\!\cdots\!11}{74\!\cdots\!44}a^{15}+\frac{70\!\cdots\!73}{37\!\cdots\!72}a^{14}+\frac{10\!\cdots\!11}{74\!\cdots\!44}a^{13}+\frac{16\!\cdots\!29}{37\!\cdots\!72}a^{12}+\frac{48\!\cdots\!75}{37\!\cdots\!72}a^{11}+\frac{99\!\cdots\!35}{18\!\cdots\!36}a^{10}+\frac{48\!\cdots\!37}{74\!\cdots\!44}a^{9}+\frac{28\!\cdots\!81}{74\!\cdots\!44}a^{8}+\frac{64\!\cdots\!83}{37\!\cdots\!72}a^{7}+\frac{28\!\cdots\!21}{18\!\cdots\!36}a^{6}+\frac{24\!\cdots\!75}{11\!\cdots\!96}a^{5}+\frac{79\!\cdots\!47}{23\!\cdots\!92}a^{4}+\frac{29\!\cdots\!13}{23\!\cdots\!92}a^{3}+\frac{93\!\cdots\!26}{28\!\cdots\!49}a^{2}+\frac{20\!\cdots\!07}{57\!\cdots\!98}a+\frac{18\!\cdots\!27}{28\!\cdots\!49}$, $\frac{50\!\cdots\!63}{37\!\cdots\!72}a^{17}+\frac{41\!\cdots\!79}{37\!\cdots\!72}a^{16}+\frac{13\!\cdots\!05}{37\!\cdots\!72}a^{15}+\frac{10\!\cdots\!57}{37\!\cdots\!72}a^{14}+\frac{16\!\cdots\!93}{37\!\cdots\!72}a^{13}+\frac{13\!\cdots\!05}{37\!\cdots\!72}a^{12}+\frac{60\!\cdots\!57}{18\!\cdots\!36}a^{11}+\frac{45\!\cdots\!53}{18\!\cdots\!36}a^{10}+\frac{52\!\cdots\!63}{37\!\cdots\!72}a^{9}+\frac{49\!\cdots\!23}{46\!\cdots\!84}a^{8}+\frac{12\!\cdots\!29}{37\!\cdots\!72}a^{7}+\frac{31\!\cdots\!97}{11\!\cdots\!96}a^{6}+\frac{40\!\cdots\!97}{92\!\cdots\!68}a^{5}+\frac{12\!\cdots\!44}{28\!\cdots\!49}a^{4}+\frac{17\!\cdots\!27}{11\!\cdots\!96}a^{3}+\frac{34\!\cdots\!91}{11\!\cdots\!96}a^{2}+\frac{20\!\cdots\!53}{57\!\cdots\!98}a+\frac{36\!\cdots\!38}{28\!\cdots\!49}$, $\frac{24\!\cdots\!85}{18\!\cdots\!36}a^{17}-\frac{24\!\cdots\!59}{18\!\cdots\!36}a^{16}+\frac{81\!\cdots\!79}{23\!\cdots\!92}a^{15}-\frac{17\!\cdots\!91}{35\!\cdots\!12}a^{14}+\frac{40\!\cdots\!31}{92\!\cdots\!68}a^{13}-\frac{15\!\cdots\!71}{18\!\cdots\!36}a^{12}+\frac{57\!\cdots\!21}{18\!\cdots\!36}a^{11}-\frac{35\!\cdots\!27}{46\!\cdots\!84}a^{10}+\frac{25\!\cdots\!35}{18\!\cdots\!36}a^{9}-\frac{51\!\cdots\!23}{11\!\cdots\!96}a^{8}+\frac{15\!\cdots\!29}{46\!\cdots\!84}a^{7}-\frac{28\!\cdots\!71}{18\!\cdots\!36}a^{6}+\frac{39\!\cdots\!83}{92\!\cdots\!68}a^{5}-\frac{13\!\cdots\!81}{46\!\cdots\!84}a^{4}+\frac{65\!\cdots\!77}{57\!\cdots\!98}a^{3}-\frac{17\!\cdots\!93}{57\!\cdots\!98}a^{2}-\frac{18\!\cdots\!03}{57\!\cdots\!98}a-\frac{39\!\cdots\!49}{28\!\cdots\!49}$, $\frac{16\!\cdots\!01}{46\!\cdots\!84}a^{17}-\frac{59\!\cdots\!45}{46\!\cdots\!84}a^{16}+\frac{51\!\cdots\!29}{57\!\cdots\!98}a^{15}-\frac{76\!\cdots\!57}{23\!\cdots\!92}a^{14}+\frac{51\!\cdots\!71}{46\!\cdots\!84}a^{13}-\frac{11\!\cdots\!70}{28\!\cdots\!49}a^{12}+\frac{92\!\cdots\!13}{11\!\cdots\!96}a^{11}-\frac{13\!\cdots\!83}{46\!\cdots\!84}a^{10}+\frac{80\!\cdots\!79}{23\!\cdots\!92}a^{9}-\frac{14\!\cdots\!21}{11\!\cdots\!96}a^{8}+\frac{95\!\cdots\!25}{10\!\cdots\!66}a^{7}-\frac{97\!\cdots\!85}{28\!\cdots\!49}a^{6}+\frac{32\!\cdots\!22}{28\!\cdots\!49}a^{5}-\frac{23\!\cdots\!65}{46\!\cdots\!84}a^{4}-\frac{17\!\cdots\!88}{28\!\cdots\!49}a^{3}-\frac{11\!\cdots\!51}{28\!\cdots\!49}a^{2}-\frac{13\!\cdots\!56}{28\!\cdots\!49}a-\frac{59\!\cdots\!62}{28\!\cdots\!49}$, $\frac{43\!\cdots\!59}{46\!\cdots\!84}a^{17}+\frac{54\!\cdots\!19}{46\!\cdots\!84}a^{16}+\frac{56\!\cdots\!75}{23\!\cdots\!92}a^{15}+\frac{71\!\cdots\!83}{23\!\cdots\!92}a^{14}+\frac{13\!\cdots\!33}{46\!\cdots\!84}a^{13}+\frac{89\!\cdots\!41}{23\!\cdots\!92}a^{12}+\frac{61\!\cdots\!19}{28\!\cdots\!49}a^{11}+\frac{12\!\cdots\!79}{46\!\cdots\!84}a^{10}+\frac{26\!\cdots\!63}{28\!\cdots\!49}a^{9}+\frac{14\!\cdots\!61}{11\!\cdots\!96}a^{8}+\frac{63\!\cdots\!60}{28\!\cdots\!49}a^{7}+\frac{94\!\cdots\!89}{28\!\cdots\!49}a^{6}+\frac{81\!\cdots\!16}{28\!\cdots\!49}a^{5}+\frac{24\!\cdots\!77}{46\!\cdots\!84}a^{4}+\frac{23\!\cdots\!09}{23\!\cdots\!92}a^{3}+\frac{46\!\cdots\!41}{11\!\cdots\!96}a^{2}+\frac{24\!\cdots\!71}{54\!\cdots\!33}a+\frac{24\!\cdots\!69}{28\!\cdots\!49}$
| 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: | \( 40934.0329443 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 163}{2\cdot\sqrt{39739057971752889532465351767}}\cr\approx \mathstrut & 0.255418396926 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512)
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 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512, 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 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512);
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 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512);
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
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 6.0.2250423.1, \(\Q(\zeta_{27})^+\) |
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 | $18$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | $18$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $9$ | $2$ | $44$ | |||
|
\(7\)
| 7.18.9.2 | $x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |