Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091)
gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 156*y^15 + 504*y^14 - 1512*y^13 + 4272*y^12 - 10422*y^11 + 24552*y^10 - 51962*y^9 + 108099*y^8 - 193914*y^7 + 351249*y^6 - 518733*y^5 + 814050*y^4 - 933768*y^3 + 1272483*y^2 - 860841*y + 992091, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091)
\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 504 x^{14} - 1512 x^{13} + 4272 x^{12} - 10422 x^{11} + \cdots + 992091 \)
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: |
\(-2322038274967832964613417227771\)
\(\medspace = -\,3^{44}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.64\) | 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: | $3^{22/9}11^{1/2}\approx 48.6399077968597$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(297=3^{3}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(67,·)$, $\chi_{297}(133,·)$, $\chi_{297}(199,·)$, $\chi_{297}(265,·)$, $\chi_{297}(10,·)$, $\chi_{297}(76,·)$, $\chi_{297}(142,·)$, $\chi_{297}(208,·)$, $\chi_{297}(274,·)$, $\chi_{297}(34,·)$, $\chi_{297}(100,·)$, $\chi_{297}(166,·)$, $\chi_{297}(232,·)$, $\chi_{297}(43,·)$, $\chi_{297}(109,·)$, $\chi_{297}(175,·)$, $\chi_{297}(241,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{53}a^{16}-\frac{17}{53}a^{15}-\frac{10}{53}a^{14}-\frac{9}{53}a^{13}-\frac{5}{53}a^{12}-\frac{18}{53}a^{11}+\frac{18}{53}a^{10}-\frac{26}{53}a^{9}+\frac{16}{53}a^{8}-\frac{7}{53}a^{7}+\frac{25}{53}a^{5}-\frac{23}{53}a^{4}-\frac{2}{53}a^{3}-\frac{20}{53}a^{2}-\frac{2}{53}a+\frac{26}{53}$, $\frac{1}{34\!\cdots\!93}a^{17}+\frac{74\!\cdots\!29}{34\!\cdots\!93}a^{16}+\frac{16\!\cdots\!93}{34\!\cdots\!93}a^{15}-\frac{10\!\cdots\!59}{34\!\cdots\!93}a^{14}+\frac{13\!\cdots\!15}{34\!\cdots\!93}a^{13}-\frac{86\!\cdots\!74}{34\!\cdots\!93}a^{12}+\frac{15\!\cdots\!69}{34\!\cdots\!93}a^{11}-\frac{16\!\cdots\!03}{34\!\cdots\!93}a^{10}-\frac{73\!\cdots\!68}{34\!\cdots\!93}a^{9}-\frac{12\!\cdots\!82}{34\!\cdots\!93}a^{8}-\frac{16\!\cdots\!79}{34\!\cdots\!93}a^{7}-\frac{10\!\cdots\!22}{34\!\cdots\!93}a^{6}+\frac{64\!\cdots\!51}{34\!\cdots\!93}a^{5}+\frac{10\!\cdots\!81}{34\!\cdots\!93}a^{4}-\frac{14\!\cdots\!42}{34\!\cdots\!93}a^{3}-\frac{91\!\cdots\!59}{34\!\cdots\!93}a^{2}+\frac{11\!\cdots\!15}{34\!\cdots\!93}a-\frac{76\!\cdots\!44}{34\!\cdots\!93}$
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
$C_{1791}$, which has order $1791$ (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: |
$\frac{10\!\cdots\!62}{20\!\cdots\!21}a^{17}-\frac{88\!\cdots\!32}{20\!\cdots\!21}a^{16}+\frac{42\!\cdots\!28}{20\!\cdots\!21}a^{15}-\frac{14\!\cdots\!49}{20\!\cdots\!21}a^{14}+\frac{45\!\cdots\!55}{20\!\cdots\!21}a^{13}-\frac{13\!\cdots\!89}{20\!\cdots\!21}a^{12}+\frac{37\!\cdots\!29}{20\!\cdots\!21}a^{11}-\frac{90\!\cdots\!09}{20\!\cdots\!21}a^{10}+\frac{21\!\cdots\!45}{20\!\cdots\!21}a^{9}-\frac{43\!\cdots\!80}{20\!\cdots\!21}a^{8}+\frac{89\!\cdots\!07}{20\!\cdots\!21}a^{7}-\frac{15\!\cdots\!43}{20\!\cdots\!21}a^{6}+\frac{27\!\cdots\!01}{20\!\cdots\!21}a^{5}-\frac{36\!\cdots\!00}{20\!\cdots\!21}a^{4}+\frac{10\!\cdots\!49}{38\!\cdots\!57}a^{3}-\frac{55\!\cdots\!71}{20\!\cdots\!21}a^{2}+\frac{60\!\cdots\!09}{20\!\cdots\!21}a-\frac{41\!\cdots\!88}{20\!\cdots\!21}$, $\frac{10\!\cdots\!22}{20\!\cdots\!21}a^{17}-\frac{39\!\cdots\!63}{20\!\cdots\!21}a^{16}+\frac{30\!\cdots\!02}{20\!\cdots\!21}a^{15}-\frac{13\!\cdots\!61}{20\!\cdots\!21}a^{14}+\frac{41\!\cdots\!13}{20\!\cdots\!21}a^{13}-\frac{12\!\cdots\!75}{20\!\cdots\!21}a^{12}+\frac{36\!\cdots\!77}{20\!\cdots\!21}a^{11}-\frac{94\!\cdots\!02}{20\!\cdots\!21}a^{10}+\frac{20\!\cdots\!38}{20\!\cdots\!21}a^{9}-\frac{45\!\cdots\!84}{20\!\cdots\!21}a^{8}+\frac{88\!\cdots\!31}{20\!\cdots\!21}a^{7}-\frac{16\!\cdots\!79}{20\!\cdots\!21}a^{6}+\frac{25\!\cdots\!85}{20\!\cdots\!21}a^{5}-\frac{42\!\cdots\!36}{20\!\cdots\!21}a^{4}+\frac{93\!\cdots\!34}{38\!\cdots\!57}a^{3}-\frac{67\!\cdots\!15}{20\!\cdots\!21}a^{2}+\frac{51\!\cdots\!17}{20\!\cdots\!21}a-\frac{59\!\cdots\!82}{20\!\cdots\!21}$, $\frac{23\!\cdots\!26}{34\!\cdots\!93}a^{17}-\frac{40\!\cdots\!98}{64\!\cdots\!81}a^{16}+\frac{87\!\cdots\!52}{34\!\cdots\!93}a^{15}-\frac{25\!\cdots\!12}{34\!\cdots\!93}a^{14}+\frac{77\!\cdots\!38}{34\!\cdots\!93}a^{13}-\frac{24\!\cdots\!02}{34\!\cdots\!93}a^{12}+\frac{60\!\cdots\!06}{34\!\cdots\!93}a^{11}-\frac{14\!\cdots\!67}{34\!\cdots\!93}a^{10}+\frac{30\!\cdots\!69}{34\!\cdots\!93}a^{9}-\frac{12\!\cdots\!83}{64\!\cdots\!81}a^{8}+\frac{11\!\cdots\!14}{34\!\cdots\!93}a^{7}-\frac{21\!\cdots\!21}{34\!\cdots\!93}a^{6}+\frac{29\!\cdots\!19}{34\!\cdots\!93}a^{5}-\frac{52\!\cdots\!06}{34\!\cdots\!93}a^{4}+\frac{41\!\cdots\!76}{34\!\cdots\!93}a^{3}-\frac{10\!\cdots\!82}{34\!\cdots\!93}a^{2}+\frac{22\!\cdots\!51}{34\!\cdots\!93}a-\frac{82\!\cdots\!30}{34\!\cdots\!93}$, $\frac{23\!\cdots\!14}{34\!\cdots\!93}a^{17}-\frac{55\!\cdots\!59}{64\!\cdots\!81}a^{16}+\frac{23\!\cdots\!74}{34\!\cdots\!93}a^{15}-\frac{10\!\cdots\!88}{34\!\cdots\!93}a^{14}+\frac{36\!\cdots\!80}{34\!\cdots\!93}a^{13}-\frac{11\!\cdots\!95}{34\!\cdots\!93}a^{12}+\frac{34\!\cdots\!42}{34\!\cdots\!93}a^{11}-\frac{98\!\cdots\!38}{34\!\cdots\!93}a^{10}+\frac{23\!\cdots\!13}{34\!\cdots\!93}a^{9}-\frac{10\!\cdots\!38}{64\!\cdots\!81}a^{8}+\frac{11\!\cdots\!52}{34\!\cdots\!93}a^{7}-\frac{22\!\cdots\!45}{34\!\cdots\!93}a^{6}+\frac{36\!\cdots\!40}{34\!\cdots\!93}a^{5}-\frac{70\!\cdots\!92}{34\!\cdots\!93}a^{4}+\frac{87\!\cdots\!13}{34\!\cdots\!93}a^{3}-\frac{13\!\cdots\!14}{34\!\cdots\!93}a^{2}+\frac{99\!\cdots\!35}{34\!\cdots\!93}a-\frac{15\!\cdots\!07}{34\!\cdots\!93}$, $\frac{11\!\cdots\!34}{34\!\cdots\!93}a^{17}-\frac{25\!\cdots\!73}{34\!\cdots\!93}a^{16}+\frac{17\!\cdots\!38}{34\!\cdots\!93}a^{15}-\frac{47\!\cdots\!46}{34\!\cdots\!93}a^{14}+\frac{52\!\cdots\!46}{34\!\cdots\!93}a^{13}-\frac{10\!\cdots\!65}{34\!\cdots\!93}a^{12}+\frac{72\!\cdots\!36}{34\!\cdots\!93}a^{11}-\frac{15\!\cdots\!26}{34\!\cdots\!93}a^{10}+\frac{70\!\cdots\!25}{34\!\cdots\!93}a^{9}-\frac{30\!\cdots\!71}{34\!\cdots\!93}a^{8}+\frac{80\!\cdots\!22}{34\!\cdots\!93}a^{7}-\frac{81\!\cdots\!89}{34\!\cdots\!93}a^{6}-\frac{17\!\cdots\!54}{34\!\cdots\!93}a^{5}+\frac{86\!\cdots\!51}{34\!\cdots\!93}a^{4}+\frac{20\!\cdots\!61}{34\!\cdots\!93}a^{3}+\frac{33\!\cdots\!96}{34\!\cdots\!93}a^{2}+\frac{19\!\cdots\!57}{34\!\cdots\!93}a-\frac{20\!\cdots\!42}{34\!\cdots\!93}$, $\frac{18\!\cdots\!64}{34\!\cdots\!93}a^{17}-\frac{11\!\cdots\!82}{34\!\cdots\!93}a^{16}+\frac{37\!\cdots\!26}{34\!\cdots\!93}a^{15}-\frac{76\!\cdots\!07}{34\!\cdots\!93}a^{14}+\frac{23\!\cdots\!85}{34\!\cdots\!93}a^{13}-\frac{68\!\cdots\!19}{34\!\cdots\!93}a^{12}+\frac{16\!\cdots\!45}{34\!\cdots\!93}a^{11}-\frac{25\!\cdots\!67}{34\!\cdots\!93}a^{10}+\frac{60\!\cdots\!75}{34\!\cdots\!93}a^{9}-\frac{95\!\cdots\!06}{34\!\cdots\!93}a^{8}+\frac{17\!\cdots\!67}{34\!\cdots\!93}a^{7}-\frac{20\!\cdots\!12}{34\!\cdots\!93}a^{6}+\frac{79\!\cdots\!06}{64\!\cdots\!81}a^{5}-\frac{97\!\cdots\!80}{34\!\cdots\!93}a^{4}+\frac{44\!\cdots\!94}{34\!\cdots\!93}a^{3}-\frac{45\!\cdots\!79}{34\!\cdots\!93}a^{2}+\frac{12\!\cdots\!66}{34\!\cdots\!93}a+\frac{71\!\cdots\!01}{34\!\cdots\!93}$, $\frac{14\!\cdots\!78}{34\!\cdots\!93}a^{17}-\frac{13\!\cdots\!20}{34\!\cdots\!93}a^{16}+\frac{50\!\cdots\!10}{34\!\cdots\!93}a^{15}-\frac{11\!\cdots\!10}{34\!\cdots\!93}a^{14}+\frac{28\!\cdots\!34}{34\!\cdots\!93}a^{13}-\frac{91\!\cdots\!47}{34\!\cdots\!93}a^{12}+\frac{24\!\cdots\!88}{34\!\cdots\!93}a^{11}-\frac{49\!\cdots\!36}{34\!\cdots\!93}a^{10}+\frac{93\!\cdots\!87}{34\!\cdots\!93}a^{9}-\frac{19\!\cdots\!28}{34\!\cdots\!93}a^{8}+\frac{70\!\cdots\!94}{64\!\cdots\!81}a^{7}-\frac{72\!\cdots\!17}{34\!\cdots\!93}a^{6}+\frac{10\!\cdots\!14}{34\!\cdots\!93}a^{5}-\frac{19\!\cdots\!93}{34\!\cdots\!93}a^{4}+\frac{24\!\cdots\!19}{34\!\cdots\!93}a^{3}-\frac{50\!\cdots\!24}{34\!\cdots\!93}a^{2}+\frac{37\!\cdots\!45}{34\!\cdots\!93}a-\frac{17\!\cdots\!63}{64\!\cdots\!81}$, $\frac{53\!\cdots\!38}{34\!\cdots\!93}a^{17}-\frac{61\!\cdots\!30}{34\!\cdots\!93}a^{16}+\frac{34\!\cdots\!44}{34\!\cdots\!93}a^{15}-\frac{10\!\cdots\!32}{34\!\cdots\!93}a^{14}+\frac{24\!\cdots\!78}{34\!\cdots\!93}a^{13}-\frac{68\!\cdots\!52}{34\!\cdots\!93}a^{12}+\frac{23\!\cdots\!52}{34\!\cdots\!93}a^{11}-\frac{54\!\cdots\!64}{34\!\cdots\!93}a^{10}+\frac{99\!\cdots\!86}{34\!\cdots\!93}a^{9}-\frac{19\!\cdots\!89}{34\!\cdots\!93}a^{8}+\frac{49\!\cdots\!90}{34\!\cdots\!93}a^{7}-\frac{80\!\cdots\!02}{34\!\cdots\!93}a^{6}+\frac{13\!\cdots\!82}{34\!\cdots\!93}a^{5}-\frac{34\!\cdots\!84}{64\!\cdots\!81}a^{4}+\frac{35\!\cdots\!32}{34\!\cdots\!93}a^{3}-\frac{31\!\cdots\!60}{34\!\cdots\!93}a^{2}+\frac{59\!\cdots\!89}{34\!\cdots\!93}a-\frac{17\!\cdots\!35}{34\!\cdots\!93}$
| 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 1791}{2\cdot\sqrt{2322038274967832964613417227771}}\cr\approx \mathstrut & 0.367142155312 \end{aligned}\] (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 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091)
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 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091, 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 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091);
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 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091);
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
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), 6.0.8732691.1, \(\Q(\zeta_{27})^+\) |
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 | $18$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(11\)
| 11.18.9.2 | $x^{18} + 99 x^{16} + 4356 x^{14} + 111804 x^{12} + 18 x^{11} + 1844782 x^{10} - 3744 x^{9} + 20287674 x^{8} - 20196 x^{7} + 148892436 x^{6} + 1280664 x^{5} + 702432669 x^{4} + 3521970 x^{3} + 1922907025 x^{2} - 17918856 x + 2360533311$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |