Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793)
gp: K = bnfinit(y^18 + 18*y^16 - 36*y^15 + 81*y^14 - 693*y^13 + 1299*y^12 - 3672*y^11 + 22167*y^10 - 25679*y^9 + 88533*y^8 - 388035*y^7 + 197658*y^6 - 1441638*y^5 + 4494087*y^4 + 2545989*y^3 + 21115836*y^2 + 5593428*y + 3447793, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793)
\( x^{18} + 18 x^{16} - 36 x^{15} + 81 x^{14} - 693 x^{13} + 1299 x^{12} - 3672 x^{11} + 22167 x^{10} + \cdots + 3447793 \)
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: |
\(-1773722731399331010404115847164903\)
\(\medspace = -\,3^{44}\cdot 23^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.33\) | 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: | $3^{70/27}23^{1/2}\approx 82.76466256650681$ | ||
| Ramified primes: |
\(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{3}-\frac{1}{9}$, $\frac{1}{9}a^{10}-\frac{1}{3}a^{4}-\frac{1}{9}a$, $\frac{1}{9}a^{11}-\frac{1}{3}a^{5}-\frac{1}{9}a^{2}$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{9}+\frac{1}{3}a^{7}+\frac{2}{9}a^{6}+\frac{1}{3}a^{4}+\frac{11}{27}a^{3}+\frac{1}{3}a+\frac{10}{27}$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{10}+\frac{1}{3}a^{8}+\frac{2}{9}a^{7}+\frac{1}{3}a^{5}+\frac{11}{27}a^{4}+\frac{1}{3}a^{2}+\frac{10}{27}a$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{11}+\frac{2}{9}a^{8}+\frac{1}{3}a^{6}+\frac{11}{27}a^{5}+\frac{1}{3}a^{3}+\frac{10}{27}a^{2}+\frac{1}{3}$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{9}-\frac{1}{3}a^{7}-\frac{10}{27}a^{6}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a-\frac{11}{27}$, $\frac{1}{27}a^{16}-\frac{1}{27}a^{10}-\frac{1}{3}a^{8}-\frac{10}{27}a^{7}-\frac{1}{3}a^{5}+\frac{4}{9}a^{4}-\frac{1}{3}a^{2}-\frac{11}{27}a$, $\frac{1}{70\!\cdots\!13}a^{17}-\frac{13\!\cdots\!62}{70\!\cdots\!13}a^{16}+\frac{33\!\cdots\!45}{23\!\cdots\!71}a^{15}+\frac{12\!\cdots\!45}{70\!\cdots\!13}a^{14}-\frac{93\!\cdots\!74}{70\!\cdots\!13}a^{13}+\frac{32\!\cdots\!53}{23\!\cdots\!71}a^{12}+\frac{11\!\cdots\!76}{23\!\cdots\!71}a^{11}+\frac{11\!\cdots\!33}{23\!\cdots\!71}a^{10}+\frac{32\!\cdots\!97}{23\!\cdots\!71}a^{9}-\frac{33\!\cdots\!21}{70\!\cdots\!13}a^{8}+\frac{24\!\cdots\!87}{70\!\cdots\!13}a^{7}+\frac{22\!\cdots\!36}{23\!\cdots\!71}a^{6}-\frac{15\!\cdots\!74}{70\!\cdots\!13}a^{5}-\frac{26\!\cdots\!86}{70\!\cdots\!13}a^{4}+\frac{13\!\cdots\!43}{23\!\cdots\!71}a^{3}+\frac{18\!\cdots\!05}{77\!\cdots\!57}a^{2}-\frac{24\!\cdots\!39}{23\!\cdots\!71}a-\frac{36\!\cdots\!82}{23\!\cdots\!71}$
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
$C_{9}$, which has order $9$ (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: |
$\frac{22\!\cdots\!92}{33\!\cdots\!71}a^{17}-\frac{86\!\cdots\!97}{99\!\cdots\!13}a^{16}+\frac{19\!\cdots\!82}{29\!\cdots\!39}a^{15}-\frac{59\!\cdots\!32}{33\!\cdots\!71}a^{14}+\frac{29\!\cdots\!33}{33\!\cdots\!71}a^{13}-\frac{74\!\cdots\!98}{29\!\cdots\!39}a^{12}+\frac{22\!\cdots\!67}{33\!\cdots\!71}a^{11}-\frac{78\!\cdots\!47}{99\!\cdots\!13}a^{10}+\frac{19\!\cdots\!25}{29\!\cdots\!39}a^{9}-\frac{26\!\cdots\!13}{33\!\cdots\!71}a^{8}+\frac{39\!\cdots\!75}{99\!\cdots\!13}a^{7}-\frac{26\!\cdots\!05}{29\!\cdots\!39}a^{6}+\frac{14\!\cdots\!82}{33\!\cdots\!71}a^{5}+\frac{21\!\cdots\!90}{33\!\cdots\!71}a^{4}+\frac{21\!\cdots\!39}{29\!\cdots\!39}a^{3}+\frac{61\!\cdots\!02}{33\!\cdots\!71}a^{2}+\frac{56\!\cdots\!65}{99\!\cdots\!13}a+\frac{16\!\cdots\!42}{29\!\cdots\!39}$, $\frac{13\!\cdots\!56}{33\!\cdots\!71}a^{17}+\frac{11\!\cdots\!94}{99\!\cdots\!13}a^{16}+\frac{13\!\cdots\!40}{29\!\cdots\!39}a^{15}+\frac{25\!\cdots\!24}{33\!\cdots\!71}a^{14}-\frac{19\!\cdots\!23}{33\!\cdots\!71}a^{13}-\frac{96\!\cdots\!68}{29\!\cdots\!39}a^{12}-\frac{15\!\cdots\!85}{33\!\cdots\!71}a^{11}+\frac{22\!\cdots\!65}{99\!\cdots\!13}a^{10}-\frac{18\!\cdots\!02}{29\!\cdots\!39}a^{9}+\frac{78\!\cdots\!77}{33\!\cdots\!71}a^{8}-\frac{57\!\cdots\!75}{99\!\cdots\!13}a^{7}+\frac{43\!\cdots\!59}{29\!\cdots\!39}a^{6}-\frac{16\!\cdots\!84}{33\!\cdots\!71}a^{5}+\frac{22\!\cdots\!88}{33\!\cdots\!71}a^{4}-\frac{11\!\cdots\!53}{29\!\cdots\!39}a^{3}+\frac{22\!\cdots\!79}{33\!\cdots\!71}a^{2}+\frac{17\!\cdots\!95}{99\!\cdots\!13}a-\frac{15\!\cdots\!09}{29\!\cdots\!39}$, $\frac{13\!\cdots\!38}{70\!\cdots\!13}a^{17}-\frac{10\!\cdots\!19}{70\!\cdots\!13}a^{16}+\frac{10\!\cdots\!50}{70\!\cdots\!13}a^{15}-\frac{96\!\cdots\!07}{70\!\cdots\!13}a^{14}+\frac{27\!\cdots\!67}{70\!\cdots\!13}a^{13}-\frac{13\!\cdots\!19}{70\!\cdots\!13}a^{12}+\frac{38\!\cdots\!62}{70\!\cdots\!13}a^{11}-\frac{84\!\cdots\!51}{70\!\cdots\!13}a^{10}+\frac{27\!\cdots\!95}{70\!\cdots\!13}a^{9}-\frac{77\!\cdots\!21}{70\!\cdots\!13}a^{8}+\frac{12\!\cdots\!40}{70\!\cdots\!13}a^{7}-\frac{41\!\cdots\!36}{70\!\cdots\!13}a^{6}+\frac{67\!\cdots\!75}{70\!\cdots\!13}a^{5}-\frac{31\!\cdots\!62}{70\!\cdots\!13}a^{4}+\frac{24\!\cdots\!58}{70\!\cdots\!13}a^{3}+\frac{20\!\cdots\!63}{70\!\cdots\!13}a^{2}+\frac{14\!\cdots\!25}{70\!\cdots\!13}a+\frac{81\!\cdots\!06}{70\!\cdots\!13}$, $\frac{12\!\cdots\!90}{70\!\cdots\!13}a^{17}+\frac{36\!\cdots\!62}{70\!\cdots\!13}a^{16}+\frac{82\!\cdots\!71}{70\!\cdots\!13}a^{15}+\frac{23\!\cdots\!33}{70\!\cdots\!13}a^{14}-\frac{34\!\cdots\!59}{70\!\cdots\!13}a^{13}+\frac{14\!\cdots\!21}{70\!\cdots\!13}a^{12}-\frac{19\!\cdots\!84}{70\!\cdots\!13}a^{11}+\frac{15\!\cdots\!90}{70\!\cdots\!13}a^{10}+\frac{44\!\cdots\!82}{70\!\cdots\!13}a^{9}+\frac{10\!\cdots\!52}{70\!\cdots\!13}a^{8}-\frac{32\!\cdots\!36}{70\!\cdots\!13}a^{7}+\frac{54\!\cdots\!15}{70\!\cdots\!13}a^{6}-\frac{27\!\cdots\!95}{70\!\cdots\!13}a^{5}+\frac{39\!\cdots\!16}{70\!\cdots\!13}a^{4}+\frac{25\!\cdots\!01}{70\!\cdots\!13}a^{3}+\frac{27\!\cdots\!42}{70\!\cdots\!13}a^{2}+\frac{55\!\cdots\!13}{70\!\cdots\!13}a+\frac{38\!\cdots\!01}{70\!\cdots\!13}$, $\frac{65\!\cdots\!80}{70\!\cdots\!13}a^{17}+\frac{14\!\cdots\!18}{70\!\cdots\!13}a^{16}+\frac{72\!\cdots\!00}{70\!\cdots\!13}a^{15}-\frac{93\!\cdots\!99}{70\!\cdots\!13}a^{14}-\frac{48\!\cdots\!51}{70\!\cdots\!13}a^{13}-\frac{29\!\cdots\!84}{70\!\cdots\!13}a^{12}+\frac{25\!\cdots\!15}{70\!\cdots\!13}a^{11}+\frac{14\!\cdots\!11}{70\!\cdots\!13}a^{10}+\frac{81\!\cdots\!93}{70\!\cdots\!13}a^{9}+\frac{30\!\cdots\!80}{70\!\cdots\!13}a^{8}-\frac{35\!\cdots\!02}{70\!\cdots\!13}a^{7}-\frac{16\!\cdots\!73}{70\!\cdots\!13}a^{6}-\frac{15\!\cdots\!24}{70\!\cdots\!13}a^{5}+\frac{51\!\cdots\!72}{70\!\cdots\!13}a^{4}+\frac{22\!\cdots\!84}{70\!\cdots\!13}a^{3}+\frac{43\!\cdots\!13}{70\!\cdots\!13}a^{2}+\frac{13\!\cdots\!23}{70\!\cdots\!13}a+\frac{63\!\cdots\!75}{70\!\cdots\!13}$, $\frac{28\!\cdots\!03}{70\!\cdots\!13}a^{17}-\frac{39\!\cdots\!52}{70\!\cdots\!13}a^{16}+\frac{28\!\cdots\!37}{70\!\cdots\!13}a^{15}-\frac{14\!\cdots\!38}{70\!\cdots\!13}a^{14}+\frac{13\!\cdots\!82}{70\!\cdots\!13}a^{13}-\frac{11\!\cdots\!59}{70\!\cdots\!13}a^{12}+\frac{53\!\cdots\!47}{70\!\cdots\!13}a^{11}-\frac{61\!\cdots\!43}{70\!\cdots\!13}a^{10}+\frac{36\!\cdots\!02}{70\!\cdots\!13}a^{9}-\frac{11\!\cdots\!99}{70\!\cdots\!13}a^{8}+\frac{64\!\cdots\!35}{70\!\cdots\!13}a^{7}-\frac{59\!\cdots\!43}{70\!\cdots\!13}a^{6}+\frac{16\!\cdots\!14}{70\!\cdots\!13}a^{5}-\frac{22\!\cdots\!89}{70\!\cdots\!13}a^{4}+\frac{64\!\cdots\!27}{70\!\cdots\!13}a^{3}-\frac{99\!\cdots\!55}{70\!\cdots\!13}a^{2}-\frac{53\!\cdots\!37}{70\!\cdots\!13}a-\frac{20\!\cdots\!18}{70\!\cdots\!13}$, $\frac{16\!\cdots\!21}{70\!\cdots\!13}a^{17}+\frac{57\!\cdots\!66}{70\!\cdots\!13}a^{16}-\frac{11\!\cdots\!87}{70\!\cdots\!13}a^{15}+\frac{60\!\cdots\!70}{70\!\cdots\!13}a^{14}-\frac{30\!\cdots\!46}{70\!\cdots\!13}a^{13}+\frac{39\!\cdots\!62}{70\!\cdots\!13}a^{12}-\frac{14\!\cdots\!77}{70\!\cdots\!13}a^{11}+\frac{73\!\cdots\!09}{70\!\cdots\!13}a^{10}-\frac{49\!\cdots\!84}{70\!\cdots\!13}a^{9}+\frac{24\!\cdots\!63}{70\!\cdots\!13}a^{8}-\frac{96\!\cdots\!37}{70\!\cdots\!13}a^{7}-\frac{70\!\cdots\!83}{70\!\cdots\!13}a^{6}-\frac{15\!\cdots\!36}{70\!\cdots\!13}a^{5}+\frac{67\!\cdots\!38}{70\!\cdots\!13}a^{4}+\frac{23\!\cdots\!19}{70\!\cdots\!13}a^{3}+\frac{23\!\cdots\!08}{70\!\cdots\!13}a^{2}+\frac{96\!\cdots\!86}{70\!\cdots\!13}a+\frac{11\!\cdots\!19}{70\!\cdots\!13}$, $\frac{81\!\cdots\!54}{70\!\cdots\!13}a^{17}+\frac{71\!\cdots\!14}{70\!\cdots\!13}a^{16}+\frac{12\!\cdots\!30}{70\!\cdots\!13}a^{15}+\frac{58\!\cdots\!65}{70\!\cdots\!13}a^{14}-\frac{19\!\cdots\!53}{70\!\cdots\!13}a^{13}-\frac{23\!\cdots\!07}{70\!\cdots\!13}a^{12}-\frac{18\!\cdots\!00}{70\!\cdots\!13}a^{11}+\frac{73\!\cdots\!47}{70\!\cdots\!13}a^{10}+\frac{39\!\cdots\!95}{70\!\cdots\!13}a^{9}+\frac{60\!\cdots\!03}{70\!\cdots\!13}a^{8}-\frac{16\!\cdots\!79}{70\!\cdots\!13}a^{7}-\frac{21\!\cdots\!31}{70\!\cdots\!13}a^{6}-\frac{12\!\cdots\!87}{70\!\cdots\!13}a^{5}+\frac{23\!\cdots\!24}{70\!\cdots\!13}a^{4}+\frac{51\!\cdots\!18}{70\!\cdots\!13}a^{3}+\frac{21\!\cdots\!54}{70\!\cdots\!13}a^{2}+\frac{59\!\cdots\!96}{70\!\cdots\!13}a+\frac{33\!\cdots\!01}{70\!\cdots\!13}$
| 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: | \( 189495677.488 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 189495677.488 \cdot 9}{2\cdot\sqrt{1773722731399331010404115847164903}}\cr\approx \mathstrut & 0.309020841610 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793)
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 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793, 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 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793);
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 + 18*x^16 - 36*x^15 + 81*x^14 - 693*x^13 + 1299*x^12 - 3672*x^11 + 22167*x^10 - 25679*x^9 + 88533*x^8 - 388035*x^7 + 197658*x^6 - 1441638*x^5 + 4494087*x^4 + 2545989*x^3 + 21115836*x^2 + 5593428*x + 3447793);
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
$C_3^2:C_{18}$ (as 18T82):
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 30 conjugacy class representatives for $C_3^2:C_{18}$ |
| Character table for $C_3^2:C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), \(\Q(\zeta_{9})^+\), 6.0.79827687.1 |
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 27 sibling: | 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
| 3.9.22.15 | $x^{9} + 15 x^{6} + 18 x^{5} + 9 x^{3} + 57$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ | |
|
\(23\)
| 23.18.9.2 | $x^{18} + 207 x^{16} + 19044 x^{14} + 1022034 x^{12} + 16 x^{11} + 35258328 x^{10} - 6956 x^{9} + 810882342 x^{8} - 80592 x^{7} + 12435414609 x^{6} + 10575816 x^{5} + 122628249325 x^{4} + 57236028 x^{3} + 705126299095 x^{2} - 630180356 x + 1800380410659$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |